1. NUMBERS
IMPORTANT FACTS AND FORMULAE
1. Numbers: In Hindu-Arabic System, we use ten symbols, namely 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. We
call them digits.
A number is denoted by a group of digits, called numeral.
Some numerals are given below in a place-value chart.
Ten
Crores
Crores
Ten
Lacs
Lacs
Ten
Thousands
Thousands
Hundreds
Tens
Units
(i)
5
3
6
7
4
9
(ii)
1
6
8
0
3
0
4
(iii)
6
0
5
1
4
0
8
9
(iv)
2
4
1
6
0
8
0
0
3
We write these numbers in words as:
(i) Five lac thirty-six thousand seven hundred forty-nine.
(ii) Sixteen lac eighty thousand three hundred four.
(iii) Six crore five lac fourteen thousand eighty-nine.
(iv) Twenty-four crore sixteen lac eight thousand three.
2. Face Value of a Digit in a Numeral: The face value of a digit in a numeral is the value of the
digit itself, wherever it may be in the place value chart.
In the numeral 536749, the face value of 7 is 7, the face value of 6 is 6, the face value of 5 is 5 and
so on.
3. Place Value or Local Value of a Digit in a Numeral: In the numeral 536749, we have
Place value of 9 = (9 × 1) = 9;
Place value of 4 = (4 × 10) = 40;
Place value of 7 = (7 × 100) = 700;
Place value of 6 = (6 × 1000) = 6000;
Place value of 3 = (3 × 10000) = 30000;
Place value of 5 = (5 × 100000) = 500000.
4. Types of Numbers:
(i) Natural Numbers: Counting numbers are called natural numbers. Thus 1, 2, 3, 4, 5, 6, ……. etc.
are all natural numbers.
(ii) Whole Numbers: All counting numbers together with zero form the set of whole numbers.
Note: (i) 0 is a whole number which is not a natural number.
(ii) Every natural number is a whole number.
Thus 0, 1, 2, 3, 4, 5, 6, …… are whole numbers.
(iii) Integers: All counting numbers, 0 and negatives of counting numbers are called integers.
(a) Positive integers: {1, 2, 3, 4, 5, ……} is the set of positive integers.
(b) Negative integers: {–1, –2, –3, –4, ……} is the set of negative integers.
(c) Non-negative integers: 0 is neither positive nor negative.
{0, 1, 2, 3, 4, 5, ……} is the set of non-negative integers.
(d) Non-positive integers: {0, –1, –2, –3, –4, ……} is the set of non-positive integers.
(iv) Even Numbers: A number divisible by 2 is called an even number. Thus 0, 2, 4, 6, 8, 10, ….. etc.
are all even numbers.
(v) Odd Numbers: A number not divisible by 2 is called an odd number. Thus 1, 3, 5, 7, 9, …… etc.
are odd numbers.
(vi) Prime Numbers: A number greater than 1 having exactly two factors, namely 1 and itself is called
a prime number.
Prime numbers upto 100 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,
43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
To test whether a given number p is prime: Let p be a given number. Find a whole number k such
that
.kp
Take all prime numbers less than or equal to k.
If no one of these divides p exactly, we say that p is prime. Otherwise, p is not prime.
Ex. 1. Test whether 191 is prime or not.
Sol. Clearly,
14 191.
Prime numbers upto 14 are 2, 3, 5, 7, 11, 13.
No one of these divides 191 exactly.
191 is a prime number.
Ex. 2. Test whether 221 is prime or not.
Sol. Clearly,
15 221.
Prime numbers upto 15 are 2, 3, 5, 7, 11, 13.
Clearly, 13 divides 221 exactly.
So, 221 is not prime.
(vii) Composite Numbers: Numbers greater than 1 which are not prime, are called composite numbers.
e.g. 4, 6, 8, 9, 10, 12, etc.
Note: (i) 1 is neither prime nor composite.
(ii) 2 is the only even number which is prime.
(viii) Co-primes: Two natural numbers a and b are said to be co-prime if their HCF is 1. e.g. (2, 3), (4,
5), (7, 9), (8, 11) etc. are pairs of co-primes.
5. Tests of Divisibility:
(i) Divisibility by 2: A number is divisible by 2, if its unit digit is any of 0, 2, 4, 6, 8. Ex.: 64892 is
divisible by 2, while 64895 is not divisible by 2.
(ii) Divisibility by 3: A number is divisible by 3, only when the sum of its digits is divisible by 3.
Ex.: (a) Consider the number 587421. Sum of its digits is 27, which is divisible by 3. So,
587421 is divisible by 3.
(b) Consider the number 689453. Sum of its digits is 35, which is not divisible by 3.
So, 689453 is not divisible by 3.
(iii) Divisibility by 4: A number is divisible by 4, if the sum of its last 2 digits is divisible by 4.
Ex.: (a) 5249376 is divisible by 4, since 76 is divisible by 4.
(b) 638214 is not divisible by 4, since 14 is not divisible by 4.
(iv) Divisibility by 5: A number is divisible by 5 if its unit digit is 5 or 0.
Ex.: (a) 328695 is divisible by 5.
(b) 947310 is divisible by 5.
(c) None of 507062, 717554, 656676, 30578 is divisible by 5.
(v) Divisibility by 6: A number is divisible by 6, if it is divisible by both 2 and 3.
Ex.: (a) 974562 is divisible by 2 as well as 3. So, it is divisible by 6.
(b) 975416 is not divisible by 6, since sum of its digits is 32, which is not divisible by
3.
(vi) Divisibility by 8: A number is divisible by 8 only when the number formed by its last 3 digits is
divisible by 8.
Ex.: (a) 6754120 is divisible by 8, since 120 is divisible by 8.
(b) 5943246 is not divisible by 8, since 246 is not divisible by 8.
(vii) Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
Ex.: (a) 594324 is divisible by 9, since the sum of its digits is 27, which is divisible by 9.
(b) 3714529 is not divisible by 9, since the sum of its digits is 31, which is not divisible
by 9.
(viii) Divisibility by 10: A number is divisible by 10 only when its unit digit is 0.
Ex.: 234780 is divisible by 10 while 234785 is not divisible by 10.
(ix) Divisibility by 11: A number is divisible by 11, if the difference of the sum of its digits at odd
places and the sum of its digits at even places is either 0 or a number divisible by 11.
Ex.: 49235714 is divisible by 11, since
(Sum of its digits at odd places) – (Sum of its digits at even places) is 11, which is divisible
by 11.
An important note: If a number N is divisible by two numbers a and b, where a and b are co-
primes, then N is divisible by ab.
6. Some Results on Division:
(i) (xn – an) is divisible by (x – a) for all values of n.
(ii) (xn – an) is divisible by (x + a) for even values of n.
(iii) (xn + an) is divisible by (x + a) for odd values of n.
7. Division Algorithm: If we divide a number by another number, then
Dividend = (Divisor × Quotient) + Remainder
8. Some Series:
(i)
1
(1 2 3 4 ...... ) ( 1)
2
n n n
(ii)
2 2 2 2 1
(1 2 3 ...... ) ( 1)(2 1)
6
n n n n
(iii)
3 3 3 3 2 2
1
(1 2 3 ...... ) ( 1) .
4
n n n
9. Basic Formulae:
(i)
2 2 2
( ) 2a b a b ab
(ii)
2 2 2
( ) 2a b a b ab
(iii)
22
( ) ( ) 4a b a b ab
(iv)
2 2 2 2
( ) ( ) 2( )a b a b a b
(v)
22
( ) ( )( )a b a b a b
(vi)
2 2 2 2
( ) 2( )a b c a b c ab bc ca
(vii)
3 3 2 2
( ) ( )( )a b a b a ab b
(viii)
3 3 2 2
( ) ( )( )a b a b a ab b
(ix)
3 3 3 2 2 2
( 3 ) ( )( )a b c abc a b c a b c ab bc ca
(x)
3 3 3
0 ( ) 3 .a b c a b c abc
EXERCISE
Mark (√) against the correct answer in each of the following:
1. The unit digit in 7105 is:
(a) 5 (b) 7
(c) 9 (d) 1
2. The unit digit in (365 × 659 × 771) is:
(a) 1 (b) 2
(c) 4 (d) 6
3. The unit digit in (795 – 358) is:
(a) 0 (b) 4
(c) 6 (d) 7
4. The digit in the unit place of the product (2137)754 is:
(a) 1 (b) 3
(c) 7 (d) 9
5. Which one of the following will divide (325 + 326 + 327 +328) completely?
(a) 11 (b) 16
(c) 25 (d) 30
6. If 17200 is divided by 18, the remainder is:
(a) 17 (b) 16
(c) 1 (d) 2
7.
2 2 2 2 2 2 2 2
(1 2 3 4 5 6 ....... 9 10 ) ?
(a) 45 (b) –45
(c) –54 (d) –55
8.
1 1 1 1 1 1
...... ?
2 6 12 20 30 ( 1)nn
(a)
1
n
(b)
1
1n
(c)
2( 1)n
n
(d)
( 1)
n
n
9.
1 1 1 1 1 ?
1.4 4.7 7.10 10.13 13.16
(a)
1
3
(b)
5
16
(c)
3
8
(d)
41
7280
10. (45 + 46 + 47 + …… + 113 + 114 + 115) = ?
(a) 5600 (b) 5656
(c) 5680 (d) 4000
11. Which of the following fractions is less than
7
8
and greater than
1
3
?
(a)
1
4
(b)
23
24
(c)
11
12
(d)
17
24
12. 7589 – ? = 3434
(a) 11023 (b) 4245
(c) 4155 (d) 11123
(e) None of these
13. 916 × ? × 3 = 214344
(a) 78 (b) 68
(c) 84 (d) 66
(e) None of these
14.
? (88 42) 16
(a) 3696 (b) 39660
(c) 43163 (d) 53361
(e) None of these
15.
5 2 4 6 ?
6 9 9 7
(a) 0.44 (b) 0.32
(c) 0.49 (d) 0.36
(e) 0.4
16. 8988 ÷ 8 ÷ 4 = ?
(a) 4494 (b) 561.75
(c) 2247 (d) 280.875
(e) None of these
17. 888888 ÷ 888 ÷ 88 = ?
(a) 88889 (b) 29
(c) 44449 (d) 11
(e) 56
18. 106 × 106 – 94 × 94 = ?
(a) 2400 (b) 2000
(c) 1904 (d) 1906
19. If n is a natural number, then the largest number dividing
3
()nn
is:
(a) 2 (b) 3
(c) 6 (d) 12
20.
()
nn
xa
is exactly divisible by (x – a):
(a) for every natural number n
(b) for every even natural number n
(c) for every odd natural number n
(d) for every prime number n
21. A, B, C, D are four consecutive odd integers and their average is 42. What is the product of B and
D?
(a) 1860 (b) 1890
(c) 1845 (d) 1677
(e) None of these
22. The sum and product of two numbers are 12 and 35 respectively. The sum of their reciprocals is:
(a)
12
35
(b)
1
35
(c)
35
8
(d)
7
32
23. By how much is
3th
4
of 52 lesser than
2rd
3
of 99?
(a) 27 (b) 33
(c) 39 (d) 66
(e) None of these
24. The difference between
3th
5
of
2rd
3
of a number and
2th
5
of
1th
4
of the same number is 288. What
is the number?
(a) 960 (b) 850
(c) 895 (d) 955
(e) None of these
25. If the difference between the reciprocal of a positive proper fraction and the fraction itself be
9,
20
the
fraction is:
(a)
3
5
(b)
3
10
(c)
4
5
(d)
5
4
26. The smallest 3-digit prime number is:
(a) 103 (b) 107
(c) 109 (d) None of these
27. The difference between the squares of two consecutive even integers is always divisible by:
(a) 3 (b) 4
(c) 6 (d) 7
28. What least number of five digits is exactly divisible by 41?
(a) 10045 (b) 10004
(c) 10041 (d) 10025
29. In a division sum, the divisor is ten times the quotient and five times the remainder. If the
remainder is 46, the dividend is:
(a) 4236 (b) 4306
(c) 4336 (d) 5336
30. The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as
quotient and 15 as remainder. The smaller number is:
(a) 240 (b) 270
(c) 295 (d) 360
31. The largest four digit number exactly divisible by each of 12, 15, 18 and 27 is:
(a) 9690 (b) 9720
(c) 9930 (d) 9960
32. In a division sum, the remainder is 6 and the divisor is 5 times the quotient and is obtained by adding 2
to the thrice of the remainder. The dividend is:
(a) 40 (b) 42
(c) 80 (d) 86
33. On dividing a certain number by 357, the remainder is 39. On dividing the same number by 17, what
will be the remainder?
(a) 0 (b) 3
(c) 5 (d) 11
34. A number when divided by 5, leaves 3 as remainder. What will be the remainder when the square
of this number is divided by 5?
(a) 0 (b) 1
(c) 2 (d) 4
35. 64239 is divided by a certain number. While dividing, the numbers 175, 114 and 213 appear as three
successive remainders. The divisor is:
(a) 184 (b) 224
(c) 234 (d) 6250
(e) None of these
36. A number divided by 56 gives 29 as remainder. If the same number is divided by 8, the remainder
will be:
(a) 4 (b) 5
(c) 6 (d) 7
37. The number 2272 and 875 are divided by a three-digit number N, giving the same remainder. The
sum of the digits of N is:
(a) 10 (b) 11
(c) 12 (d) 13
38. It is given that
32
(2 1)
is exactly divisible by a certain number. Which of the following is also
definitely divisible by the same number?
(a)
16
(2 1)
(b)
16
(2 1)
(c)
33
72
(d)
96
21
39. The numbers 1, 3, 5, ……, 25 are multiplied together. The number of zeros at the right end of the
product is:
(a) 0 (b) 1
(c) 2 (d) 3
40. The number
2
(6 6 )nn
for natural number n is always divisible by:
(a) 6 only (b) 6 and 12
(c) 12 only (d) 18 only
41. On multiplying a number by 7, all the digits in the product appear as 3’s. The smallest such
number is:
(a) 47619 (b) 47719
(c) 48619 (d) 47649
42. There are four prime numbers written in ascending order. The product of first three is 385 and that of
the last three is 1001. The first number is:
(a) 5 (b) 7
(c) 11 (d) 17
43. The difference between the squares of two consecutive odd integers is always divisible by:
(a) 3 (b) 6
(c) 7 (d) 8
44. A positive number which when added to 1000, gives a sum which is greater than when it is
multiplied by 1000. This positive integer is:
(a) 1 (b) 3
(c) 5 (d) 7
45. In a question on division with zero remainder, a candidate took 12 as divisor, instead of 21. The
quotient obtained by him was 35. The correct quotient is:
(a) 0 (b) 12
(c) 13 (d) 20
46. A number when divided by 6 leaves a remainder 3. When the square of the same number is
divided by 6, the remainder is:
(a) 0 (b) 1
(c) 2 (d) 3
47. When 2525 is divided by 26, the remainder is:
(a) 1 (b) 2
(c) 24 (d) 25
ANSWERS
1. (b) 2. (c) 3. (b) 4. (d) 5. (d) 6. (c)
7. (d) 8. (d) 9. (b) 10. (c) 11. (d) 12. (c)
13. (a) 14. (d) 15. (c) 16. (d) 17. (d) 18. (a)
19. (c) 20. (a) 21. (c) 22. (a) 23. (a) 24. (a)
25. (c) 26. (c) 27. (b) 28. (b) 29. (d) 30. (b)
31. (b) 32. (d) 33. (c) 34. (d) 35. (c) 36. (b)
37. (a) 38. (d) 39. (a) 40. (b) 41. (a) 42. (a)
43. (d) 44. (a) 45. (d) 46. (d) 47. (d)
2. H.C.F. AND L.C.M.
IMPORTANT FACTS AND FORMULAE
1. Factors and Multiples: If a divides b exactly, we say that a is a factor of b and also we say that b
is a multiple of a.
Thus, 3 is a factor of 6; 5 is a factor of 15 etc.
Likewise, 6 is a multiple of 3; 15 is a multiple of 5 etc.
2. H.C.F. (or G.C.D. or G.C.M.): The H.C.F. of two or more than two numbers is the greatest
number that divides each of them exactly.
3. Methods of Finding H.C.F.
(i) Factorization Method: Express each one of the given numbers as the product of prime factors. The
product of common prime factors with least powers gives H.C.F.
(ii) Division Method: Suppose we have to find the H.C.F. of two given numbers. Divide the larger by
the smaller one. Now, divide the divisor by the remainder. Repeat this process till the remainder is
zero. The last divisor is the required H.C.F.
4. Finding H.C.F. of more than Two Numbers: H.C.F. of 3 numbers = H.C.F. of {(H.C.F. of any
two) and 3rd number}. Similarly, H.C.F. of more than 3 numbers can be obtained.
5. L.C.M.: The least number which is exactly divisible by each one of the given numbers is called
their L.C.M.
6. Methods of Finding L.C.M.:
(i) Factorization Method: Resolve each one of the given numbers into a product of prime factors.
L.C.M. is the product of terms of highest powers of all factors.
(ii) Short-cut Method: Arrange the given numbers in a row in any order. Divide by a number which
divides exactly at least two of the given numbers and carry forward the numbers which are not
divisible. Repeat the above process till no two of the numbers are divisible by the same number
except 1.
The product of the divisors and the undivided numbers is the required L.C.M. of the given
numbers.
7. An Important Result:
Product of Two Numbers = (Their H.C.F.) + (Their L.C.M.)
8. Co-primes: Two numbers are said to be co-prime if their H.C.F. is 1. Thus, 6 and 11 are co-prime.
9. H.C.F. and L.C.M. of Fractions:
(i)
H.C.F. of numerators
H.C.F. = L.C.M. of denominators
(ii)
L.C.M. of numerators
L.C.M. = H.C.F. of denominators
10. H.C.F. and L.C.M. of Decimals: In given decimals, make same number of decimal places by
annexing zeros in some numbers, if necessary.
Find H.C.F. and L.C.M. of these numbers without decimal points. In the result mark off as many
decimal places as are there in each of the given numbers.
11. Find L.C.M. of the denominators of the given fractions. Convert each one of them into an
equivalent fraction with this as denominator. In resultant fractions, the one having larger
numerator is larger.
EXERCISE
Mark (√) against the correct answer in each of the following:
1. The simplest form of
69
92
is:
(a)
2
3
(b)
3
4
(c)
13
24
(d) None of these
2. The simplest form of
561
748
is:
(a)
13
14
(b)
11
14
(c)
23
24
(d)
3
4
3.
1095
1168
in simplest form is:
(a)
13
16
(b)
17
26
(c)
15
16
(d)
25
26
4.
128352
238368
when reduced to its lowest terms is:
(a)
3
4
(b)
7
13
(c)
9
13
(d)
5
13
5. The H.C.F. of 595 and 252 is:
(a) 1 (b) 7
(c) 11 (d) 17
6. The H.C.F. of 1485 and 4356 is:
(a) 189 (b) 89
(c) 99 (d) 83
7. The H.C.F. of 42, 63, 140 is:
(a) 14 (b) 9
(c) 21 (d) 7
8. 252 can be expressed as a product of primes as:
(a) 2 × 2 × 2 × 3 × 7 (b) 3 × 3 × 3 × 3 × 7
(c) 2 × 2 × 3 × 3 × 7 (d) 2 × 3 × 3 × 3 × 7
9. H.C.F. of
2 3 5 3 2 2
(2 3 5 ), (2 3 5 7)
and
4 4 2
(2 3 5 11 7 )
is:
(a) 180 (b) 360
(c) 540 (d) 1260
10. The H.C.F. of 1134, 1344 and 1512 is:
(a) 24 (b) 42
(c) 44 (d) 64
11. The H.C.F. of
1 2 3 4
,,,
2 3 4 5
is:
(a) 1 (b) 12
(c)
4
5
(d)
1
60
12. The L.C.M. of 24, 36 and 40 is:
(a) 120 (b) 240
(c) 360 (d) 480
13. The L.C.M. of 26, 56, 104, 182 is:
(a) 456 (b) 728
(c) 748 (d) 1274
14. The L.C.M. of two numbers is 2310 and their H.C.F. is 30. If one of these numbers is 210, the
second number is:
(a) 330 (b) 1470
(c) 2100 (d) 16170
15. The L.C.M. of two numbers is 12 times their H.C.F. The sum of H.C.F. and L.C.M. is 403. If one
number is 93, then the other is:
(a) 128 (b) 124
(c) 134 (d) None of these
16. The sum of two numbers is 45. Their difference is
1
9
of their sum. Their L.C.M. is:
(a) 100 (b) 150
(c) 200 (d) 250
17. The H.C.F. and L.C.M. of two numbers are 21 and 4641 respectively. If one of the numbers lies
between 200 and 300, then the two numbers are:
(a) 273, 363 (b) 273, 359
(c) 273, 361 (d) 273, 357
18. If the H.C.F. of two numbers (each greater than 13) be 13 and L.C.M. 273, then the sum of the
numbers will be:
(a) 286 (b) 130
(c) 288 (d) 290
19. The sum of the H.C.F. and L.C.M. of two numbers is 680 and the L.C.M. is 84 times the H.C.F. If
one of the numbers is 56, the other is:
(a) 8 (b) 12
(c) 84 (d) 96
20. Two 3-digit numbers have their H.C.F. 29 and L.C.M. 4147. The sum of the numbers is:
(a) 666 (b) 669
(c) 696 (d) 966
21. If two numbers are in the ratio 5 : 7 and their L.C.M. is 315, then their product is:
(a) 2358 (b) 2385
(c) 2538 (d) 2835
(e) None of these
22. The capacity of two pots is 120 litres and 56 litres respectively. The capacity of a container which can
exactly measure the contents of the two pots, is:
(a) 7500 c.c. (b) 7850 c.c.
(c) 8000 c.c. (d) 9500 c.c.
23. What is the greatest possible length of a scale that can be used to measure exactly the lengths 3 m,
5 m 10 cm and 12 m 90 cm?
(a) 10 cm (b) 20 cm
(c) 25 cm (d) 30 cm
24. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108
seconds respectively. If they all change simultaneously at 8:20 hours, then at what time will they again
change simultaneously?
(a) 8 : 27 : 12 hrs (b) 8 : 27 : 24 hrs
(c) 8 : 27 : 36 hrs (d) 8 : 27 : 48 hrs
25. Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12 seconds respectively.
In 30 minutes, how many times do they toll together?
(a) 4 (b) 10
(c) 15 (d) 16
26. An electronic device makes a deep after every 60 sec. Another device makes a beep after every 62
sec. They beeped together at 10 a.m. The next time, when they would beep together at the earliest,
is:
(a) 10:30 a.m. (b) 10:31 a.m.
(c) 10:59 a.m. (d) 11 a.m.
27. A, B and C start at the same time in the same direction to run around a circular stadium. A
completes a round in 252 seconds, B in 308 seconds and C in 198 seconds, all starting at the same
point. After what time will they meet again at the starting point?
(a) 26 min 18 sec (b) 42 min 36 sec
(c) 46 min 12 sec (d) 45 min
28. Three sets of English, Mathematics and Science books containing 336, 240 and 96 books
respectively have to be stacked in such a way that all the books are stored subject-wise and the
height of each stack is the same. Total number of stacks will be:
(a) 14 (b) 21
(c) 22 (d) 48
29. What is the greatest number of 3 digits which when divided by 6, 9, 12 leaves a remainder 3 in
each case?
(a) 903 (b) 939
(c) 975 (d) 996
30. Which greatest number will divide 3026 and 5053 leaving remainders 11 and 13 respectively?
(a) 15 (b) 30
(c) 45 (d) 60
ANSWERS
1. (b) 2. (d) 3. (c) 4. (b) 5. (b) 6. (c)
7. (d) 8. (c) 9. (a) 10. (b) 11. (d) 12. (c)
13. (b) 14. (a) 15. (b) 16. (a) 17. (d) 18. (b)
19. (d) 20. (c) 21. (d) 22. (c) 23. (d) 24. (a)
25. (d) 26. (b) 27. (c) 28. (a) 29. (c) 30. (c)
3. DECIMAL FRACTIONS
IMPORTANT FACTS AND FORMULAE
1. Decimal Fractions: Fractions in which denominators are powers of 10 are called decimal
fractions.
1 2 9
.1, .2,......, .9
10 10 10
1 2 9 10 1 11 99
.01, .02,....., .09, .1, .11,......, .99
100 100 100 100 10 100 100
1 2 9 17 999
.001, .002,......, .009, .017,....., .999
1000 1000 1000 1000 1000
Converting a Decimal into a Vulgar Fraction
Rule: In the denominator, put 1 under the decimal point and annex with it as many zeros as is the
number of digits after the decimal point. Remove the decimal point and reduce the fraction to its
lowest terms.
Ex. 1. Convert each of the following into a vulgar fraction:
(i) 0.56 (ii) 0.0024 (iii) 2.08
Sol. We have
(i)
56 14
0.56 .
100 25
(ii)
24 3
0.0024 .
10000 1250
(iii)
8 2 2
2.08 2 0.08 2 2 2 .
100 25 25
Remark 1. Annexing zeros to the right of a decimal fraction does not change its value. Thus, 0.6 =
0.60 = 0.600 etc.
Remark 2. If the numerator and denominator of a fraction contains the same number of decimal
places, then we may remove each of the decimal signs.
For example: (i)
2.71 271
3.41 341
(ii)
14.4 144 12.
15.6 156 13
Addition of Decimals:
Rule: The given numbers are so placed under each other that the decimal points lie in one column. The
numbers so arranged can now be added or subtracted in a usual way.
Ex. 2. (i) 21.3 + .213 + 3.21 + .021 + 2.0031 = ?
(ii) 345.9 + 34.59 + 3.459 + .3459 = ?
Sol. We have
( ) 21.3 ( ) 345.9
.213 34.59
3.21 3.459
.021 .3459
2.0031 384.2949
26.7471
i ii
Ex. 3. Subtract: (i) 16.5628 from 23.004 (ii) 8.2863 from 14
Sol. We have
( ) 23.004 ( ) 14.0000
16.5628 8.2863
6.4412 5.7137
i ii
Multiplication of Two or More Decimals:
Rule: Multiply the given numbers considering them without the decimal point. Now, in the product,
decimal point is marked to obtain as many places of decimal as is the sum of the number of
decimal places in the given numbers.
Ex. 4. Find each of the following products:
(i) 2.3 × 0.12 (ii) 1.11 × 2.1 × 3.2 (iii) 1.1 × .11 × .011
Sol. (i) 23 × 12 = 276
Sum of decimal places = (1 + 2) = 3
2.3 × 0.12 = .276
(ii) 111 × 21 × 32 = 74592
Sum of decimal places = (2 + 1 + 1) = 4
1.11 × 2.1 × 3.2 = 7.4592
(iii) 11 × 11 × 11 = 1331
Sum of decimal places = (1 + 2 + 3) = 6
1.1 × .11 × .011 = 0.001331
Dividing a Decimal Fraction by a Counting Number:
Rule: Divide the given decimal without considering the decimal point by the given counting number. In
the quotient so obtained put the decimal point to give as many places of decimal as are there in the
dividend.
Ex. 5. Divide (i) 0.63 by 9 (ii) 0.0221 by 17 (iii) .00042 by 7
Sol. We have
(i)
63 0.63
7 .07 ( )
99
two places of decimal
(ii)
221 .0221
13 .0013 ( )
17 17 four places of decimal
(iii)
42 0.00042
6 .00006 ( )
77 five places of decimal
Dividing a Decimal by a Decimal:
Rule: Multiply both the dividend and divisor by a suitable multiple of 10 to make divisor a whole
number. Now, proceed as above.
Ex. 6. Divide (i) 35 by .007 (ii) 0.00042 by 0.06
Sol. We have
(i)
35 35 1000 35000 5000.
.007 .007 1000 7
(ii)
0.00042 0.00042 100 .042 .007.
0.06 0.06 100 6
Recurring Decimals:
I. Pure Recurring Decimals: A decimal fraction in which all the figures after the decimal point are
repeated is called a pure recurring decimal. e.g.
—
0.5,0.32,
etc.
Converting a Pure Recurring Decimal into a Vulgar Fraction:
Rule: Write the repeated figure only once in the numerator and take as many nines in the denominator as
is the number of repeating figures.
Ex. 7. Express each of the following as a vulgar fraction:
(i)
–
0.3
(ii)
—
0.27
(iii)
0.062
Sol. We have
(i)
–31
0.3 = = .
93
(ii)
—27 3
0.27 = = .
99 11
(iii)
62
0.062 .
999
II. Mixed Recurring Decimals: A decimal fraction in which some figures do not repeat and some of
them repeat, is called a mixed recurring decimal e.g.
—–
0.5342,0.078,
etc.
Converting a Mixed Recurring Decimal into a Vulgar Fraction:
Rule: For converting a mixed recurring decimal into a vulgar fraction take in the numerator the
difference between the number formed by all the digits after decimal point (taking the repeated
digits only once) and that formed by non-repeating digits. In the denominator, take as many nines
as there are repeating digits and annex as many zeros as is the number of non-repeating digits.
Ex. 8. Express each of the following as a vulgar fraction:
(i)
—
0.26
(ii)
—
0.3268
(iii)
–
3.467
Sol. We have
(i)
—(26 2) 24 4
0.26 = = = .
90 90 15
(ii)
—(3268 32) 3236 809
0.3268 = = = .
9900 9900 2475
(iii)
––
(467 46) 421 421
3.467 3 0.467 3 3 3 .
900 900 900
EXERCISE
Mark (√) against the correct answer in each of the following:
1. 4 + 4.44 + 44.4 + 4.04 + 444 = ?
(a) 472.88 (b) 495.22
(c) 500.88 (d) 577.2
(e) None of these
2. 3.75 + 2.832 – 1.001 + 1.803 = ?
(a) 4.009 (b) 5.01
(c) 7.384 (d) 8.385
(e) None of these
3. 1027.05 – 314.005 + 112.25 = ?
(a) 825.095 (b) 825.295
(c) 825.305 (d) 825.395
4. 27.06 × 25 – ? = 600
(a) 66.5 (b) 66.75
(c) 76.3 (d) 76.5
(e) None of these
5. 16.02 × 0.001 = ?
(a) 0.001602 (b) 0.01602
(c) 0.1602 (d) 1.6021
(e) None of these
6. 3.14 × 106 = ?
(a) 314 (b) 3140
(c) 3140000 (d) None of these
7. When simplified (2.43 × 2.43 + 2.43 × 7.57 × 2 + 7.57 × 7.57) is equal to:
(a) 10 (b) 100
(c) 101.89 (d) 200.59
8. (6.5 × 6.5 – 45.5 + 3.5 × 3.5) = ?
(a) 6 (b) 7
(c) 9 (d) 10
9.
12.1 ?
19.8
(a)
7
9
(b)
11
18
(c)
13
17
(d)
11
19
10.
1999 ?
2111
(a) 0.946 (b) 0.904
(c) 0.893 (d) 0.981
11. (256)0.16 × (16)0.18 = ?
(a) 4 (b) 16
(c) 64 (d) 256.25
12. 0.9 ÷ 0.75 = ?
(a) 1.4 (b) 1.8
(c) 1.1 (d) 1.35
(e) None of these
13. 33.5 × 212 × 422.5 ÷ 22.5 × 73.5 = (21)?
(a) 8 (b) 10
(c) 12.5 (d) 6.5
(e) None of these
14. If 0.13 ÷ p2 = 13, then p = ?
(a) 10 (b) 0.01
(c) 0.1 (d) 100
15. 20.40 × ? = 12.24
(a) 0.6 (b) 0.06
(c) 6.60 (d) 0.66
(e) None of these
16.
1
6 0.25 0.75 0.3125 ?
4
(a) 5.9375 (b) 4.2968
(c) 2.1250 (d) 2
17. If 1.5x = 0.04y, then the value of
yx
yx
is:
(a)
730
77
(b)
73
77
(c)
73
770
(d)
703
77
18. If
0.25,
b
a
then
22
?
29
ab
ab
(a)
4
9
(b)
5
9
(c) 1 (d) 2
19.
0.25 0.25 0.24 0.24 ?
0.49
(a) 0.0006 (b) 0.49
(c) 0.01 (d) 0.1
20.
0.07 0.07 0.07 0.05 0.05 0.05 ?
0.07 0.07 0.07 0.05 0.05 0.05
(a) 0.002 (b) 0.02
(c) 0.2 (d) 0.0002
21.
33
22
(0.73) (0.27) ?
(0.73) (0.27) 0.73 0.27
(a) 1 (b) 0.4087
(c) 0.73 (d) 0.46
(e) None of these
22.
3
2
(2.3) 0.027 ?
(2.3) 0.69 0.09
(a) 2.6 (b) 2
(c) 2.33 (d) 2.8
23.
22
22
(6.4) (5.4) ?
(8.9) (8.9 2.2) (1.1)
(a) 0.118 (b) 0.92
(c) 1.5 (d) 0.61
24.
2 2 2
2 2 2
(0.06) (0.47) (0.079) ?
(0.006) (0.047) (0.0079)
(a) 0.1 (b) 10
(c) 100 (d) 1000
25. The value of
3 3 3
2 2 2
(1.5) (4.7) (3.8) 3 1.5 4.7 3.8
(1.5) (4.7) (3.8) 1.5 4.7 4.7 3.8 1.5 3.8
is:
(a) 0 (b) 1
(c) 10 (d) 30
26. The value of {(.98)3 + (.02)3 + 3 × .98 × .02 – 1} is:
(a) 0 (b) 1
(c) 1.09 (d) 1.98
27.
3
2
(0.013) 0.000000343 ?
(0.013) 0.000091 0.000049
(a) 0.002 (b) 0.02
(c) 0.021 (d) 0.023
(e) None of these
28.
2.3 2.3 2.3 1 ?
2.3 2.3 2.3 1
(a) 0.3 (b) 1.3
(c) 2.2 (d) 3.3
29. The value of
22
(0.09) (0.03)
0.09 0.03
is:
(a) 0.012 (b) 0.06
(c) 0.12 (d) 0.6
30. Which of the following sets of numbers is in ascending order?
(a)
9 7 5
,,
11 8 7
(b)
7 5 9
,,
8 7 11
(c)
5 9 7
,,
7 11 8
(d)
5 7 9
,,
7 8 11
31. 0.72 ÷ 3.6 = ?
(a) 0.02 (b) 0.2
(c) 2 (d) 20
32. (25.732)2 – (15.732)2 = ?
(a) 4.1464 (b) 41.464
(c) 414.64 (d) 4164.4
33.
(0.142857 0.285714) ?
(a)
1
2
(b)
1
3
(c) 2 (d) 10
34.
2.136 ?
(a)
47
220
(b)
68
495
(c)
11
227
(d)
3
222
35.
(0.3467 0.1333) ?
(a) 0.48 (b)
0.48
(c)
0.48
(d)
0.4801
36.
(0.63 0.37 0.80) ?
(a)
1.79
(b) 1.80
(c)
1.80
(d)
1.81
37. 0.393939….. = ?
(a)
39
100
(b)
83
100
(c)
93
100
(d)
39
99
38.
2.8768 ?
(a)
878
2999
(b)
9
210
(c)
292
2333
(d)
4394
24995
39.
0.2956
when expressed as a vulgar fraction, is:
(a)
2956
1000
(b)
2956
10000
(c)
2927
9900
(d) None of these
40.
(0.2 0.3 0.32) ?
(a)
0.77
(b)
0.82
(c)
0.86
(d)
0.87
41.
0.423 ?
(a)
94
99
(b)
49
99
(c)
491
990
(d)
419
990
42. Which of the following fractions lies between
3
5
and
2
3
?
(a)
2
5
(b)
1
3
(c)
1
15
(d)
31
50
43. Which of the following is the smallest?
(a)
15
16
(b)
8
3
(c)
11
12
(d)
7
8
44. If
10.2689,
3.718
then
1?
0.0003718
(a) 0.2689 (b) 2.689
(c) 2689 (d) 26890
45. If 47.2506 = 4A +
7
B
+ 2C +
5
D
+ 6E, then the value of (5A + 3B + 6C + D + 3E) is:
(a) 53.6003 (b) 53.603
(c) 153.6003 (d) 213.0003
46. If
3 3 3
1 2 ...... 9 2025,
then the value of
3 3 3
(0.11) (0.22) ..... (0.99)
is close to:
(a) 0.2695 (b) 0.3695
(c) 2.695 (d) 3.695
47. If
4.965, 2.343ab
and
2.622,c
then
3 3 3
( 3 ) ?a b c abc
(a) 6 (b) 8
(c) 10 (d) 0
48. The value of
2 2 2
2 2 2
(0.1) (0.01) (0.009)
(0.01) (0.001) (0.0009)
is:
(a) 0.01 (b) 0.1
(c) 10 (d) 100
49. Which of the following fractions is greater than
3
4
but less than
5?
6
(a)
1
2
(b)
2
3
(c)
4
5
(d)
9
10
50. The rational numbers lying between
1
4
and
3
4
are:
(a)
9 31
and
40 41
(b)
13 264
and
50 350
(c)
63 187
and
250 250
(d)
262 752
and
1000 1000
(e) None of these
51. If 1.125 × 10k = .001125, then k = ?
(a) –4 (b) –3
(c) –2 (d) –1
52. The value of
0.943 0.943 0.943 0.057 0.057 0.057
0.943 0.943 0.943 0.057 0.057 0.057
is:
(a) 0.32 (b) 0.886
(c) 1.1286 (d) None of these
53. The number whose square is equal to the difference of the squares of 75.15 and 60.12 is:
(a) 45.09 (b) 46.09
(c) 47.09 (d) 48.09
ANSWERS
1. (c) 2. (c) 3. (b) 4. (d) 5. (b) 6. (c)
7. (b) 8. (c) 9. (b) 10. (a) 11. (a) 12. (e)
13. (a) 14. (b) 15. (a) 16. (d) 17. (b) 18. (c)
19. (c) 20. (b) 21. (a) 22. (b) 23. (a) 24. (c)
25. (c) 26. (a) 27. (b) 28. (b) 29. (b) 30. (c)
31. (b) 32. (c) 33. (a) 34. (c) 35. (d) 36. (d)
37. (d) 38. (c) 39. (c) 40. (d) 41. (d) 42. (d)
43. (d) 44. (c) 45. (c) 46. (c) 47. (d) 48. (c)
49. (c) 50. (c) 51. (b) 52. (d) 53. (a)
4. SIMPLIFICATION
IMPORTANT FACTS AND FORMULAE
1. While simplifying an expression, remember the word ‘VBODMAS’, where V, B, O, D, M, A, S
stand respectively for:
(i) Virnaculum or Bar (ii) Bracket (iii) of
(iv) Division (v) Multiplication (vi) Addition
(vii) Subtraction
We must follow the above order very strictly.
2. Modulus of a real number x is its positive value, denoted by |x|.
Thus, |5| = 5 and |–5| = 5.
FORMULAE
1. (i)
2 2 2
( ) ( 2 )a b a b ab
(ii)
2 2 2
( ) ( 2 )a b a b ab
(iii)
22
( ) ( ) 4a b a b ab
(iv)
2 2 2 2
( ) ( ) 2( )a b a b a b
(v)
22
( )( ) ( )a b a b a b
(vi)
2 2 2 2
( ) ( ) 2( )a b c a b c ab bc ca
2. (i)
3 3 3
( ) 3 ( )a b a b ab a b
(ii)
3 3 3
( ) 3 ( )a b a b ab a b
(iii)
3 3 2 2
( ) ( )( )a b a b a b ab
(iv)
3 3 2 2
( ) ( )( )a b a b a b ab
EXERCISE
Mark (√) against the correct answer in each of the following:
1. (35614 – 26889) ÷ 25 = ?
(a) 317 (b) 349
(c) 356 (d) 363
(e) None of these
2. 254 × ? × 8 = 95504
(a) 47 (b) 49
(c) 51 (d) 53
(e) None of these
3. 4500 × ? = 3375
(a)
2
5
(b)
3
4
(c)
1
4
(d)
3
5
(e) None of these
4. 5852 ÷ 28 × ? – 1653 = 1064
(a) 9 (b) 13
(c) 15 (d) 18
(e) None of these
5. 2567 ÷ 17 × 3 = ? + 180
(a) 51 (b) 73
(c) 271 (d) 273
(e) None of these
6. [(125)2 ÷ 50 × 20] ÷ 25 = ?
(a) 11 (b) 100
(c) 150 (d) 250
(e) None of these
7. 24 × 15 ÷ 12 + ? = 165
(a) 65 (b) 85
(c) 135 (d) 158
(e) None of these
8.
11 4 21
16 3 18 ?
13 9 26
(a)
23
11234
(b)
11
23 234
(c)
49
32 234
(d)
31
49 234
(e) None of these
9.
1 5 2 4
12 10 7 1 ?
3 6 3 7
(a)
13
1114
(b)
11
1314
(c)
13
1314
(d)
11
1413
10.
5 3 1
5 6 5 ?
6 7 2
(a) 40 (b) 40.5
(c) 42.5 (d) 43
(e) None of these
11.
48 12 3 9 ?
12 9 3
(a)
1
43
(b) 3
(c)
1
23
(d) 21
12. By how much does
6
7 /8
exceed
6 / 7
8
?
(a)
1
68
(b)
3
64
(c)
3
74
(d)
5
76
13.
995
999 999 ?
999
(a) 990809 (b) 998996
(c) 998999 (d) 999824
14.
2 3 1 7
9 1 of 3 5 of ?
9 11 7 9
(a)
1
14
(b) 8
(c)
32
881
(d) 9
15.
23?
5 2 1
3 of 1
6 3 4
(a)
1
2
(b)
2
3
(c) 1 (d) 2
16.
1 4 5
3 of 21
4 5 6 1 of 1 ?
1 1 3 1 32
4 21
3 5 10 5
(a) 9 (b)
1
112
(c) 13 (d)
1
15 2
17.
1 2 3 4 5 6
999 999 999 999 999 999 ?
777777
(a) 2997 (b) 5979
(c) 5997 (d) 5994
18.
1
1 2 1 2 1 ?
3
(a)
4
15
(b)
1
24
(c)
1
45
(d)
1
54
19.
1 1 1 1 ?
123 234 345 456
(a)
1
30
(b)
7
30
(c)
11
30
(d)
13
30
20.
2
12 5
11
73125 ?
2
11 5
12
7315
(a)
3
4
(b)
24
25
(c) 1 (d)
1
124
21.
7 [3 {8 (4 10 )}] ?m n m n m
(a)
11 5mn
(b)
11 11nm
(c)
11 7nm
(d)
11 7mn
22.
14
1 1 ?
17
11
13
(a) 1 (b)
1
13
(c)
1
14
(d)
1
17
23.
1
12?
1
11
14
(a)
1
14
(b)
1
12
(c)
5
6
(d) 1
24.
5?
3
32
13
(a)
3
5
(b)
2
13
(c)
5
12
(d) 5
25.
9
514 ?
3
51
33
5
(a) 1 (b)
1
12
(c) 2 (d)
1
22
26.
1 1 1 (1 1 1) ?
(a) 0 (b) 1
(c) 2 (d) 3
27.
1 3 5 999
2 2 2 ..... 2 ?
3 5 7 1001
(a)
999
1001
(b)
1003
3
(c)
1001
3
(d) None of these
28.
1 1 1 1
1 1 1 1
1 1 1 1
10 10 10 10
10 10 10 10
11
1 1 simplifies to:
11
10 10
10 10
(a)
20
101
(b)
90
101
(c)
100
101
(d)
101
100
29.
2 2 2 2 2 2
{(1998) (1997) (1996) (1995) (1994) (1993) } ?
(a) 11953 (b) 11958
(c) 11963 (d) 11973
30.
1 1 1 1
1 1 1 1 ?
1 2 3 4x x x x
(a)
5
1
x
x
(b)
1
5
xx
(c)
1
( 5)x
(d)
6
5
x
x
31.
1 1 1 1 1 ?
1.4 4.7 7.10 10.13 13.16
(a)
1
3
(b)
5
16
(c)
3
8
(d)
41
7280
32.
1 1 1 1 1 1
..... ?
2 6 12 20 30 ( 1)nn
(a)
1
n
(b)
1
1n
(c)
1
n
n
(d)
2( 1)n
n
33. If
1,
3
x
y
then
22
22
?
xy
xy
(a)
10
9
(b)
5
4
(c)
5
4
(d)
5
3
34.
2
22
()
?
()
ab
ab
(a)
()
ab
ab
(b)
2
()
ab
ab
(c)
()
()
ab
ab
(d) None of these
35. If
6,
5
x
y
then
65 ?
75
xy
xy
(a)
1
7
(b)
2
7
(c)
3
7
(d)
4
7
36. If
4,
3
a
b
then
(3 2 ) ?
(3 2 )
ab
ab
(a) –1 (b) 3
(c) 5 (d) 6
37. If
11ab
and
11,bc
then
1?ca
(a) 0 (b)
1
2
(c) 1 (d) 2
38. If
( 2 ) 6ab
and ab = 4, then
21 ?
ab
(a)
1
2
(b)
1
3
(c)
3
2
(d) 2
(e)
5
2
39. If
,
3 4 7
a b c
then
()
?
abc
c
(a)
1
7
(b)
1
2
(c) 2 (d) 7
40.
885 885 885 115 115 115 ?
885 885 115 115 885 115
(a) 115 (b) 770
(c) 885 (d) 1000
41.
147 147 147 143 143 143 ?
147 147 147 143 143 143
(a)
1
290
(b) 290
(c)
1
4
(d) 4
42.
1 1 1 1 1 1 1 1 1 1 1 1
. . . . 3. . . . .
3 3 3 4 4 4 3 4 5 5 5 5 ?
1 1 1 1 1 1 1 1 1 1 1 1
. . . . . .
3 3 4 4 5 5 3 4 4 5 3 5
(a)
2
3
(b)
3
4
(c)
47
60
(d)
49
60
43.
22
(469 174) (469 174) ?
469 174
(a) 2 (b) 4
(c) 295 (d) 643
44. If
0,abc
then
2 2 2 ?
a b c
bc ca ab
(a) –1 (b) 0
(c) 1 (d) 3
45.
2 2 2 2 2 2
2 2 2 2 2 2
( ) ( ) ( ) ?
( ) ( ) ( )
x y z y x z z x y
x z y x y z y z x
(a) –1 (b) 0
(c) 1 (d) None of these
46. If
2 2 2 2 2 2
( ) 4,( ) 9,( ) 36,x y z y z x z x y
then
( ) ?x y z
(a) 0 (b)
1
(c)
3
(d)
7
47. If
( ) 0,ab bc ca
then
2 2 2
111
?
a bc b ca c ab
(a) 0 (b) 1
(c) 3 (d)
abc
48. A boy was asked to multiply a number by 12. By mistake he multiplied the number by 21 and got
his answer 63 more than the correct answer. The number was:
(a) 7 (b) 8
(c) 9 (d) 12
49. A farmer divides his herd of cows among his four sons so that first son gets one-half of the herd,
the second son one-fourth, the third son one-fifth and the fourth son 7 cows. The total number of
cows in the herd is:
(a) 100 (b) 140
(c) 180 (d) 240
50. A boy read
3th
8
of a book on one day and
4th
5
of the remainder on another day. If there were 30
pages unread, how many pages did the book contain?
(a) 240 (b) 300
(c) 600 (d) None of these
51. In a classroom there are certain number of benches. If 6 students are made to sit on a bench, then
to accommodate all of them, one more bench is needed. However, if 7 students are made to sit on a
bench, then after accommodating all of them, space for 5 students is left. What is the total number
of students in the class?
(a) 30 (b) 42
(c) 72 (d) None of these
52. In a class, there are two sections A and B. If 10 students of section B shift over to section A, the
strength of A becomes three times the strength of B. But, if 10 students shift over from A to B,
both A and B are equal in strength. How many students are there in sections A and B?
(a) 50 and 30 (b) 45 and 15
(c) 90 and 40 (d) 80 and 40
53. Each boy contributed rupees equal to the number of girls and each girl contributed rupees equal to
the number of boys in a class of 60 students. If the total contribution thus collected is Rs 1600,
how many boys are there in the class?
(a) 25 (b) 30
(c) 50 (d) Data inadequate
54. Some students planned a picnic. The budget for food was Rs 500. But, 5 of them failed to go and
thus the cost of food for each member increased by Rs 5. How many students attended the picnic?
(a) 15 (b) 20
(c) 25 (d) 30
55. In an objective examination of 90 questions, 5 marks are allotted for every correct answer and 2
marks are deducted for every wrong answer. After attempting all the 90 questions, a student got a
total of 387 marks. The number of questions attempted wrong were:
(a) 9 (b) 18
(c) 27 (d) 36
56. After measuring 120 metres of a rope, it was discovered that the metre rod was 3 cm longer. The
true length of the rope measured is:
(a) 116 m 40 cm (b) 121 m 20 cm
(c) 123 m (d) 123 m 60 cm
57. I paid
3
5
of a bill. If Rs 400 of the bill is still due, what was the total amount of the bill?
(a) Rs 1000 (b) Rs 1200
(c) Rs 1500 (d) Rs 1800
58. A man has in all Rs 640 in the denominations of one-rupee, five-rupee and ten-rupee notes. The
number of each type of notes are equal. What is the total number of notes he has?
(a) 90 (b) 100
(c) 120 (d) 150
59. A bag contains three types of coins—rupee-coins, 50 p coins and 25 p coins, totaling 175 coins. If
the total value of the coins of each kind be the same, the total amount in the bag is:
(a) Rs 75 (b) Rs 126
(c) Rs 175 (d) Rs 300
60. Ram went to a shop to buy 50 kg of rice. He buys two varieties of rice which cost him Rs 4.50 per
kg and Rs 5 per kg. He spends a total of Rs 240. What was the quantity of cheaper rice bought by
him?
(a) 20 kg (b) 25 kg
(c) 30 kg (d) None of these
61. On the Children’s Day sweets were distributed equally among 540 children. But, on that day 120
children were absent. Hence each child got 4 more sweets. How many sweets were to be
distributed to each child originally?
(a) 14 (b) 18
(c) 20 (d) 25
(e) None of these
62. A part of monthly expenses of a family is constant and the remaining varies with the price of
wheat. When the rate of wheat is Rs 250 per quintal, the total monthly expenses of the family are
Rs 1000 and when it is Rs 240 per quintal, the total monthly expenses are Rs 980. The total
monthly expenses of the family when the cost of wheat is Rs 350 per quintal, are:
(a) Rs 800 (b) Rs 1000
(c) Rs 1200 (d) Rs 1400
63. Students of a class are made to stand in rows. If 4 students are extra in each row, there would be 2
rows less. If 4 students are less in each row, there would be 4 more rows. The number of students
in the class is:
(a) 90 (b) 92
(c) 94 (d) 96
64. When the capacity of the bucket is 13.5 litres, 12 buckets of water will fill a tank. How many
buckets will be needed to fill the same tank, if the capacity of each bucket is 9 litres?
(a) 8 (b) 15
(c) 16 (d) 18
65. How many digits are required for numbering the pages of a book having 300 pages?
(a) 299 (b) 492
(c) 789 (d) 792
66. A total of 324 coins of 20 paise and 25 paise make a sum of Rs 71. The number of 25 paise coins is:
(a) 120 (b) 124
(c) 144 (d) 200
67. A sink contains exactly 12 litres of water. If water is drained from the sink until it holds 6 litres of
water less than the quantity drained away, then how many litres of water were drained away?
(a) 2 (b) 3
(c) 6 (d) 9
68. A class starts at 10 a.m. and lasts till 1.27 p.m. Four periods are held during this interval. After
every period, 5 minutes are given free to the students. The exact duration of each period is:
(a) 42 min (b) 48 min
(c) 51 min (d) 53 min
ANSWERS
1. (b) 2. (a) 3. (b) 4. (b) 5. (d) 6. (d)
7. (c) 8. (c) 9. (c) 10. (d) 11. (c) 12. (b)
13. (b) 14. (b) 15. (d) 16. (c) 17. (c) 18. (a)
19. (b) 20. (d) 21. (d) 22. (a) 23. (c) 24. (c)
25. (a) 26. (a) 27. (b) 28. (a) 29. (d) 30. (a)
31. (b) 32. (c) 33. (c) 34. (c) 35. (a) 36. (b)
37. (c) 38. (c) 39. (c) 40. (d) 41. (c) 42. (c)
43. (b) 44. (d) 45. (c) 46. (d) 47. (a) 48. (a)
49. (b) 50. (a) 51. (c) 52. (a) 53. (d) 54. (b)
55. (a) 56. (d) 57. (a) 58. (c) 59. (a) 60. (a)
61. (a) 62. (c) 63. (d) 64. (d) 65. (d) 66. (b)
67. (d) 68. (b)
5. SQUARE ROOTS AND CUBE ROOTS
IMPORTANT FACTS AND FORMULAE
1. Square Root: If
2,yx
then square root of y is x and we write,
.yx
Thus,
9 3, 16 4, 25 5, 144 12
, etc.
2. Cube Root: If
3,yx
then cube root of y is x and we write,
3.yx
11
3 3 3
33
8 2, 27 (3 3 3) 3, 64 (4 4 4) 4.
3. (i)
ab a b
(ii)
.
aa
bb
EXERCISE
Mark (√) against the correct answer in each of the following:
1.
17161 ?
(a) 129 (b) 119
(c) 121 (d) 141
(e) None of these
2.
2025 ? 81
(a) 7 (b) 9
(c) 5 (d) 11
3.
( 529 36) 48 ? 5847.75
(a) 346 (b) 339
(c) 317 (d) 325
(e) None of these
4.
2592 324
?
(a) 144 (b) 64
(c) 16 (d) 8
5.
6250 625
?
(a) 10 (b) 95
(c) 110 (d) 105
(e) None of these
6.
2
1296 (?)
(a) 6 (b) 1296
(c) 625 (d) 36
7.
535.9225 ?
(a) 23.45 (b) 28.25
(c) 23.15 (d) 24.15
8.
6?
50 200
(a) 8 (b) 576
(c) 49 (d) 24
9.
2
2500 961 (?)
(a) 81 (b) 3
(c) 6561 (d) 9
(e) None of these
10. Given that
3 1.732,
the value of
36
(5 3 2 12 32 50)
is:
(a) 4.899 (b) 2.551
(c) 1.414 (d) 1.732
11.
112 576 256 ?
12 8
196
(a) 8 (b) 12
(c) 16 (d) 32
12. If
4 1024,
n
then the value of n is:
(a) 5 (b) 8
(c) 10 (d) 12
13.
2 2 2
2 2 2
(0.27) (0.21) (0.29) ?
(0.027) (0.021) (0.029)
(a) 0.1 (b) 0.01
(c) 10 (d) 100
14. The square root of
1.44 0.81
0.9 3.6
is:
(a) 0.6 (b) 0.5
(c) 1 (d) 0.75
15.
0.729 ?
0.00841
(a)
1
929
(b)
2
929
(c)
3
929
(d)
9
929
16.
31684 ?
(a) 168 (b) 172
(c) 178 (d) 182
(e) None of these
17.
88209 ?
(a) 187 (b) 244
(c) 257 (d) 297
(e) None of these
18.
9216 12544 ?
(a) 196 (b) 200
(c) 206 (d) 218
(e) None of these
19.
4225 1225 6 ?
(a) 1156 (b) 1296
(c) 1444 (d) 1600
(e) None of these
20. If
6 2.45,
then the value of
2
33
is:
(a) 0.271 (b) 0.272
(c)
0.272
(d) None of these
21. Given
4 10 5.
2
4 10
xx
xx
The value of x is:
(a)
17
21
(b) 1
(c)
263
20
(d)
331
5
22. If
0.05 0.5 0.5 0.05 ,ab
then
?
a
b
(a) 0.0025 (b) 0.025
(c) 0.25 (d) 25
23.
48 ? 32 ? 320
(a) 4 (b) 8
(c) 16 (d) 24
(e) None of these
24.
75?
75
(a) 1 (b)
6 35
(c) 2 (d)
6 35
25. A General of Army wants to form a solid square from 36562 armies. After arrangement, he found
some armies left. How many armies were left?
(a) 36 (b) 65
(c) 81 (d) 97
26. Three-fifth of the square of a certain number is 126.15. What is the number?
(a) 14.5 (b) 75.69
(c) 145 (d) 210.25
27. Which smallest number must be added to 2203 to get a perfect square?
(a) 1 (b) 3
(c) 6 (d) 8
28. What is the smallest positive integer which when multiplied by 392, gives a perfect square?
(a) 2 (b) 3
(c) 5 (d) 6
29. What is the least number to be added to 631 to make it a perfect square?
(a) 25 (b) 30
(c) 36 (d) 45
(e) None of these
30. If
(3 8),x
then
22
1?xx
(a) 30 (b) 34
(c) 36 (d) 38
31. If
3,
2
a
then the value of
( 1 1 )aa
is:
(a)
3
(b)
3
2
(c)
(2 3)
(d)
(2 3)
32. Number of digits in the square root of 625686734489 is:
(a) 4 (b) 5
(c) 6 (d) 7
33. If
3 3 3 ..... ,a
then which of the following is true?
(a) a = 3 (b) 3 < a < 4
(c) a > 3 (d) 2 < a < 3
34.
7 5 7 5 ?
7 5 7 5
(a) 2 (b)
2( 7 5)
(c) 12 (d)
2( 7 5)
35. The square root of
44
22
11
34
43
11
34
43
is:
(a)
7
112
(b)
1
112
(c)
5
512
(d)
1
712
36.
11
?
( 2 3 5) ( 2 3 5)
(a) 0 (b)
1
2
(c) 1 (d)
2
37.
1 1 1 1
..... ?
(1 2) ( 2 3) ( 3 4) ( 99 100)
(a) 1 (b) 5
(c) 9 (d) 10
38.
0.2 ?
(a) 0.02 (b) 0.2
(c) 0.446 (d) 0.632
39.
2
3250047 (56) 7 ?
(a) 455 (b) 475
(c) 521 (d) 547
(e) None of these
40.
30.000216 ?
(a) 0.0006 (b) 0.006
(c) 0.06 (d) 0.6
41.
319683 ? 3
(a) 3 (b) 18
(c) 27 (d) 90
(e) None of these
ANSWERS
1. (e) 2. (c) 3. (b) 4. (b) 5. (e) 6. (a)
7. (c) 8. (b) 9. (b) 10. (d) 11. (d) 12. (c)
13. (c) 14. (a) 15. (d) 16. (c) 17. (d) 18. (e)
19. (b) 20. (c) 21. (c) 22. (b) 23. (c) 24. (b)
25. (c) 26. (a) 27. (c) 28. (a) 29. (d) 30. (b)
31. (a) 32. (c) 33. (d) 34. (c) 35. (c) 36. (b)
37. (c) 38. (c) 39. (e) 40. (c) 41. (e)
6. SURDS AND INDICES
IMPORTANT FACTS AND FORMULAE
1. Laws of Indices:
0
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) 1
m
m n m n m n m n mn
n
nn
n n n n
a
i a a a ii a iii a a
a
aa
iv ab a b v vi a
bb
2. Surds: Let a be a rational number and n be a positive integer such that
1n
n
aa
is irrational.
Then,
na
is called a surd of order n.
3. Laws of Surds:
1
1
( ) ( ) ( )
( ) ( )
n
n n n n
nnn
mm
nm
nn
nn
aa
i a a ii ab a b iii bb
iv a a v a a a
EXERCISE
Mark (√) against the correct answer in each of the following:
1.
2
3
8?
(a)
1
33
(b) 4
(c)
1
52
(d)
1
213
2.
3
4
16 ?
(a)
22
(b)
42
(c) 8 (d) 16
3.
1
3
8?
(a) 2 (b) 4
(c)
2
(d)
22
4.
1
2
81 ?
169
(a)
3
169
(b)
9
169
(c)
9
13
(d)
13
9
5.
4
5
32 ?
243
(a)
4
9
(b)
9
4
(c)
16
81
(d)
81
16
6.
1
6
(36) ?
(a) 1 (b) 6
(c)
6
(d)
36
7.
2
3
1?
343
(a)
1
49
(b)
1
49
(c) –49 (d) 49
8.
9 24
(1000) 10 ?
(a) 10 (b) 100
(c) 1000 (d) 10000
(e) None of these
9.
?
410000 (100)
(a)
1
2
(b)
1
4
(c)
1
8
(d) 2
(e) None of these
10. If
4 1024,
n
then n = ?
(a) 5 (b) 8
(c) 10 (d) 12
11. If
2 64,
n
then n = ?
(a) 2 (b) 4
(c) 6 (d) 12
12.
537?
22
27 9 3 3
(a)
7
2
(b)
11
2
(c)
21
2
(d) 14
(e) None of these
13.
0.16 0.18
(256) (16) ?
(a) 4 (b) 16
(c) 64 (d) 256.25
14.
7.5 2.5 2.5 ?
10 5 2 10
(a) 9.5 (b) 10
(c) 11.5 (d) 12.5
(e) None of these
15.
4.2 2.1 8.4 3.5 ?
8 64 7 56 (56)
(a) 9.8 (b) 11.9
(c) 12.6 (d) 18.2
(e) None of these
16.
8.6 3.9 4.4 3.9 8.6 ?
9 8 72 9 8 72
(a) 15.1 (b) 17.9
(c) 20.9 (d) 29.4
(e) None of these
17. If
24
2 16 ,
xx
then
3?x
(a) 2 (b) 4
(c) 8 (d) 16
18.
( ) ( ) ( )
( ) .( ) .( ) ?
b c b c c a c a a b a b
x x x
(a) 0 (b) 1
(c) x (d)
2 2 2
a b c
x
19. If
55
5 1,
x
then x = ?
(a) –1 (b)
4
5
(c) 0 (d) 1
20. If
22 1,
x
a
where a is a positive real number other than 1, then x = ?
(a) –2 (b) –1
(c) 0 (d) 1
21.
?
1
42
(2 ) 256
(a) 1 (b) 2
(c) 4 (d) 8
(e) None of these
22. If
3,
2
a
then
1 1 ?aa
(a)
(2 3)
(b)
(2 3)
(c)
3
2
(d)
3
23. If
51
51
a
and
51
,
51
b
then
22
22
()
?
()
a ab b
a ab b
(a)
3
4
(b)
4
3
(c)
3
5
(d)
5
3
24. If
,
ab
x y y z
and
,
c
zx
then abc = ?
(a) 4 (b) 3
(c) 2 (d) 1
25. The greatest of
3
64
2, 3, 4, 5
is:
(a)
2
(b)
34
(c)
45
(d)
63
26. The greatest among
7 5 , 5 3 , 9 7 , 11 9
is:
(a)
11 9
(b)
97
(c)
75
(d)
53
27. If
7 4 3,x
then
1?xx
(a) 1 (b) 2
(c) 3 (d) 4
28.
1
3
(42 229) (9261) ?
(a) 448 (b) 452
(c) 456 (d) 458
(e) None of these
29.
35
is a surd of order:
(a) 2 (b) 3
(c) 5 (d) None of these
30.
3 2 5
8 8 8 ?
(a) 0 (b) 1
(c) 8 (d) None of these
ANSWERS
1. (b) 2. (c) 3. (c) 4. (d) 5. (d) 6. (d)
7. (d) 8. (c) 9. (a) 10. (c) 11. (d) 12. (a)
13. (a) 14. (b) 15. (b) 16. (e) 17. (c) 18. (b)
19. (a) 20. (b) 21. (c) 22. (d) 23. (b) 24. (d)
25. (b) 26. (d) 27. (d) 28. (d) 29. (b) 30. (b)
7. PROBLEMS ON NUMBERS
Mark (√) against the correct answer in each of the following:
1.
3th
4
of
2rd
3
of a number is 782. What is
3th
5
of
1th
4
of the same number?
(a) 231 (b) 246.5
(c) 236.6 (d) 234.6
(e) None of these
2. 75% of a number is 304.5 more than 40% of the same number. What is 25% of that number?
(a) 220 (b) 217.5
(c) 219.4 (d) 215
(e) None of these
3. If 25% of
13
12
of a certain number is 520, then what is 60% of that number?
(a) 1252 (b) 1152
(c) 1552 (d) 1225
(e) None of these
4. A number is 25 more than its
2th
5
. The number is:
(a) 30 (b) 60
(c)
125
3
(d)
125
7
5. If the difference between the reciprocal of a positive proper fraction and the fraction itself be
9,
20
then the fraction is:
(a)
5
4
(b)
4
5
(c)
3
5
(d)
3
10
6. The sum and product of two numbers are 12 and 35 respectively. The sum of their reciprocals will
be:
(a)
12
35
(b)
1
35
(c)
35
8
(d)
7
32
7. The product of two fractions is
14
15
and their quotient is
35.
24
The greater of the fractions is:
(a)
4
5
(b)
7
3
(c)
7
6
(d)
7
4
8. If the numerator of a fraction is increased by 150% and the denominator of the fraction is
increased by 300%, the resultant fraction is
5.
18
What is the original fraction?
(a)
4
9
(b)
4
5
(c)
8
9
(d)
8
11
(e) None of these
9. If the sum of three consecutive numbers is more than the middle number by 130, then the middle
number is:
(a) 64 (b) 65
(c) 66 (d) 60
10. The sum of two numbers is 2490. If 6.5% of one number is equal to 8.5% of the other, the greater
number is:
(a) 1079 (b) 1380
(c) 1411 (d) 1250
11. If the difference of two numbers is 3 and the difference of their squares is 39, then the larger
number is:
(a) 8 (b) 9
(c) 12 (d) 13
12. If the product of two numbers is 5 and one of the numbers is
3,
2
then the sum of the two numbers
is:
(a)
1
43
(b)
2
43
(c)
5
46
(d)
1
56
(e)
1
62
13. The product of two successive numbers is 1980. Which is the smaller number?
(a) 34 (b) 35
(c) 44 (d) 45
(e) None of these
14. The difference between a number and its two-fifth is 270 more than two-fifth of the number. The
number is:
(a) 675 (b) 810
(c) 1350 (d) Cannot be determined
(e) None of these
15. If one number is
2rd
3
of other number and their sum is 60, then the first number is:
(a) 18 (b) 24
(c) 36 (d) 42
16. If the sum and difference of two numbers are 20 and 8 respectively, then the difference of their
squares is:
(a) 12 (b) 28
(c) 80 (d) 160
17. The sum of the squares of two positive integers is 100 and the difference of their squares is 28. The
sum of the numbers is:
(a) 12 (b) 13
(c) 14 (d) 15
18. Two-fifth of one-fifth of one-third of a number is 10. What is the number?
(a) 150 (b) 275
(c) 300 (d) 625
(e) None of these
19. Thrice the square of a natural number decreased by 4 times the number is equal to 50 more than
the number. The number is:
(a) 4 (b) 5
(c) 6 (d) 10
20. The difference between two positive numbers is 3. If the sum of squares of these numbers is 369,
then the sum of the numbers is:
(a) 25 (b) 27
(c) 33 (d) 81
21. If the sum of two numbers is 22 and the sum of their squares is 404, then the product of the
numbers is:
(a) 40 (b) 44
(c) 80 (d) 88
22. The product of two positive numbers is 11520 and their quotient is
9.
5
Their difference is:
(a) 60 (b) 64
(c) 70 (d) 74
23. The denominator of a fraction is 1 more than its numerator. If 1 is deducted from both the
numerator and the denominator, the fraction becomes equivalent to 0.5. The fraction is:
(a)
3
4
(b)
4
5
(c)
2
3
(d)
7
8
24. A fraction becomes
1
2
when denominator is increased by 4. The same fraction becomes
1
8
when
the numerator is reduced by 5. The fraction is:
(a)
8
12
(b)
6
8
(c)
3
5
(d)
5
8
25. A two-digit number is three times the sum of its digits. If 45 is added to the number, its digits are
interchanged. The sum of the digits of the number is:
(a) 5 (b) 7
(c) 9 (d) 11
26. A two-digit number is 7 times the sum of its digits. The number formed by reversing its digits is 18
less than the original number. What is the number?
(a) 42 (b) 52
(c) 62 (d) 72
27. When the digits of a two-digit number are interchanged, the number obtained is more than the
original number by 36. If the sum of the digits of the original number is 12, what is the original
number?
(a) 39 (b) 48
(c) 57 (d) 75
(e) None of these
28. In a two-digit positive number, the unit digit is equal to the square of tens digit. The difference
between the original number and the number formed by interchanging the digits is 54. What is
40% of the original number?
(a) 15.6 (b) 24
(c) 37.2 (d) 39
(e) None of these
29. The difference between a two-digit number and the number obtained by interchanging the
positions of its digits is 36. The digit at unit place is one-third of the digit at tens place in the
original number. What is the original number?
(a) 64 (b) 73
(c) 84 (d) Cannot be determined
(e) None of these
30. In a three-digit number, the digit in the unit/place is 75% of the digit in the tens place. The digit in
the tens place is greater than the digit in the hundred’s place by 1. If the sum of the digits in the
tens place and hundreds place is 15, what is the number?
(a) 687 (b) 786
(c) 795 (d) Cannot be determined
(e) None of these
31. In a three-digit number, the digit in the units place is four times the digit in the hundreds place. If
the digit in the units place and the tens place are interchanged, the new number so formed is 18
more than the original number. If the digits in the hundreds place is one-third of the digit in the
tens place, what is 25% of the original number?
(a) 67 (b) 84
(c) 137 (d) Cannot be determined
(e) None of these
32. The number that should be added to both the numerator and denominator of
2
2
4
9
so that the fraction
becomes
4
9
is:
(a) 0 (b) 16
(c) 36 (d) 81
ANSWERS
1. (d) 2. (b) 3. (b) 4. (c) 5. (b) 6. (a)
7. (c) 8. (a) 9. (b) 10. (c) 11. (a) 12. (c)
13. (c) 14. (c) 15. (b) 16. (d) 17. (c) 18. (e)
19. (b) 20. (b) 21. (a) 22. (b) 23. (c) 24. (b)
25. (c) 26. (a) 27. (b) 28. (a) 29. (e) 30. (b)
31. (a) 32. (c)
1. AVERAGE
IMPORTANT FACTS AND FORMULAE
1. For observations
1 2 3
, , ,......., ,
n
x x x x
we have
1 2 3 .......
average = n
x x x x
n
2. A man covers a distance d at x km/hr and returns back to the starting point at y km/hr. Then,
average speed during whole journey =
2km/hr
()
xy
xy
Mark (√) against the correct answer in each of the following:
1. If 16a + 16b = 48, what is the average of a and b?
(a) 1.5 (b) 2.5
(c) 3 (d) 5
(e) None of these
2. If the average of m numbers is n2 and that of n numbers is m2, then the average of (m + n) numbers
is:
(a) (m – n) (b) mn
(c) (m + n) (d) (m/n)
3. The average of four numbers A, B, C, D is 40. The average of four numbers A, B, E, F is also 40.
Which of the following must be true?
(a)
(A + B) (C + D)
(b) (C + D) = (E + F)
(c) (C = E or F) and (D = F or E) (d) C = E and D = F
(e) None of these
4. The average of 100 numbers is 44. The average of these 100 numbers and four other new numbers
is 50. The average of the four new numbers will be:
(a) 800 (b) 200
(c) 176 (d) 24
5. The average of 6 observations is 45.5. If one new observation is added to the previous
observations, then the new average becomes 47. The new observation is:
(a) 58 (b) 56
(c) 50 (d) 46
6. The average of marks scored by the students of a class is 68. The average of marks of the girls in
the class is 80 and that of boys is 60. What is the percentage of boys in the class?
(a) 40% (b) 60%
(c) 65% (d) 70%
7. The average age of 3 friends is 23. Even if the age of the 4th friend is added, the average remains
23. What is the age of the 4th friend?
(a) 21 years (b) 23 years
(c) 32 years (d) Cannot be determined
(e) None of these
8. Out of three numbers, the first is twice the second and is half of the third. If the average of three
numbers is 56, the difference of first and third numbers is:
(a) 12 (b) 20
(c) 24 (d) 48
9. The average of three numbers is 28. The first number is half of the second and the third number is
twice and second. The third number is:
(a) 18 (b) 24
(c) 36 (d) 48
10. The average age of 24 boys and their teacher is 15 years. When the teacher’s age is excluded, the
average age decreases by 1 year. The age of the teacher is:
(a) 38 years (b) 39 years
(c) 40 years (d) 41 years
11. The mean temperature of Monday to Wednesday was 37°C and that of Tuesday to Thursday was
34°C. If the temperature on Thursday was
4
5
of that of Monday, then what was the temperature on
Thursday?
(a) 36.5°C (b) 36°C
(c) 35.5°C (d) 34°C
12. The average weight of 8 men is increased by 1.5 kg when one of the men who weighs 65 kg is
replaced by a new man. The weight of the new man is:
(a) 70 kg (b) 74 kg
(c) 76 kg (d) 77 kg
13. The average of marks in Mathematics for five students was found to be 50. Later on, it was
discovered that in case of one student, the marks 48 were misread as 84. The correct average is:
(a) 40.2 (b) 40.8
(c) 42.8 (d) 48.2
14. The average age of 40 students of a class is 15 years. When 10 new students are admitted, the
average age is increased by 0.2 year. The average age of the new students is:
(a) 15.2 years (b) 16 years
(c) 16.2 years (d) 16.4 years
15. The average monthly salary of the workers in a workshop is Rs. 8500. If the average monthly
salary of 7 technicians is Rs. 10000 and average monthly salary of the rest is Rs. 7800, the total
number of workers in the workshop is:
(a) 18 (b) 20
(c) 22 (d) 24
16. The average age of 11 players of a cricket team is decreased by 2 months when two of them aged
17 years and 20 years are replaced by two new players. The average age of the new players is:
(a) 17 years 1 month (b) 17 years 7 months
(c) 17 years 11 months (d) 18 years 3 months
17. The average of marks of 28 students in Mathematics was 50; 8 students left the school and then the
average increased by 5. What is the average of marks obtained by the students who left the school?
(a) 37.5 (b) 42.5
(c) 45 (d) 50.5
18. A train moves with a speed of 30 km/hr for 12 minutes and for next 8 minutes at a speed of 45
km/hr. The average speed of the train is:
(a) 37.5 km/hr (b) 36 km/hr
(c) 48 km/hr (d) 30 km/hr
19. A man covers half of his journey at 6 km/hr and the remaining half at 3 km/hr. His average speed
is:
(a) 9 km/hr (b) 4.5 km/hr
(c) 4 km/hr (d) 3 km/hr
20. Three years ago the average age of A and B was 18 years. With C joining them now, the average
becomes 22 years. How old is she now?
(a) 24 years (b) 27 years
(c) 28 years (d) 30 years
21. When the average age of a husband and wife and their son was 42 years, the son got married and a
child was born just one year after their marriage. When child turned to be 5 years, then the average
of the family became 36 years. What was the age of daughter-in-law at the time of marriage?
(a) 26 years (b) 25 years
(c) 24 years (d) 23 years
22. Of the three numbers, second is twice the first and is also thrice the third. If the average of three
numbers is 44, the largest number is:
(a) 24 (b) 36
(c) 72 (d) 108
23. The average of
1 2 3 4
, , ,x x x x
is 16. Half the sum of
234
,,x x x
is 23. What is the value of
1
x
?
(a) 17 (b) 18
(c) 19 (d) 20
24. The sum of five numbers is 555. The average of first two numbers is 75 and the third number is
115. What is the average of last two numbers?
(a) 145 (b) 150
(c) 265 (d) 290
(e) None of these
25. The average of five consecutive natural numbers is m. If the next three natural numbers are also
included, how much more than m will the average of these 8 numbers be?
(a) 1 (b) 1.5
(c) 1.4 (d) 2
26. If the total monthly income of 16 persons is Rs. 80800 and the income of one of them is 120% of
the average income, then his income is:
(a) Rs. 5050 (b) Rs. 6060
(c) Rs. 6160 (d) Rs. 6600
ANSWERS
1. (a) 2. (b) 3. (b) 4. (b) 5. (b) 6. (b)
7. (b) 8. (d) 9. (d) 10. (b) 11. (b) 12. (d)
13. (c) 14. (b) 15. (c) 16. (b) 17. (a) 18. (b)
19. (c) 20. (a) 21. (b) 22. (c) 23. (b) 24. (a)
25. (b) 26. (b)
2. RATIO AND PROPORTION
IMPORTANT FACTS AND FORMULAE
1. Ratio: The ratio of two quantities a and b in the same units is a fraction that one quantity is of the
other. Thus,
:.
aab
b
In a : b, we call a as antecedent and b as consequent.
Rule: The multiplication or division of each term of a ratio by the same non-zero number does not
affect the ratio.
Thus, 3 : 5 is the same as 6 : 10 or 9 : 15 or 12 : 20 etc.
2. Proportion: The equality of two ratios is called proportion.
As 2 : 3 = 6 : 9, we write, 2 : 3 :: 6 : 9 and we say that 2, 3, 6, 9 are in proportion. Here 2 and 9 are
called extremes while 3 and 6 are called means.
In a proportion, Product of extremes = Product of means.
3. (i) If a : b :: c : d, then d is called the fourth proportional to a, b, c.
(ii) If a : b :: b : c, then c is called the third proportional to a, b.
(iii) Mean proportional between a and b
.ab
EXERCISE
Mark (√) against the correct answer in each of the following:
1. If a : b = 3 : 4 and b : c = 8 : 9, then a : c = ?
(a) 1 : 2 (b) 3 : 2
(c) 1 : 3 (d) 2 : 3
2. If A : B : C = 2 : 3 : 4, then
A B C
: : ?
B C A
(a) 8 : 9 : 16 (b) 8 : 9 : 12
(c) 8 : 9 : 24 (d) 4 : 9 : 16
3. If
1 1 1
: : 2:3:5,
x y z
then x : y : z = ?
(a) 2 : 3 : 5 (b) 15 : 10 : 6
(c) 5 : 3 : 2 (d) 6 : 10 : 15
4. If ab = 36, then which of the following is correct?
(a) a : 9 = 4 : b (b) 9 : a = 4 : b
(c) a : 18 = b : 3 (d) a : 6 = b : 6
5. If 20% of (P + Q) = 50% of (P – Q), then P : Q = ?
(a) 5 : 7 (b) 7 : 5
(c) 7 : 3 (d) 7 : 8
6. If a : b = c : d = e : f = 1 : 2, then (3a + 5c + 7e) : (3b + 5d + 7f) = ?
(a) 1 : 2 (b) 1 : 4
(c) 2 : 1 (d) 8 : 7
7. If a : b = b : c, then (a4 : b4) = ?
(a) b2 : ac (b) c2 : a2
(c) a2 : c2 (d) ac : b2
8. What number has to be added to each term of 3 : 5 to make the ratio 5 : 6?
(a) 6 (b) 7
(c) 12 (d) 13
9. The ratio between two numbers is 3 : 4. If each number is increased by 6, the ratio becomes 4 : 5.
The difference between the numbers is:
(a) 1 (b) 3
(c) 6 (d) 8
10. In a school, the ratio of boys and girls is 4 : 5. When 100 girls leave the school, the ratio becomes
6 : 7. How many boys are there in the school?
(a) 1300 (b) 1500
(c) 1600 (d) Cannot be determined
(e) None of these
11. Rs 6400 are divided among three workers in the ratio
35
: 2 : .
53
The share of the second worker is:
(a) Rs 2560 (b) Rs 3000
(c) Rs 3200 (d) Rs 3840
12. Instead of dividing Rs 117 among P, Q, R in the ratio
111
: : ,
234
by mistake it was divided in the
ratio 2 : 3 : 4. Who gained in the transaction?
(a) Only P (b) Only Q
(c) Only R (d) Both Q and R
13. The monthly incomes of A and B are in the ratio 2 : 3 and their monthly expenses are in the ratio 5
: 9. If each of them saves Rs 600 per month, then their monthly incomes are:
(a) Rs 1500, Rs 2250 (b) Rs 1200, Rs 1800
(c) Rs 1600, Rs 2400 (d) Rs 1400, Rs 2100
14. If 32 students in a class are females and the ratio of female to male students is 16 : 9, then what
percentage of the class is female?
(a) 32% (b) 36%
(c) 56.25% (d) 64%
(e) 72%
15. In a class, the number of girls is 20% more than that of the boys. The strength of the class is 66. If
4 more girls are admitted to the class, what will be the ratio of the number of boys to that of the
girls?
(a) 1 : 2 (b) 1 : 4
(c) 3 : 4 (d) 3 : 5
16. Two numbers are in the ratio
12
1 :2 .
23
When each one of these is increased by 15, their ratio
becomes
21
1 :2 .
32
The larger of the numbers is:
(a) 27 (b) 36
(c) 48 (d) 64
17. Two numbers are respectively 20% and 50% more than a third number. These two numbers are in the ratio:
(a) 2 : 5 (b) 4 : 5
(c) 3 : 2 (d) 5 : 2
18. When 20% of a number is added to another number, the number increases by 50%. What is the
ratio between the first and the second number?
(a) 3 : 2 (b) 2 : 3
(c) 5 : 2 (d) Cannot be determined
(e) None of these
19. Which of the following is the lowest ratio?
(a) 7 : 13 (b) 17 : 25
(c) 7 : 15 (d) 15 : 23
20. Three numbers are in the ratio 3 : 4 : 5. The sum of the largest and the smallest equals the sum of
the third and 52. The smallest number is:
(a) 20 (b) 27
(c) 39 (d) 52
21. The sum of the squares of three numbers which are in the ratio 2 : 3 : 4 is 725. What are these
numbers?
(a) 10, 15, 20 (b) 14, 21, 28
(c) 20, 15, 30 (d) 20, 30, 40
22. If A exceeds B by 40% and B is less than C by 20%, then A : C = ?
(a) 3 : 1 (b) 3 : 2
(c) 26 : 25 (d) 28 : 25
23. Rs 1300 are divided among A, B, C and D in such a way that:
A's share B's share C's share 2
B's share C's share D's share 3
What is A’s share?
(a) Rs 320 (b) Rs 240
(c) Rs 160 (d) Rs 140
24. The ratio of the ages of two students is 3 : 2. One is older than the other by 5 years. What is the age
of the younger student?
(a) 2 years (b)
1
22
years
(c) 10 years (d) 15 years
25. A bag contains Rs 216 in the form of one-rupee, 50-paise and 25-paise coins in the ratio of 2 : 3 :
4. The number of 50-paise coins is:
(a) 96 (b) 114
(c) 141 (d) 144
26. If y varies directly as (x + 3) and y = 8 when x = 1. What is the value of y when x = 2?
(a) 6 (b) 10
(c) 12 (d) 16
ANSWERS
1. (d) 2. (c) 3. (b) 4. (a) 5. (c) 6. (a)
7. (c) 8. (b) 9. (c) 10. (e) 11. (b) 12. (d)
13. (c) 14. (d) 15. (c) 16. (c) 17. (b) 18. (c)
19. (c) 20. (c) 21. (a) 22. (d) 23. (c) 34. (c)
25. (d) 26. (b)
3. TIME AND DISTANCE
IMPORTANT FACTS AND FORMULAE
1. (i)
Distance
Speed = Time
(ii)
Distance
Time = Speed
(iii) Distance = (Speed) × (Time)
2. (i)
5
km/hr = m/sec
18
xx
(ii)
18
m/sec = km/hr.
5
xx
3. Suppose a man covers a distance at x km/hr and an equal distance at y km/hr. Then, average speed
during whole journey
2
= km/hr.
()
xy
xy
4. If the ratio of speeds of A and B is a : b, then the ratio of times taken by them to cover the same
distance is
11
:.
ab
EXERCISE
Mark (√) against the correct answer in each of the following:
1. A train is moving with a speed of 180 km/hr. Its speed is:
(a) 5 m/sec (b) 30 m/sec
(c) 40 m/sec (d) 50 m/sec
2. An athlete runs 200 metres race in 24 seconds. His speed is:
(a) 20 km/hr (b) 24 km/hr
(c) 28.5 km/hr (d) 30 km/hr
3. A man riding his bicycle covers 150 metres in 25 seconds. What is his speed in km per hour?
(a) 20 km/hr (b) 21.6 km/hr
(c) 23 km/hr (d) 25 km/hr
4. A man completes 30 km of a journey at 6 km/hr and the remaining 40 km of the journey in 5
hours. His average speed for the whole journey is:
(a)
4
611
km/hr (b) 7 km/hr
(c)
1
72
km/hr (d) 8 km/hr
5. A man covers half of his journey at 6 km/hr and the remaining half at 3 km/hr. His average speed
is:
(a) 3 km/hr (b) 4 km/hr
(c) 4.5 km/hr (d) 9 km/hr
6. The speeds of A and B are in the ratio 3 : 4. A takes 20 minutes more than B to reach a destination.
In what time does A reach the destination?
(a)
1
13
hours (b) 2 hours
(c)
2
13
hours (d)
2
23
hours
7. A is twice as fast as B and B is thrice as fast as C. The journey covered by C in 42 minutes will be
covered by A in:
(a) 7 minutes (b) 14 minutes
(c) 28 minutes (d) 63 minutes
8. The ratio between the speeds of two trains is 7 : 8. If the second train runs 400 km in 5 hours, the
speed of the first train is:
(a) 70 km/hr (b) 200 km/hr
(c) 250 km/hr (d) 350 km/hr
9. Two trains approach each other at 30 km/hr and 27 km/hr from two places 342 km apart. After
how many hours will they meet?
(a) 5 hours (b) 6 hours
(c) 7 hours (d) 12 hours
10. If a student walks from his house to school at 5 km/hr, he is late by 30 minutes. However, if he
walks at 6 km/hr, he is late by 5 minutes only. The distance of his school from his house is:
(a) 2.5 km (b) 3.6 km
(c) 5.5 km (d) 12.5 km
11. A car travelling with
5
7
of its usual speed covers 42 km in 1 hour 40 min 48 sec. What is the usual
speed of the car?
(a)
6
17 7
km/hr (b) 25 km/hr
(c) 30 km/hr (d) 35 km/hr
12. A certain distance is covered at a certain speed. If half of this distance is covered in double the
time, the ratio of the two speeds is:
(a) 4 : 1 (b) 1 : 4
(c) 2 : 1 (d) 1 : 2
13. Two cars start at the same time from one point and move along two roads at right angles to each
other. Their speeds are 36 km/hr and 48 km/hr respectively. After 15 seconds, the distance
between them will be:
(a) 150 m (b) 250 m
(c) 300 m (d) 400 m
14. A and B start simultaneously from a certain point in North and South directions on motor cycles.
The speed of A is 80 km/hr and that of B is 65 km/hr. What is the distance between A and B after
12 minutes?
(a) 14.5 km (b) 29 km
(c) 36.2 km (d) 39 km
15. R and S start walking towards each other at 10 A.M. at speeds of 3 km/hr and 4 km/hr
respectively. They were initially 17.5 km apart. At what time do they meet?
(a) 11:30 A.M. (b) 12:30 P.M.
(c) 1:30 P.M. (d) 2:30 P.M.
16. A boy is running at a speed of p km/hr to cover a distance of 1 km. But, due to slippery ground, his
speed is reduced by q km/hr (p > q). If he takes r hours to cover the distance, then
(a)
1()pq
r
(b)
()r p q
(c)
1()pq
r
(d)
()r p q
17. A star is 8.1 × 1013 km away from the earth. Suppose light travels at the speed of 3.0 × 105 km per
second. How long will it take light from star to reach the earth?
(a) 7.5 × 103 hrs (b) 7.5 × 104 hrs
(c) 2.7 × 1010 sec (d) 2.7 × 1011 sec
18. The speeds of three cars are in the ratio 2 : 3 : 4. The ratio of the times taken by these cars to travel
the same distance is:
(a) 2 : 3 : 4 (b) 4 : 3 : 2
(c) 4 : 3 : 6 (d) 6 : 4 : 3
19. Sunil covers a distance by walking for 6 hours. While returning his speed decreases by 1 km/hr
and he takes 9 hours to cover the same distance. What was his speed in return journey?
(a) 2 km/hr (b) 3 km/hr
(c) 5 km/hr (d) Cannot be determined
(e) None of these
20. A takes 2 hours more than B to walk d km. If A doubles his speed then he can make it in 1 hour
less than B. How much time does B require for walking d km?
(a)
2
d
hrs (b) 3 hrs
(c) 4 hrs (d)
2
3
d
hrs
21. A constable is 114 m behind a thief. The constable runs 21 m and the thief 15 m in a minute. In
what time will the constable catch the thief?
(a) 16 min (b) 17 min
(c) 18 min (d) 19 min
22. An express train travelled at an average speed of 100 km/hr, stopping for 3 minutes after 75 km. A
local train travelled at a speed of 50 km/hr, stopping for 1 minute after every 25 km. If the trains
began travelling at the same time, how many kms did the local train travel in the time it took the
express train to travel 600 km?
(a) 307.5 km (b) 900 km
(c) 1000 km (d) 1200 km
ANSWERS
1. (d) 2. (d) 3. (b) 4. (b) 5. (b) 6. (a)
7. (a) 8. (a) 9. (b) 10. (d) 11. (d) 12. (a)
13. (b) 14. (b) 15. (b) 16. (a) 17. (b) 18. (d)
19. (a) 20. (c) 21. (d) 22. (a)
4. BOATS AND STREAMS
IMPORTANT FACTS AND FORMULAE
1. In water, the direction along the stream is called downstream. And, the direction against the
stream is called upstream.
2. If the speed of a boat in still water is u km/hr and the speed of the stream is v km/hr, then:
Speed downstream = (u + v) km/hr;
Speed upstream = (u – v) km/hr.
3. If the speed downstream is a km/hr and the speed upstream is b km/hr, then:
Speed in still water =
1()
2ab
km/hr.
Rate of stream =
1()
2ab
km/hr.
EXERCISE
Mark (√) against the correct answer in each of the following:
1. A boat is rowed downstream at 15.5 km/hr and upstream at 8.5 km/hr. The speed of the steam is:
(a) 3.5 km/hr (b) 5.75 km/hr
(c) 6.5 km/hr (d) 7 km/hr
2. A boat running downstream covers 24 kms in 4 hours, while for covering the same distance
upstream it takes 6 hours. What is the speed of the boat in still water?
(a) 3.5 km/hr (b) 5.5 km/hr
(c) 6 km/hr (d) Data inadequate
(e) None of these
3. A boat goes 24 km downstream in 10 hours. It takes 2 hours more to cover the same distance
against the stream. What is the speed of the boat in still water?
(a) 2 km/hr (b) 2.8 km/hr
(c) 4 km/hr (d) 4.2 km/hr
(e) None of these
4. A steamer goes downstream from one port to another in 4 hours. It covers the same distance
upstream in 5 hours. If the speed of the stream is 2 km/hr, the distance between the two ports is:
(a) 50 km (b) 60 km
(c) 70 km (d) 80 km
5. A boat running downstream covers a distance of 10 km in 2 hours. While coming back upstream
the boat takes 5 hours to cover the same distance. If the speed of the current is 1.5 kmph, what is
the speed of the boat in still water?
(a) 2.5 km/hr (b) 3.5 km/hr
(c) 4.5 km/hr (d) Cannot be determined
(e) None of these
6. A boat covers a distance of 30 km in
1
22
hrs running downstream. While returning, it covers the
same distance in
3
34
hrs. What is the speed of the boat in still water?
(a) 8 km/hr (b) 12 km/hr
(c) 14 km/hr (d) 15 km/hr
(e) None of these
7. A boat can row 1 km with stream in 10 minutes and 1 km against stream in 20 minutes. What is
the speed of the boat in still water?
(a) 1.5 km/hr (b) 3 km/hr
(c) 3.4 km/hr (d) 4.5 km/hr
8. The speed of a boat in still water is 15 km/hr. It can go 30 km upstream and return of downstream
to the original point in 4 hrs 30 min. The speed of the stream is:
(a) 5 km/hr (b) 8 km/hr
(c) 10 km/hr (d) 15 km/hr
9. The speed of a motor boat is that of the current of water as 36 : 5. The boat goes along with the
current in 5 hours 10 min. It will come back in:
(a) 5 hrs 50 min. (b) 6 hours
(c) 6 hours 50 min. (d) 12 hrs 10 min.
10. A man can row at 5 km/hr in still water. If the river is running at 1 km/hr, it takes him 75 minutes
to row to a place and back. How far is the place?
(a) 2.5 km (b) 3 km
(c) 4 km (d) 5 km
11. In a river, a man takes 3 hours in rowing 3 km upstream or 15 km downstream. What is the speed
of the current?
(a) 2 km/hr (b) 4 km/hr
(c) 6 km/hr (d) 9 km/hr
12. A boat running upstream takes 8 hours 48 min to cover a certain distance, while it takes 4 hours to
cover the same distance running downstream. What is the ratio between the speed of the boat and
the speed of water current respectively?
(a) 2 : 1 (b) 3 : 1
(c) 8 : 3 (d) Cannot be determined
(e) None of these
13. A boat goes 6 km in an hour in still water. It takes thrice as much time in covering the same
distance against the current. Speed of the current is:
(a) 2 km/hr (b) 3 km/hr
(c) 4 km/hr (d) 5 km/hr
14. A streamer goes downstream and covers the distance between two ports in 4 hours while it covers
the same distance upstream in 5 hours. If the speed of the stream be 2 km/hr, the speed of the
steamer in still water is:
(a) 16 km/hr (b) 18 km/hr
(c) 24 km/hr (d) 36 km/hr
15. A boat goes 24 km upstream and 28 km downstream in 6 hours. It goes 30 km upstream and 21
km downstream in 6 hours and 30 minutes. The speed of the boat in still water is:
(a) 4 km/hr (b) 6 km/hr
(c) 10 km/hr (d) 14 km/hr
16. A fisherman can row 2 km against the stream in 20 minutes and return in 15 minutes. What is the
speed of the current?
(a) 1 km/hr (b) 2 km/hr
(c) 3 km/hr (d) None of these
17. A man can row three-quarters of a kilometer against the stream in
1
114
minutes and return in
1
72
minutes. The speed of the man in still water is:
(a) 2 km/hr (b) 3 km/hr
(c) 4 km/hr (d) 5 km/hr
18. A man can row
1
72
kms an hour in still water and he finds that it takes him twice as long to row
down the river. The rate of the stream is:
(a) 2.4 km/hr (b) 2.5 km/hr
(c) 3.4 km/hr (d) 3.5 km/hr
(e) None of these
19. A boatman goes 2 km against the current of stream in 40 minutes and returns to the same spot in
30 minutes. What is his rate of rowing in still water?
(a) 5 km/hr (b) 6 km/hr
(c) 7 km/hr (d) 9 km/hr
20. A boatman goes 2 km against the current of stream in 1 hour and goes 1 km along the current in 10
minutes. How long will he take to go 5 km in stationary water?
(a) 40 minutes (b) 1 hour
(c) 1 hr 15 min (d) 1 hr 30 min
21. The current of a stream runs at 4 km an hour. A boat goes 6 km and back to the starting point in 2
hours. The speed of the boat in still water is:
(a) 6 km/hr (b) 7.5 km/hr
(c) 8 km/hr (d) 6.8 km/hr
22. A man rows to a place 48 km distant and back in 14 hours. He finds that he can row 4 km with the
stream in the same time as 3 km against the stream. The rate of the stream is:
(a) 0.5 km/hr (b) 1 km/hr
(c) 3.5 km/hr (d) 1.8 km/hr
23. The current of a stream runs at 1 km/hr. A motor boat goes 35 km upstream and back again to the
starting point in 12 hours. The speed of the motor boat in still water is:
(a) 6 km/hr (b) 7 km/hr
(c) 8.5 km/hr (d) 8 km/hr
24. A boat takes half time in moving a certain distance downstream than upstream. What is the ratio
between rate in still water and rate of current?
(a) 1 : 2 (b) 3 : 1
(c) 2 : 1 (d) 1 : 3
ANSWERS
1. (a) 2. (e) 3. (e) 4. (d) 5. (b) 6. (e)
7. (d) 8. (a) 9. (c) 10. (b) 11. (a) 12. (c)
13. (c) 14. (b) 15. (c) 16. (a) 17. (d) 18. (b)
19. (c) 20. (c) 21. (c) 22. (b) 23. (a) 24. (b)
5. TIME AND WORK
IMPORTANT FACTS AND FORMULAE
1. If A can do a piece of work in n days, then A’s 1 day’s work =
1.
n
2. If A’s 1 day’s work =
1,
n
then A can finish the work in n days.
3. If A is twice as good a workman as B, then
ratio of work done by A and B in the same time = 2 : 1
ratio of times taken by A and B in doing the same work = 1 : 2
EXERCISE
Mark (√) against the correct answer in each of the following:
1. A can do a piece of work in 8 hours while B alone can do it in 12 hours. Both A and B working
together can finish the work in:
(a) 10 hours (b) 4 hours
(c)
1
54
hours (d)
4
45
hours
2. Ram and Shyam together can finish a job in 8 days. Ram can do the same job on his own in 12
days. How long will Shyam take to do the job by himself?
(a) 16 days (b) 20 days
(c) 24 days (d) 30 days
3. To complete a work, A takes 50% more time than B. If together they take 18 days to complete the
work, how much time shall B take to do it?
(a) 30 days (b) 35 days
(c) 40 days (d) 45 days
4. A does 20% less work than B. If A can complete a piece of work in
1
72
hours, then B can do it in:
(a) 5 hours (b)
1
52
hours
(c) 6 hours (d)
1
62
hours
5. Kamal can do a work in 15 days. Bimal is 50% more efficient than Kamal. The number of days,
Bimal will take to do the same piece of work, is:
(a) 10 (b)
1
10 2
(c) 12 (d) 14
6. 3 men or 5 women can do a work in 12 days. How long will 6 men and 5 women take to finish the
work?
(a) 4 days (b) 10 days
(c) 15 days (d) 20 days
7. A can do a work in 4 days, B can do it in 5 days and C can do it in 10 days. A, B and C together
can do the work in:
(a)
3
15
days (b)
9
111
days
(c)
5
26
days (d) 3 days
8. A and B together can complete a piece of work in 12 days, B and C can do it in 20 days and C and
A can do it in 15 days. A, B and C together can complete it in:
(a) 6 days (b) 9 days
(c) 10 days (d)
1
10 2
days
9. A and B can complete a piece of work in 12 days and 18 days respectively. A begins to do the
work and they work alternately one at a time for one day each. The whole work will be completed
in:
(a)
1
14 3
days (b)
2
15 3
days
(c)
1
16 3
days (d)
2
18 3
days
10. 9 men working 7 hours a day can complete a piece of work in 15 days. In how many days can 6
men working for 9 hours a day, complete the same piece of work?
(a)
3
15 4
days (b) 16 days
(c)
3
16 4
days (d)
1
17 2
days
11. 10 women can complete a work in 8 days and 10 children take 12 days to complete the work. How
many days will 6 women and 3 children together take to complete the work?
(a) 7 (b) 8
(c) 9 (d) 12
(e) None of these
12. A and B undertook to do a piece of work for Rs 4500. A alone could do it in 8 days and B alone in
12 days. With the assistance of C, they finished the work in 4 days. C’s share of money is:
(a) Rs 375 (b) Rs 750
(c) Rs 1500 (d) Rs 2250
13. A alone can finish a work in 10 days which B alone can finish in 15 days. If they work together
and finish it, then out of a total wages of Rs 3000, A will get:
(a) Rs 1200 (b) Rs 1500
(c) Rs 1800 (d) Rs 2000
14. A works twice as fast as B. If both of them can together finish a piece of work in 12 days, then B
alone can do it in:
(a) 24 days (b) 27 days
(c) 36 days (d) 48 days
15. A is thrice as good a workman as B and therefore is able to finish a job in 60 days less than B.
Working together, they can do it in:
(a) 20 days (b)
1
22 2
days
(c) 25 days (d) 30 days
16. 10 women can complete a work in 7 days and 10 children take 14 days to complete the work. How
many days will 5 women and 10 children take to complete the work?
(a) 3 (b) 5
(c) 7 (d) Cannot be determined
(e) None of these
17. A, B and C completed a piece of work costing Rs 1800. A worked for 6 days, B for 4 days and C
for 9 days. If their daily wages are in the ratio 5 : 6 : 4, how much amount will be received by A?
(a) Rs 800 (b) Rs 600
(c) Rs 900 (d) Rs 750
ANSWERS
1. (d) 2. (c) 3. (a) 4. (c) 5. (a) 6. (a)
7. (b) 8. (c) 9. (a) 10. (d) 11. (e) 12. (b)
13. (c) 14. (c) 15. (b) 16. (c) 17. (b)
6. PERCENTAGE
IMPORTANT FACTS AND FORMULAE
1. By a certain percent, we mean that many hundredths.
x percent = x%
100
x
Thus,
25 1 3 3
25% ,0.3 100 % 30%
100 4 10 10
2. (i) If A is R% more than B, then B is less than A by
100 %
(100 )
R
R
(ii) If A is R% less than B, then B is less than A by
100 %
(100 )
R
R
3. RESULTS ON POPULATION
Let P be the population of a town now. Let it increase at R% per annum.
Then,
(i) Population after n years
1100
n
R
P
(ii) Population n years ago
1100
n
P
R
4. RESULTS ON DEPRECIATION: Let the present value of a machine be Rs. P and let it
depreciate at R% per annum. Then,
(i) Value of the machine after n years
=Rs 1 100
n
R
P
(ii) Value of the machine n years ago
=Rs
1100
n
P
R
5. (i) If the price of a commodity increases by R%, then reduction in consumption so as not to
increase the expenditure is given by
Reduction %
100 %
(100 )
R
R
(ii) If the price of a commodity decreases by R%, then in order to have no change in
expenditure, we have
Increase % in consumption
100 %
(100 )
R
R
EXERCISE
Mark (√) against the correct answer in each of the following:
1. 150% of 15 + 75% of 75 = ?
(a) 75.75 (b) 78.75
(c) 135 (d) 281.25
(e) None of these
2. 37% of 150 – 0.05% of 1000 = ?
(a) 50 (b) 55
(c) 55.5 (d) 55.55
(e) None of these
3. (43% of 2750) – (38% of 2990) = ?
(a) 43.6 (b) 46.3
(c) 44.7 (d) 49.3
(e) None of these
4. (9% of 386) × (6.5% of 144) = ?
(a) 328.0065 (b) 333.3333
(c) 325.1664 (d) 340.1664
(e) None of these
5. 40% of ? = 240
(a) 60 (b) 6000
(c) 960 (d) 600
(e) None of these
6. (0.5% of 640) × (1.3% of 350) = ?
(a) 12.56 (b) 13.44
(c) 14.44 (d) 14.56
(e) None of these
7. (39% of 640) × (?% of 590) = 414.8
(a) 24 (b) 27
(c) 31 (d) 32
(e) None of these
8. 60 = ?% of 400
(a) 6 (b) 12
(c) 15 (d) 20
(e) None of these
9. 30% of ? = 180
(a) 60 (b) 600
(c) 960 (d) 6000
(e) None of these
10. 23% of 8040 + 42% of 545 = ?% of 3000
(a) 56.17 (b) 63.54
(c) 69.27 (d) 71.04
(e) None of these
11. 80% of 50% of 250% of 34 = ?
(a) 38 (b) 40
(c) 42.5 (d) 43
(e) None of these
12. 53% of 120 + 25% of 862 = ?% of 500
(a) 42.50 (b) 55.82
(c) 38.89 (d) 63.68
(e) None of these
13. (0.9% of 450) ÷ (0.2% of 250) = ?
(a) 5.04 (b) 7.5
(c) 8.1 (d) 9.1
(e) None of these
14. (180% of ?) ÷ 2 = 504
(a) 400 (b) 480
(c) 560 (d) 600
(e) None of these
15. (0.08% of 363 + 0.6% of 241) × 500 = ?
(a) 84.62 (b) 86.82
(c) 846.2 (d) 868.2
(e) None of these
16. (31% of 260) × ? = 12896
(a) 140 (b) 150
(c) 160 (d) 180
(e) None of these
17. How is
1%
2
expressed as a decimal fraction?
(a) 0.0005 (b) 0.005
(c) 0.05 (d) 0.5
18. How is
3
4
expressed as percentage?
(a) 0.75% (b) 7.5%
(c) 60% (d) 75%
19. Two numbers are less than a third number by 30% and 37% respectively. How much percent is the
second number less than the first?
(a) 10% (b) 15%
(c) 20% (d) 25%
20. 605 sweets were distributed equally among children in such a way that the number of sweets
received by each child is 20% of the total number of children. How many sweets did each child
receive?
(a) 11 (b) 24
(c) 45 (d) Cannot be determined
(e) None of these
21. Rs 395 are divided among A, B and C in such a manner that B gets 25% more than A and 20%
more than C. The share of A is:
(a) Rs 198 (b) Rs 120
(c) Rs 180 (d) Rs 195
22. 1 litre of water is added to 5 litres of alcohol-water solution containing 40% alcohol strength. The
strength of alcohol in the new solution will be:
(a) 30% (b)
1
33 %
3
(c)
2
33 %
3
(d) 33%
23. In an examination, 65% of the total examinees passed. If the number of failures is 420, the total
number of examinees is:
(a) 500 (b) 1200
(c) 1000 (d) 1625
24. Out of an earning of Rs 720 Ram spends 65%. How much does he save?
(a) Rs 350 (b) Rs 390
(c) Rs 252 (d) Rs 316
25. The ratio of the number of boys and girls is 3:2. If 20% of the boys and 30% of the girls are
scholarship holders, then the percentage of students who do not get scholarship, is:
(a) 50 (b) 72
(c) 75 (d) 76
26. The expenses on rice, fish and oil of a family are in the ratio 12:17:3. The prices of these articles
are increased by 20%, 30% and 50% respectively. The total expenses of the family on these
articles are increased by:
(a)
1
14 %
8
(b)
1
7%
8
(c)
1
56 %
8
(d)
1
28 %
8
27. In 50 gm alloy of gold and silver, the gold is 80% by weight. How much gold should be mixed to
this alloy so that the weight of gold would become 95%?
(a) 200 gm (b) 150 gm
(c) 50 gm (d) 10 gm
28. A reduction of 20% in the price of sugar enables a purchaser to obtain 3 kg more for Rs 120. The
original price of sugar per kg is:
(a) Rs 15 (b) Rs 12
(c) Rs 10 (d) Rs 8
29. 5% of income of A is equal to 15% of income of B and 10% of income of B is equal to 20% of
income of C. If the income of C is Rs 2000, what is the total income of A, B and C?
(a) Rs 14000 (b) Rs 16000
(c) Rs 18000 (d) Rs 12400
30. A and B are two fixed points 5 cm apart and C is a point on AB such that AC = 3 cm. If the length
of AC is increased by 6%, the length of CB is decreased by:
(a) 6% (b) 7%
(c) 8% (d) 9%
31. The length of a rectangle is increased by 10% and breadth decreased by 10%. Then, the area of
new rectangle is:
(a) neither increased nor decreased
(b) increased by 1%
(c) decreased by 1%
(d) decreased by 10%
32. A typist uses a paper 30 cm by 15 cm. He leaves a margin of 2.5 cm at the top as well as at the
bottom and 1.25 cm on either side. What percentage of paper area is approximately available for
typing?
(a) 65% (b) 70%
(c) 80% (d) 60%
33. If the side of a square is increased by 25%, then its area is increased by:
(a) 25% (b) 55%
(c) 40.5% (d) 56.25%
34. Two numbers are respectively 20% and 50% more than a third number. These two numbers are in
the ratio:
(a) 2:5 (b) 4:5
(c) 6:7 (d) 3:5
35. When a number is first increased by 10% and then reduced by 10%, the number:
(a) does not change (b) decreases by 1%
(c) increases by 1% (d) None of these
36. A period of 4 hrs 30 min is what percent of a day?
(a)
3
18 %
4
(b) 20%
(c)
2
16 %
3
(d) 19%
37. A bucket contains 2 litres more water when it is filled 80% in comparison when it is filled
2
66 %
3
.
What is the capacity of the bucket?
(a) 10 litres (b) 15 litres
(c)
2
66 3
litres (d) 20 litres
38. A dishonest dealer claims to sell his goods at the cost price but uses a false weight of 900 gm for 1
kg. What is his gain percent?
(a) 13% (b)
1
11 %
9
(c) 11.25% (d)
1
12 %
9
39. In a school, 40% of the students play football and 50% play cricket. If 18% of the students play
neither football nor cricket, the percentage of students playing both is:
(a) 40% (b) 32%
(c) 22% (d) 8%
40. If the price of eraser is reduced by 25%, a person can buy 2 more erasers for a rupee. How many
erasers are available for a rupee?
(a) 8 (b) 6
(c) 4 (d) 2
ANSWERS
1. (b) 2. (b) 3. (b) 4. (c) 5. (d) 6. (d)
7. (e) 8. (c) 9. (b) 10. (c) 11. (e) 12. (b)
13. (c) 14. (c) 15. (d) 16. (c) 17. (b) 18. (d)
19. (a) 20. (a) 21. (b) 22. (b) 23. (b) 24. (c)
25. (d) 26. (d) 27. (b) 28. (c) 29. (c) 30. (d)
31. (c) 32. (b) 33. (d) 34. (b) 35. (b) 36. (a)
37. (b) 38. (b) 39. (d) 40. (b)
7. PROFIT AND LOSS
IMPORTANT FACTS AND FORMULAE
1. (i) Cost Price (C.P.): The price at which an article is purchased, is called its cost price.
(ii) Selling Price (S.P.): The price at which an article is sold, is called its selling price.
(iii) Profit or Gain = (S.P.) – (C.P.)
(iv) Loss = (C.P.) – (S.P.)
(v) Gain or loss is always reckoned on C.P.
2. Formulae
(Gain × 100) (Loss × 100)
( ) Gain% = ( ) Loss% =
C.P. C.P.
(100 + Gain%) (100 – Loss%)
( ) S.P. = (C.P.) ( ) S.P. = (C.P.)
100 100
100 100
( ) C.P. = (S.P.) ( ) C.P. = (S.P.)
(100 + Gain%) (100 – Loss%)
i ii
iii iv
v vi
Examples:
(i) If an article is sold at a gain of 20%, then S.P. = (120% of C.P.)
(ii) If an article is sold at a loss of 10%, then S.P. = (90% of C.P.)
EXERCISE
Mark (√) against the correct answer in each of the following:
1. If a man were to sell his chair for Rs. 720, he would lose 25%. To gain 25%, he should sell it for:
(a) Rs. 1200 (b) Rs. 1000
(c) Rs. 960 (d) Rs. 900
2. A man sells his typewriter at 5% loss. If he sells it for Rs. 80 more, he gains 5%. The cost price of
the typewriter is:
(a) Rs. 1600 (b) Rs. 1200
(c) Rs. 1000 (d) Rs. 800
3. A bookseller sells a book at a gain of 10%. If he had bought it at 4% less and sold it for Rs. 6
more, he would have gained
3
18 %.
4
The cost price of the book is:
(a) Rs. 130 (b) Rs. 140
(c) Rs. 150 (d) Rs. 160
4. Ram bought 1600 eggs at Rs. 3.75 per dozen. He sold 900 of them at 2 for Re 1 and the remaining
at for Rs. 2. His gain percent is:
(a) 40% (b) 45%
(c) 42% (d) 46%
5. A merchant has 1000 kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He
gains 14% on the whole. The quantity sold at 18% profit is:
(a) 560 kg (b) 600 kg
(c) 400 kg (d) 640 kg
6. A man sold an article at a loss of 20%. If he sells the article of Rs. 12 more, he would have gained
10%. The cost price of the article is:
(a) Rs. 60 (b) Rs. 40
(c) Rs. 30 (d) Rs. 22
7. The radio is sold for Rs. 990 at a profit of 10%. What would have been the gain or loss percent,
had it been sold for Rs. 890?
(a) loss, 10% (b) gain,
1
1%
9
(c) loss,
1
1%
9
(d) loss, 1%
8. A man sold two pipes at Rs. 12 each. On one he gained 20% and on the other lost 20%. On the
whole, he:
(a) neither gained nor lost (b) gained Re 1
(c) lost Re 1 (d) gained Rs. 2
9. If an article is sold at a gain of 5% instead of being sold at a loss of 5%, one gets Rs. 5 more. What
is the cost price of the article?
(a) Rs. 100 (b) Rs. 105
(c) Rs. 110 (d) Rs. 50
10. By selling 33 m of cloth Ramesh gained the cost price of 11 m. The gain percent is:
(a) 10% (b) 20%
(c) 25% (d) 50%
(e) None of these
11. By selling 100 pencils, a shopkeeper gains the selling price of 20 pencils. His gain percent is:
(a) 25% (b) 20%
(c) 15% (d) 12%
12. Mohan bought 20 dining tables for Rs. 12000 and sold them at a profit equal to the selling price of
4 dining tables. The selling price of each dining table is:
(a) Rs. 700 (b) Rs. 750
(c) Rs. 725 (d) Rs. 775
13. Ravi buys some toffees at 2 for a rupee and sells them at 5 for a rupee. His loss percent is:
(a) 120% (b) 90%
(c) 30% (d) 60%
14. A fruit seller buys lemons at 2 for a rupee and sells them at 5 for 3 rupees. His gain percent is:
(a) 10% (b) 15%
(c) 20% (d) 25%
15. Oranges are bought at 5 for Rs. 10 and sold at 6 for Rs. 15. His gain percent is:
(a) 50% (b) 40%
(c) 35% (d) 25%
16. A person sells an article for Rs. 75 and gains as much percent as the cost price of the article in
rupees. The cost price of the article is:
(a) Rs. 37.50 (b) Rs. 40
(c) Rs. 50 (d) Rs. 150
17. By selling a tape-recorder for Rs. 950, I lose 5%. What percent shall I gain by selling it for Rs.
1040?
(a) 4% (b) 4.5%
(c) 5% (d) 9%
18. A trader buys some goods for Rs. 150. If the overhead expenses be 12% of cost price, then at what
price should it be sold to earn 10%?
(a) Rs. 184.80 (b) Rs. 185.80
(c) Rs. 187.80 (d) Rs. 188.80
19. The profit earned after selling an article for Rs. 625 is the same as loss incurred after selling the
article for Rs. 435. The cost price of the article is:
(a) Rs. 520 (b) Rs. 530
(c) Rs. 540 (d) Rs. 550
(e) None of these
20. If a watch is sold at Rs. 60, there is a loss of 15%. For a profit of 2%, the watch is to be sold at:
(a) Rs. 70 (b) Rs. 72
(c) Rs. 75 (d) Rs. 85
21. A sofa set carrying a sale price ticket of Rs. 5000 is sold at a discount of 4%, thereby gaining 20%.
The trader’s cost price of the sofa set is:
(a) Rs. 3600 (b) Rs. 3800
(c) Rs. 4000 (d) Rs. 4200
22. A dealer offers a discount of 10% on the marked price of an article and still makes a profit of 20%.
If its marked price is Rs. 800, then the cost price is:
(a) Rs. 600 (b) Rs. 700
(c) Rs. 800 (d) Rs. 900
23. The marked price of a shirt and trousers are in the ratio 1:2. The shopkeeper gives 40% discount
on the shirt. If the total discount on both is 30%, the discount offered on the trousers is:
(a) 15% (b) 20%
(c) 25% (d) 30%
24. A single discount equivalent to successive discounts of 30%, 20% and 10% is:
(a) 50% (b) 51%
(c) 49.4% (d) 49.6%
25. A shopkeeper offers his customers 10% discount and still makes a profit of 26%. What is the
actual cost of an article for him, marked Rs. 280?
(a) Rs. 175 (b) Rs. 200
(c) Rs. 215 (d) Rs. 225
26. A fan is listed at Rs. 1500 and a discount of 20% is offered on the list price. What additional
discount must be offered to the customer to bring the net price to Rs. 1104?
(a) 8% (b) 10%
(c) 12% (d) 15%
27. By selling a table for Rs. 350 instead of Rs. 400, loss percent increases by 5%. The cost price of
the table is:
(a) Rs. 435 (b) Rs. 417.50
(c) Rs. 1000 (d) Rs. 1050
ANSWERS
1. (a) 2. (d) 3. (c) 4. (d) 5. (b) 6. (b)
7. (c) 8. (c) 9. (d) 10. (e) 11. (a) 12. (b)
13. (d) 14. (c) 15. (d) 16. (c) 17. (a) 18. (a)
19. (b) 20. (b) 21. (c) 22. (a) 23. (c) 24. (d)
25. (b) 26. (a) 27. (c)
8. SIMPLE INTEREST
IMPORTANT FACTS AND FORMULAE
1. Principal: The money borrowed or lent out for a certain period is called the principal or the sum.
2. Interest: Extra money paid for using other’s money is called interest.
3. Simple Interest (S.I.): If the interest on a sum borrowed for a certain period is reckoned uniformly
(i.e. on the same principal throughout the loan period) then it is called simple interest.
Let Principal = P, Rate = R% per annum (p.a.) and Time = T years. Then,
(i) S.I.
P × R × T .
100
(ii)
100 × S.I. 100 × S.I. 100 × S.I.
P ;R and T .
R × T P × T P × R
EXERCISE
Mark (√) against the correct answer in each of the following:
1. How much simple interest will Rs. 2000 earn in 18 months at 6% per annum?
(a) Rs. 120 (b) Rs. 180
(c) Rs. 216 (d) Rs. 240
2. A man deposited Rs. 400 for 2 years, Rs. 550 for 4 years and Rs. 1200 for 6 years. He received Rs.
1020 as the total simple interest. The rate of interest per annum is:
(a) 8% (b) 10%
(c) 15% (d) 20%
3. A sum of money amounts to Rs. 5200 in 5 years and to Rs. 5680 in 7 years at simple interest. The
rate of interest per annum is:
(a) 3% (b) 4%
(c) 5% (d) 6%
4. At which sum the simple interest at the rate of
3
3%
4
per annum will be Rs. 210 in
1
2%
3
years?
(a) Rs. 1580 (b) Rs. 2400
(c) Rs. 2800 (d) None of these
5. A person invests money in three different schemes for 6 years, 10 years and 12 years at 10%, 12%
and 15% simple interest respectively. At the completion of each scheme, he gets the same interest.
The ratio of his investments is:
(a) 2 : 3 : 4 (b) 3 : 4 : 2
(c) 3 : 4 : 6 (d) 6 : 3 : 2
6. Simple interest on Rs. 500 for 4 years at 6.25% per annum is equal to the simple interest on Rs.
400 at 5% per annum for a certain period of time. The period of time is:
(a) 4 years (b) 5 years
(c)
1
6%
4
years (d)
2
8%
3
years
7. A borrows Rs. 800 at the rate of 12% per annum simple interest and B borrows Rs. 910 at the rate
of 10% per annum simple interest. In how many years will their amounts of debts be equal?
(a) 18 years (b) 20 years
(c) 22 years (d) 24 years
8. Rs. 6000 becomes Rs. 7200 in 4 years at a certain rate of interest. If the rate becomes 1.5 times of
itself, the amount of the same principal in 5 years will be:
(a) Rs. 8000 (b) Rs. 8250
(c) Rs. 9000 (d) Rs. 9250
9. If the simple interest for 6 years be equal to 30% of the principal, it will be equal to the principal
after
(a) 10 years (b) 20 years
(c) 22 years (d) 30 years
10. At what rate of simple interest a certain sum will be doubled in 15 years?
(a)
1
5%
2
p.a. (b) 6% p.a.
(c)
2
6%
3
p.a. (d)
1
7%
2
p.a.
11. A certain sum is invested on simple interest. If it trebles in 10 years, what is the rate of interest?
(a) 18% p.a. (b) 20% p.a.
(c) 22% p.a. (d) 25% p.a.
12. In how many years, a sum will be thrice of it at the rate of 10% per annum?
(a) 15 years (b) 20 years
(c) 30 years (d) 40 years
13. A man invests
1
3
of his capital at 7% p.a.,
1
4
at 8% p.a. and the remainder at 10% p.a. If his annual
income is Rs. 561, the capital is:
(a) Rs. 5400 (b) Rs. 6000
(c) Rs. 6600 (d) Rs. 7200
14. The sum of money that will give Re. 1 as simple interest per day at the rate of 5% per annum, is:
(a) Rs. 730 (b) Rs. 3650
(c) Rs. 7300 (d) Rs. 36500
15. A certain sum of money becomes three times of itself in 20 years at simple interest. In how many
years does it become double of itself at the same rate?
(a) 8 years (b) 10 years
(c) 12 years (d) 14 years
16. If the simple interest on a certain sum of money for 15 months at
1
7%
2
p.a. exceeds the simple interest
on the same sum for 8 months at
1
12 %
2
p.a. by Rs. 32.50. The sum is:
(a) Rs. 312 (b) Rs. 312.50
(c) Rs. 3120 (d) Rs. 3120.50
17. Rs. 2000 amount to Rs. 2600 in 5 years at simple interest. If the interest rate were increased by
3%, it would amount to how much?
(a) Rs. 2900 (b) Rs. 3200
(c) Rs. 3600 (d) None of these
18. If the annual rate of simple interest increases from 10% to
1
12 %,
2
a man’s annual income
increases by Rs. 1250. The principal is:
(a) Rs. 45000 (b) Rs. 50000
(c) Rs. 60000 (d) Rs. 65000
19. The least number of years in which the simple interest on Rs. 2600 at
2
6%
3
simple interest will be
exact number of rupees, is:
(a) 2 years (b) 3 years
(c) 4 years (d) 5 years
20. The simple interest on a sum for 5 years is two-fifth of the sum. The rate percent per annum is:
(a) 10% (b) 8%
(c) 6% (d)
1
12 %
2
ANSWERS
1. (b) 2. (b) 3. (d) 4. (b) 5. (d) 6. (c)
7. (c) 8. (b) 9. (b) 10. (c) 11. (b) 12. (b)
13. (c) 14. (c) 15. (b) 16. (c) 17. (a) 18. (b)
19. (b) 20. (b)
9. COMPOUND INTEREST
IMPORTANT FACTS AND FORMULAE
1. Let Principal = Rs. P, Rate = R% p.a., Time = t years.
(i) When interest is compounded annually:
Amount after t years =
t
R
P 1+ .
100
(ii) When interest is compounded half-yearly:
Principal = Rs. P, Rate = R% p.a. = (R/2)% per half-yearly,
Time = t years = (2t) half-years.
Amount after t years =
2t
R / 2
P 1+ .
100
(iii) When interest is compounded quarterly:
Principal = Rs. P, Rate = R% p.a. = (R/4)% per quarter,
Time = t years = (4t) quarters.
Amount after t years =
4t
(R / 4)
P 1+ .
100
(iv) When time is fraction of a year, say
3
24
years.
Principal = Rs. P, Rate = R% p.a., Time =
3
24
years.
Amount after
3
24
years =
2
R (3 / 4)R
P 1+ × 1+ .
100 100
iv) When Rate of Interest is R1% during first year, R2% during 2nd year, R3% during third year.
Principal = Rs. P.
Amount after3 years =
3
12
R
RR
Rs. P 1+ 1+ 1+ .
100 100 100
2. P.W. of a sum of Rs x due n years hence is given by:
n
x
P.W.= Rs .
R
1+100
EXERCISE
Mark (√) against the correct answer in each of the following:
1. The difference between the simple interest and the compound interest on Rs. 5000 at 10% p.a. for
3 years is:
(a) Rs. 145 (b) Rs. 150
(c) Rs. 165 (d) Rs. 180
(e) None of these
2. The difference between compound interest and simple interest on Rs. 8000 at 5% p.a. for 3 years
is:
(a) Rs. 50 (b) Rs. 60
(c) Rs. 61 (d) Rs. 600
3. A deposited Rs. 6000 in a bank at 5% per annum simple interest. B deposited Rs. 5000 at 8% p.a.
compound interest. After 2 years, the difference between their interests will be:
(a) Rs. 230 (b) Rs. 232
(c) Rs. 600 (d) Rs. 832
4. What will be the compound interest on Rs. 25000 after 3 years at 12% per annum?
(a) Rs. 9000.30 (b) Rs. 9720
(c) Rs. 10123.20 (d) Rs. 10483.20
(e) None of these
5. How much would a sum of Rs. 16000 approximately amount to in 2 years at 10% p.a.
compounded half-yearly?
(a) Rs. 17423 (b) Rs. 18973
(c) Rs. 19448 (d) Rs. 19880
6. The compound interest on Rs. 16000 for 9 months at 20% p.a. compounded quarterly, is:
(a) Rs. 2518 (b) Rs. 2520
(c) Rs. 2522 (d) Rs. 2524
7. If the interest is payable annually, then the principal on which the compound interest for 3 years at
10% p.a. is Rs. 331, is given by:
(a) Rs. 900 (b) Rs. 1000
(c) Rs. 1050 (d) Rs. 1100
8. The difference between compound interest and simple interest on a sum for 2 years at 8% p.a. is
Rs. 768. The sum is:
(a) Rs. 100000 (b) Rs. 110000
(c) Rs. 120000 (d) Rs. 170000
9. A sum is invested at compound interest payable annually. The interest in two successive years was
Rs. 500 and Rs. 540. The sum is:
(a) Rs. 3750 (b) Rs. 5000
(c) Rs. 5600 (d) Rs. 6250
10. A sum of money amounts to Rs. 9680 in 2 years and Rs. 10648 in 3 years. The rate of interest per
annum, is
(a) 5% (b) 10%
(c) 15% (d) 20%
11. A sum amounts to Rs. 2916 in 2 years and Rs. 3149.28 in 3 years at compound interest. The sum
is:
(a) Rs. 1500 (b) Rs. 2000
(c) Rs. 2500 (d) Rs. 3000
12. A sum of money placed at compound interest doubles itself in 5 years. In how many years, it
would amount to 8 times of itself at the same rate of interest?
(a) 7 years (b) 10 years
(c) 15 years (d) 20 years
13. A sum of money amounts to Rs. 6690 after 3 years and to Rs. 10035 after 6 years on compound
interest. The sum is:
(a) Rs. 4400 (b) Rs. 4445
(c) Rs. 4460 (d) Rs. 4520
14. If the rate of interest be 4% per annum for first year, 5% per annum for second year and 6% per
annum for third year, then the compound interest of Rs. 10000 for 3 years will be:
(a) Rs. 1575.20 (b) Rs. 1600
(c) Rs. 1625.80 (d) Rs. 2000
15. The effective annual rate of interest, corresponding to a nominal rate of 6% per annum, payable
half-yearly, is:
(a) 6.06% (b) 6.07%
(c) 6.08% (d) 6.09%
16. In how many years will a sum of Rs. 800 at 10% p.a. compounded semi-annually become Rs.
926.10?
(a)
1
13
years (b)
1
12
years
(c)
1
23
years (d)
1
22
years
17. At what rate of interest will Rs. 20000 become Rs. 24200 after 2 years when interest is
compounded annually?
(a) 5% (b) 6%
(c) 10% (d) 15%
18. On a sum of money the difference between simple interest and compound interest for 2 years is Rs.
160 and the simple interest for 2 years is Rs. 2880. The rate percent per annum is:
(a)
5
5%
9
(b)
1
11 %
9
(c)
1
12 %
2
(d) 9%
19. The value of a machine depreciates every year at the rate of 10% on its value at the beginning of that
year. If the present value of the machine is Rs. 7290, its worth 3 years ago was:
(a) Rs. 9471 (b) Rs. 8000
(c) Rs. 10000 (d) Rs. 7508.70
20. A tree increases annually by
1
8
of its height. By how much will it increase after 2 years, if it stands
today 64 cm high?
(a) 72 cm (b) 74 cm
(c) 75 cm (d) 81 cm
21. The difference between compound interest and simple interest at the same rate on Rs. 5000 for 2
years is Rs. 72. The rate of interest per annum is:
(a) 6% (b) 8%
(c) 10% (d) 12%
22. The sum on which the compound interest for second year at 10% p.a. is Rs. 132, is given by:
(a) Rs. 1000 (b) Rs. 1200
(c) Rs. 1320 (d) None of these
23. The present worth of Rs. 1690 due 2 years hence at 4% p.a. is:
(a) Rs. 1505 (b) Rs. 1547.50
(c) Rs. 1562.50 (d) Rs. 1580
ANSWERS
1. (e) 2. (c) 3. (b) 4. (c) 5. (c) 6. (c)
7. (b) 8. (c) 9. (d) 10. (b) 11. (c) 12. (c)
13. (c) 14. (a) 15. (d) 16. (b) 17. (c) 18. (b)
19. (c) 20. (d) 21. (d) 22. (b) 23. (c)
MENSURATION PRACTICE SET-1
PERIMETER
Mark (√) against the correct answer in each of the following:
Directions (Questions 1-6): Find the perimeter of each of the following figures :
1.
(a) 10 cm (b) 11 cm
(c) 12 cm (d) 13 cm
2.
(a) 130 cm (b) 131 cm
(c) 132 cm (d) 133 cm
3.
(a) 40 cm (b) 50 cm
(c) 60 cm (d) 70 cm
4.
(a) 15 cm (b) 17 cm
(c) 18 cm (d) 20 cm
5.
(a) 10 cm (b) 15 cm
(c) 20 cm (d) 25 cm
6.
(a) 50 cm (b) 51 cm
(c) 52 cm (d) 53 cm
7. A rectangular piece of land measures 0.7 km by 0.5 km. Each side is to be fenced with 4 rows of
wires. What is the length of the wire needed?
(a) 9.6 km (b) 10.7 km
(c) 12 km (d) 13.6 km
Directions (Questions 8-11): What is the perimeter of each of the following figures?
8.
(a) 100 cm (b) 101 cm
(c) 102 cm (d) 103 cm
9.
(a) 98 cm (b) 100 cm
(c) 101 cm (d) 103 cm
10.
(a) 90 cm (b) 100 cm
(c) 110 cm (d) 120 cm
11.
(a) 70 cm (b) 80 cm
(c) 90 cm (d) 100 cm
Directions (Questions 12-15): Read the detail given below and answer the questions that follow by
selecting the most appropriate option.
Avneet buys 9 square paying slabs, each with a side of
1 m.
2
He lays them in the form of a square.
12. What is the perimeter of his arrangement [Fig. (i)]?
(a) 2 m (b) 4 m
(c) 6 m (d) 8 m
13. Shari does not like his arrangement. She gets him to lay them out like a cross. What is the
perimeter of her arrangement [Fig. (ii)]?
(a) 10 m (b) 20 m
(c) 30 m (d) 40 m
14. Which has greater perimeter?
(a) Cross has greater perimeter (b) Square has greater perimeter
(c) Star has greater perimeter (d) Circle has greater perimeter
15. Avneet wonders if there is a way of getting an even greater perimeter. Can you find a way of doing
this? (The paving slabs must meet along complete edges i.e. they cannot be broken.)
(a) Yes, we can rearrange the slabs to get the perimeter of the arrangement more than 10 cm
(b) No, we cannot rearrange the slabs to get the perimeter of the arrangement more than 10 cm
(c) Cannot be determined
(d) None of these
AREA
Directions (Questions 16-29): Find the areas of the following figures by counting square:
16.
(a) 8 sq units (b) 9 sq units
(c) 10 sq units (d) 11 sq units
17.
(a) 5 sq units (b) 6 sq units
(c) 8 sq units (d) 9 sq units
18.
(a) 4 sq units (b) 5 sq units
(c) 6 sq units (d) 7 sq units
19.
(a) 6 sq units (b) 7 sq units
(c) 8 sq units (d) 9 sq units
20.
(a) 8 sq units (b) 9 sq units
(c) 10 sq units (d) 11 sq units
21.
(a) 2 sq units (b) 4 sq units
(c) 6 sq units (d) 8 sq units
22.
(a) 2 sq units (b) 4 sq units
(c) 6 sq units (d) 8 sq units
23.
(a) 4 sq units (b) 5 sq units
(c) 6 sq units (d) 7 sq units
24.
(a) 6 sq units (b) 7 sq units
(c) 8 sq units (d) 9 sq units
25.
(a) 2 sq units (b) 4 sq units
(c) 6 sq units (d) 8 sq units
26.
(a) 2 sq units (b) 3 sq units
(c) 4 sq units (d) 5 sq units
27.
(a) 6 sq units (b) 7 sq units
(c) 8 sq units (d) 9 sq units
28.
(a) 12 sq units (b) 14 sq units
(c) 16 sq units (d) 18 sq units
29.
(a) 12 sq units (b) 14 sq units
(c) 16 sq units (d) 18 sq units
30. The area of a rectangular garden 50 m long is 300 sq m. Find the width of the garden.
(a) 6 m (b) 7 m
(c) 8 m (d) 9 m
31. A table-top measures 2 m by 1 m 50 cm. What is its area in square metres?
(a) 2 sq m (b) 3 sq m
(c) 4 sq m (d) 5 sq m
32. A floor is 5 m long and 4 m wide. A square carpet of sides 3 m is laid on the floor. Find the area of
the floor that is not carpeted.
(a) 10 sq m (b) 11 sq m
(c) 12 sq m (d) 13 sq m
33. Five square flower beds each of sides 1 m are dug on a piece of land 5 m long and 4 m wide. What
is the area of the remaining part of the land?
(a) 13 sq m (b) 14 sq m
(c) 15 sq m (d) 16 sq m
Directions (Questions 34-35): By splitting the following figures into rectangles, find their areas (The
measures are given in centimetres).
34.
(a) 28 sq cm (b) 30 sq cm
(c) 32 sq cm (d) 34 sq cm
35.
(a) 8 sq cm (b) 9 sq cm
(c) 10 sq cm (d) 11 sq cm
Directions (Questions 36-38): Split the following shapes into rectangles and find their areas. (The
measures are given in centimetres).
36.
(a) 10 sq cm (b) 20 sq cm
(c) 30 sq cm (d) 40 sq cm
37.
(a) 240 sq cm (b) 245 sq cm
(c) 250 sq cm (d) 255 sq cm
38.
(a) 2 sq cm (b) 4 sq cm
(c) 7 sq cm (d) 9 sq cm
39. How many tiles whose length and breadth are 12 cm and 5 cm respectively will be needed to fit in
a rectangular region whose length and breadth is respectively : 100 cm and 144 cm?
(a) 240 tiles (b) 242 tiles
(c) 244 tiles (d) 246 tiles
40. How many tiles whose length and breadth are 12 cm and 5 cm respectively will be needed to fit in
a rectangular region whose length and breadth is respectively : 70 cm and 36 cm?
(a) 40 tiles (b) 42 tiles
(c) 44 tiles (d) 46 tiles
ANSWERS
1. (c) 2. (d) 3. (c) 4. (d) 5. (b) 6. (c)
7. (a) 8. (a) 9. (b) 10. (b) 11. (d) 12. (c)
13. (a) 14. (a) 15. (b) 16. (b) 17. (a) 18. (a)
19. (c) 20. (c) 21. (b) 22. (c) 23. (b) 24. (d)
25. (b) 26. (d) 27. (c) 28. (b) 29. (d) 30. (a)
31. (b) 32. (b) 33. (c) 34. (a) 35. (b) 36. (d)
37. (b) 38. (d) 39. (a) 40. (b)
PERIMETER AND AREA PRACTICE SET-2
SQUARES AND RECTANGLES
Mark (√) against the correct answer in each of the following:
1. Find the breadth of a rectangular plot of land, if its area is 440 m2 and the length is 22 m. Also find
its perimeter.
(a) 10 m (b) 20 m
(c) 30 m (d) 40 m
2. The perimeter of a rectangular sheet is 100 cm. If the length is 35 cm, find its breadth. Also find
the area.
(a) 15 cm; 525 cm2 (b) 15 cm; 425 cm2
(c) 16 cm; 525 cm2 (d) 16 cm; 425 cm2
3. The area of a square park is the same as of a rectangular park. If the side of the square park is 60 m
and the length of the rectangular park is 90 m, find the breadth of the rectangular park.
(a) 40 cm (b) 50 cm
(c) 60 cm (d) 70 cm
4. A wire is in the shape of a rectangle. Its length is 40 cm and breadth is 22 cm. If the same wire is
rebent in the shape of a square, what will be the measure of each side. Also find which shape
encloses more area?
(a) 30 cm; Square (b) 31 cm; Square
(c) 30 cm; Rectangle (d) 31 cm; Rectangle
AREA OF A PARALLELOGRAM AND TRIANGLE
Directions (Questions 5-9): Find the area of each of the following parallelograms :
5.
(a) 26 cm2 (b) 27 cm2
(c) 28 cm2 (d) 29 cm2
6.
(a) 15 cm2 (b) 16 cm2
(c) 17 cm2 (d) 18 cm2
7.
(a) 8 cm2 (b) 8.75 cm2
(c) 9 cm2 (d) 9.75 cm2
8.
(a) 20 cm2 (b) 21 cm2
(c) 23 cm2 (d) 24 cm2
9.
(a) 8 cm2 (b) 8.8 cm2
(c) 9 cm2 (d) 9.8 cm2
Directions (Questions 10-13): Find the missing values:
10.
Base
Height
Area of the Parallelogram
20 cm
246 cm2
(a) 10.3 cm (b) 11.3 cm
(c) 12.3 cm (d) 13.3 cm
11.
Base
Height
Area of the Parallelogram
15 cm
154.5 cm2
(a) 10.3 cm (b) 11.3 cm
(c) 12.3 cm (d) 13.3 cm
12.
Base
Height
Area of the Parallelogram
8.4 cm
48.72 cm2
(a) 5.3 cm (b) 5.8 cm
(c) 6.3 cm (d) 6.8 cm
13.
Base
Height
Area of the Parallelogram
15.6 cm
16.38 cm2
(a) 0.03 cm (b) 1.05 cm
(c) 2.05 cm (d) 3.06 cm
Directions (Questions 14-15): PQRS is a parallelogram (Fig.). QM is the height from Q to SR and QN is
the height from Q to PS. If SR = 12 cm and QM = 7.6 cm. Find:
14. the area of the parallelogram PQRS
(a) 88.2 cm2 (b) 89.2 cm2
(c) 90.2 cm2 (d) 91.2 cm2
15. QN, if PS = 8 cm
(a) 11.4 cm (b) 12.4 cm
(c) 13.4 cm (d) 14.4 cm
16. DL and BM are the heights on sides AB and AD respectively of parallelogram ABCD (Fig.). If the
area of the parallelogram is 1470 cm2, AB = 35 cm and AD = 49 cm, find the length of BM and
DL.
(a) length of BM = 20 cm; length of DL = 30 cm
(b) length of BM = 30 cm; length of DL = 42 cm
(c) length of BM = 20 cm; length of DL = 42 cm
(d) length of BM = 30 cm; length of DL = 30 cm
Directions (Questions 17-20): Find the area of each of the following triangles:
17.
(a) 4 cm2 (b) 5 cm2
(c) 6 cm2 (d) 7 cm2
18.
(a) 5 cm2 (b) 6 cm2
(c) 7 cm2 (d) 8 cm2
19.
(a) 4 cm2 (b) 5 cm2
(c) 6 cm2 (d) 7 cm2
20.
(a) 1 cm2 (b) 2 cm2
(c) 3 cm2 (d) 4 cm2
Directions (Questions 21-23): Find the missing values:
21.
Base
Height
Area of Triangle
15 cm
—
87 cm2
(a) 11.6 cm (b) 12.6 cm
(c) 13.6 cm (d) 14.6 cm
22.
Base
Height
Area of Triangle
—
31.4 mm
1256 mm2
(a) 60 cm (b) 70 cm
(c) 80 cm (d) 90 cm
23.
Base
Height
Area of Triangle
22 cm
—
170.5 cm2
(a) 15 cm (b) 15.5 cm
(c) 16 cm (d) 16.5 cm
24.
ABC
is right angled at A (Fig.). AD is perpendicular to BC. If AB = 5 cm, BC = 13 cm and AC =
12 cm, Find the area of
ABC
. Also find the length of AD.
(a) Area of
ABC
= 30 cm2; length of AD
60
13
cm
(b) Area of
ABC
= 40 cm2; length of AD
60
13
cm
(c) Area of
ABC
= 30 cm2; length of AD
50
12
cm
(d) Area of
ABC
= 40 cm2; length of AD
50
12
cm
25.
ABC
is isosceles with AB = AC = 7.5 cm and BC = 9 cm (Fig.). The height AD from A to BC,
is 6 cm. Find the area of
ABC
. What will be the height from C to AB i.e., CE?
(a) Area of
ABC
= 28 cm2; length of CE = 7.2 cm
(b) Area of
ABC
= 27 cm2; length of CE = 7.2 cm
(c) Area of
ABC
= 27 cm2; length of CE = 8.2 cm
(d) Area of
ABC
= 28 cm2; length of CE = 8.2 cm
CIRCLES
Directions (Questions 26-28): Find the circumference of the circles with the following radius:
22
Take 7
26. 14 cm
(a) 80 cm (b) 88 cm
(c) 90 cm (d) 98 cm
27. 28 mm
(a) 172 mm (b) 174 mm
(c) 176 mm (d) 178 mm
28. 21 cm
(a) 130 cm (b) 132 cm
(c) 134 cm (d) 136 cm
29. A gardener wants to fence a circular garden of diameter 21 m. Find the length of the rope he needs
to purchase, if he makes 2 rounds of fence. Also find the costs of the rope, if it cost Rs. 4 per
meter.
22
Take 7
(a) 132 m; Rs. 528 (b) 142 m; Rs. 528
(c) 132 m; Rs. 628 (d) 142 m; Rs. 628
30. Find the cost of polishing a circular table-top of diameter 1.6 m, if the rate of polishing is Rs.
15/m2.
(Take 3.14)
(a) Rs. 20.14 (approx.) (b) Rs. 30.14 (approx.)
(c) Rs. 40.14 (approx.) (d) Rs. 50.14 (approx.)
31. From a circular card sheet of radius 14 cm, two circles of radius 3.5 cm and a rectangle of length 3
cm and breadth 1 cm are removed. (as shown in the adjoining figure). Find the area of the
remaining sheet.
22
Take 7
(a) 536 cm2 (b) 636 cm2
(c) 746 cm2 (d) 846 cm2
32. A circular flower garden has an area of 314 m2. A sprinkler at the centre of the garden can cover
an area that has a radius of 12 m. Will the sprinkler water the entire garden?
(Take 3.14)
(a) Yes (b) No
(c) Cannot be determined (d) None of these
33. Find the circumference of the inner and the outer circles, shown in the adjoining figure?
(Take 3.14)
(a) 118.32 m; 46.52 m (b) 118.32 m; 56.52 m
(c) 119.32 m; 46.52 m (d) 119.32 m; 56.52 m
CONVERSION OF UNITS
34. A garden is 90 m long and 75 m broad. A path 5 m wide is to be built outside and around it. Find
the area of the path. Also find the area of the garden in hectare.
(a) 1740 m2; 0.675 ha (b) 1750 m2; 0.675 ha
(c) 1840 m2; 0.575 ha (d) 1850 m2; 0.575 ha
35. A 3 m wide path runs outside and around a rectangular park of length 125 m and breadth 65 m.
Find the area of the path.
(a) 1176 m2 (b) 1276 m2
(c) 1376 m2 (d) 1476 m2
Directions (Questions 36-37): A path 1 m wide is built along the border and inside a square garden of
side 30 m. Find:
36. the area of the path
(a) 114 m2 (b) 115 m2
(c) 116 m2 (d) 117 m2
37. the cost of planting grass in the remaining portion of the garden at the rate of Rs. 40 per m2.
(a) Rs. 31,360 (b) Rs. 32,260
(c) Rs. 33,360 (d) Rs. 34,260
38. Pragya wrapped a cord around a circular pipe of radius 4 cm (adjoining figure) and cut off the
length required of the cord. Then she wrapped it around a square box of side 4 cm (also shown).
Did she have any cord left?
Take 3.14
(a) Yes, 9.12 m cord is left (b) Yes, 10.12 m cord is left
(c) No (d) Cannot be determined
Directions (Questions 39-42): The adjoining figure represents a rectangular lawn with a circular flower
bed in the middle. Find:
39. the area of whole land
(a) 40 m2 (b) 50 m2
(c) 60 m2 (d) 70 m2
40. the area of the flower bed
(a) 10.56 m2 (b) 11.56 m2
(c) 12.56 m2 (d) 13.56 m2
41. the area of the lawn excluding the area of the flower bed
(a) 32.44 m2 (b) 34.44 m2
(c) 35.44 m2 (d) 37.44 m2
42. the circumference of the flower bed
(a) 10.56 m (b) 11.56 m
(c) 12.56 m (d) 13.56 m
Directions (Questions 43-44): In the following figures, find the area of the shaded portions:
43.
(a) 100 cm2 (b) 110 cm2
(c) 120 cm2 (d) 130 cm2
44.
(a) 150 cm2 (b) 160 cm2
(c) 170 cm2 (d) 180 cm2
45. Find the area of the quadrilateral ABCD.
Here, AC = 22 cm, BM = 3 cm,
DN = 3 cm, and
BM AC, DN AC
(a) 64 cm2 (b) 65 cm2
(c) 66 cm2 (d) 67 cm2
ANSWERS
1. (b) 2. (a) 3. (a) 4. (b) 5. (c) 6. (a)
7. (b) 8. (d) 9. (b) 10. (c) 11. (a) 12. (b)
13. (b) 14. (d) 15. (a) 16. (b) 17. (c) 18. (d)
19. (c) 20. (c) 21. (a) 22. (c) 23. (b) 24. (a)
25. (b) 26. (b) 27. (c) 28. (b) 29. (a) 30. (b)
31. (a) 32. (a) 33. (d) 34. (b) 35. (a) 36. (c)
37. (a) 38. (a) 39. (b) 40. (c) 41. (d) 42. (c)
43. (b) 44. (a) 45. (c)
MENSURATION PRACTICE SET-3
RECTANGLES
Mark (√) against the correct answer in each of the following:
1. A square and a rectangular field with measurements as given in the figure have the same
perimeter. Which field has a larger area?
(A) (B)
(a) Fig. (A) (b) Fig. (B)
(c) Both (d) Cannot be determined
2. Mrs. Kaushik has a square plot with the measurement as shown in the figure. She wants to
construct a house in the middle of the plot. A garden is developed around the house. Find the total
cost of developing a garden around the house at the rate of Rs. 55 per m2.
(a) Rs. 15,875 (b) Rs. 16,875
(c) Rs. 17,875 (d) Rs. 18,875
3. The shape of a garden is rectangular in the middle and semi circular at the ends as shown in the
diagram. Find the area and the perimeter of this garden [Length of rectangle is 20 – (3.5 + 3.5)
metres].
(a) Area = 109.5 m2; Perimeter = 48 m
(b) Area = 129.5 m2; Perimeter = 38 m
(c) Area = 109.5 m2; Perimeter = 38 m
(d) Area = 129.5 m2; Perimeter = 48 m
4. A flooring tile has the shape of a parallelogram whose base is 24 cm and the corresponding height
is 10 cm. How many such tiles are required to cover a floor of area 1080 m2? (If required you can
split the tiles in whatever way you want to fill up the corners).
(a) 25000 tiles (b) 45000 tiles
(c) 65000 tiles (d) 85000 tiles
5. An ant is moving around a few food pieces of different shapes scattered
on the floor. For which food-piece would the ant have to take a longer round?
Remember, circumference of a circle can be obtained by using the
expression
2,cr
where r is the radius of the circle.
(A) (B) (C)
(a) Fig. (A) (b) Fig. (B)
(c) Fig. (C) (d) None of these
AREA OF A TRAPEZIUM, GENERAL QUADRILATERAL AND POLYGON
6. The shape of the top surface of a table is a trapezium. Find its area if its parallel sides are 1 m and
1.2 m and perpendicular distance between them is 0.8 m.
(a) 0.88 m2 (b) 0.98 m2
(c) 0.108 m2 (d) 0.118 m2
7. The area of a trapezium is 34 cm2 and the length of one of the parallel sides is 10 cm and its height
is 4 cm. Find the length of the other parallel side.
(a) 5 cm (b) 6 cm
(c) 7 cm (d) 8 cm
8. The diagonal of a quadrilateral shaped field is 24 m and the perpendiculars dropped on it from the
remaining opposite vertices are 8 m and 13 m. Find the area of the field.
(a) 250 m2 (b) 252 m2
(c) 254 m2 (d) 256 m2
9. Mohan wants to buy a trapezium shaped field. Its side along the river is parallel to and twice the
side along the road. If the area of this field is
10500 m2 and the perpendicular distance between the two parallel sides is
100 m, find the length of the side along the river.
(a) 110 m (b) 120 m
(c) 130 m (d) 140 m
10. Top surface of a raised platform is in the shape of a regular octagon as shown in the figure. Find
the area of the octagonal surface.
(a) 109 m2 (b) 119 m2
(c) 129 m2 (d) 139 m2
11. There is a pentagonal shaped park as shown in the figure. For finding its area Jyoti and Kavita
divided it in two different ways.
Find the area of this park using both ways. Can you suggest some other way of finding its area?
(a) Area using Jyoti’s way
22
1 15
2 (30 15)m 337.5m ,
22
Area using Kavita’s way
2
115 15 15 15 337.5m
2
(b) Area using Jyoti’s way
2
115 15 15 15 337.5m
2
Area using Kavita’s way
2
115 15 15 15 337.5m
2
(c) Cannot be determined
(d) None of these
12. Diagram of the adjacent picture frame has outer dimensions = 24 cm × 28 cm and inner
dimensions 16 cm × 20 cm. Find the area of each section of the frame, if the width of each section
is same.
(a) 70 cm2, 86 cm2, 70 cm2, 86 cm2
(b) 80 cm2, 96 cm2, 80 cm2, 96 cm2
(c) 90 cm2, 106 cm2, 90 cm2, 106 cm2
(d) 100 cm2, 96 cm2, 100 cm2, 96 cm2
SOLID SHAPES
13. There are two cuboidal boxes as shown in the adjoining figure. Which box requires the lesser
amount of material to make?
(A) (B)
(a) Fig. (A) (b) Fig. (B)
(c) Both (d) Cannot be determined
14. A suitcase with measures 80 cm × 48 cm × 24 cm is to be covered with a tarpaulin cloth. How
many metres of tarpaulin of width 96 cm is required to cover 100 such suitcases?
(a) 124 m (b) 144 m
(c) 164 m (d) 184 m
15. Find the side of a cube whose surface area is 600 cm2.
(a) 2 cm (b) 6 cm
(c) 8 cm (d) 10 cm
16. Rukhsar painted the outside of the cabinet of measure 1 m × 2 m × 1.5 m. How much surface area
did she cover if she painted all except the bottom of the cabinet.
(a) 11 m2 (b) 12 m2
(c) 13 m2 (d) 14 m2
17. Daniel is painting the walls and ceiling of a cuboidal hall with length, breadth and height of 15 m,
10 m and 7 m respectively. From each can of paint 100 m2 of area is painted. How many cans of
paint will she need to paint the room?
(a) 2 cans (b) 3 cans
(c) 4 cans (d) 5 cans
18. Describe how the two figures at the right are alike and how they are different. Which box has
larger lateral surface area?
(a) Similarity—Both have same heights.
Difference—One is cylinder, the other is cube.
The cube has larger lateral surface area
(b) Similarity—Both are cylinders.
Difference—No difference
The cylinder has larger lateral surface area
(c) Cannot be determined
(d) None of these
19. A road roller takes 750 complete revolutions to move once over to level a road. Find the area of
the road if the diameter of a road roller is 84 cm and length is 1 m.
(a) 1880 m2 (b) 1980 m2
(c) 2080 m2 (d) 2180 m2
20. A company packages its milk powder in cylindrical container whose base has a diameter of 14 cm
and height 20 cm. Company places a label around the surface of the container (as shown in the
figure). If the label is placed 2 cm from top and bottom, what is the area of the label.
(a) 701 cm2 (b) 702 cm2
(c) 703 cm2 (d) 704 cm2
VOLUME AND CAPACITY
Directions (Questions 21-23): Given a cylindrical tank, in which situation will you find surface area and
in which situation volume.
21. To find how much it can hold.
(a) Surface area (b) Volume
(c) Cannot be determined (d) None of these
22. Number of cement bags required to plaster it.
(a) Surface area (b) Volume
(c) Cannot be determined (d) None of these
23. To find the number of smaller tanks that can be filled with water from it.
(a) Surface area (b) Volume
(c) Cannot be determined (d) None of these
24. Diameter of cylinder A is 7 cm, and the height is 14 cm. Diameter of cylinder B is 14 cm and
height is 7 cm. Without doing any calculations can you suggest whose volume is greater? Verify it
by finding the volume of both the cylinders. Check whether the cylinder with greater volume also
has greater surface area?
(a) Volume of cylinder B is greater; Surface area of cylinder B is greater.
(b) Volume of cylinder A is greater; Surface area of cylinder A is greater.
(c) Volume of cylinder A is greater; Surface area of cylinder B is greater.
(d) Volume of cylinder B is greater; Surface area of cylinder A is greater.
25. A milk tank is in the form of cylinder whose radius is 1.5 m and length is 7 m. Find the quantity of
milk in litres that can be stored in the tank?
(a) 39500 L (b) 49500 L
(c) 59500 L (d) 69500 L
Directions (Questions 26-27): If each edge of a cube is doubled,
26. how many times will its surface area increase?
(a) 2 times (b) 4 times
(c) 6 times (d) 8 times
27. how many times will its volume increase?
(a) 2 times (b) 4 times
(c) 6 times (d) 8 times
ANSWERS
1. (a) 2. (c) 3. (d) 4. (b) 5. (b) 6. (a)
7. (c) 8. (b) 9. (d) 10. (b) 11. (a) 12. (b)
13. (a) 14. (b) 15. (d) 16. (a) 17. (d) 18. (a)
19. (b) 20. (d) 21. (b) 22. (a) 23. (b) 24. (a)
25. (b) 26. (b) 27. (d)
SURFACE AREAS AND VOLUMES PRACTICE SET-4
SURFACE AREA OF A CUBOID AND A CUBE
Mark (√) against the correct answer in each of the following:
Directions (Questions 1-2): A plastic box 1.5 m long, 1.25 m wide and 65 cm deep is to be made. It is
opened at the top. Ignoring the thickness of the plastic sheet, determine:
1. The area of the sheet required for making the box.
(a) 5.45 m2 (b) 6.45 m2
(c) 7.45 m2 (d) 8.45 m2
2. The cost of sheet for it, if a sheet measuring 1 m2 costs Rs. 20.
(a) Rs. 106 (b) Rs. 107
(c) Rs. 108 (d) Rs. 109
3. The floor of a rectangular hall has a perimeter 250 m. If the cost of painting the four walls at the
rate of Rs. 10 per m2 is Rs. 15000, find the height of the hall.
[Hint: Area of the four walls = Lateral surface area.]
(a) 2 m (b) 4 m
(c) 6 m (d) 8 m
Directions (Questions 4-5): A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm
long, 10 cm wide and 8 cm high.
4. Which box has the greater lateral surface area and by how much?
(a) Lateral surface area of cubical box is greater by 20 cm2
(b) Lateral surface area of cubical box is greater by 30 cm2
(c) Lateral surface area of cubical box is greater by 40 cm2
(d) None of these
5. Which box has the smaller total surface area and by how much?
(a) Total surface area of cuboidal box is greater by 10 cm2
(b) Total surface area of cuboidal box is greater by 20 cm2
(c) Total surface area of cuboidal box is greater by 30 cm2
(d) None of these
Directions (Questions 6-7): A small indoor greenhouse (herbarium) is made entirely of glass panes
(including base) held together with tape. It is 30 cm long, 25 cm wide and 25 cm high.
6. What is the area of the glass?
(a) 1250 cm2 of glass (b) 2250 cm2 of glass
(c) 3250 cm2 of glass (d) 4250 cm2 of glass
7. How much of tape is needed for all the 12 edges?
(a) 320 cm of tape (b) 420 cm of tape
(c) 520 cm of tape (d) 620 cm of tape
8. Parveen wanted to make a temporary shelter for her car, by making a box-like structure with
tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can
be rolled up). Assuming that the stitching margins are very small, and therefore negligible, how
much tarpaulin would be required to make the shelter of height 2.5 m, with base dimensions
4 m × 3 m?
(a) 45 m2 (b) 46 m2
(c) 47 m2 (d) 48 m2
SURFACE AREA OF A RIGHT CIRCULAR CYLINDER
Directions (Questions 9-11): A metal pipe is 77 cm long. The inner diameter of a cross section is 4 cm,
the outer diameter being 4.4 cm (See Fig.). Find its
9. inner curved surface area
(a) 968 cm2 (b) 1068 cm2
(c) 1168 cm2 (d) 1268 cm2
10. outer curved surface area
(a) 964.8 cm2 (b) 1064.8 cm2
(c) 1164.8 cm2 (d) 1264.8 cm2
11. total surface area
(a) 2038.08 cm2 (b) 2138.08 cm2
(c) 2238.08 cm2 (d) 2338.08 cm2
12. A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved
surface of the pillar at the rate of Rs. 12.50 per m2.
(a) Rs. 68.75 (b) Rs. 69.75
(c) Rs. 70.75 (d) Rs. 71.75
Directions (Questions 13-14): The inner diameter of a circular well is 3.5 m. It is
10 m deep. Find
13. its inner curved surface area
(a) 100 m2 (b) 110 m2
(c) 120 m2 (d) 130 m2
14. the cost of plastering this curved surface at the rate of Rs. 40 per m2.
(a) Rs. 4400 (b) Rs. 5400
(c) Rs. 6400 (d) Rs. 7400
Directions (Questions 15-16): Find:
15. the lateral or curved surface area of a closed cylindrical petrol storage tank that is 4.2 m in
diameter and 4.5 m high.
(a) 57.4 m2 (b) 58.4 m2
(c) 59.4 m2 (d) 60.4 m2
16. how much steel was actually used, if
1
12
of the steel actually used was wasted in making the tank.
(a) 92.04 m2 (b) 93.04 m2
(c) 94.04 m2 (d) 95.04 m2
17. In Fig., you see the frame of a lampshade. It is to be covered with a decorative cloth. The frame
has a base diameter of 20 cm and height of 30 cm. A margin of 2.5 cm is to be given for folding it
over the top and bottom of the frame. Find how much cloth is required for covering the lampshade.
(a) 2200 cm2 (b) 2300 cm2
(c) 2400 cm2 (d) 2500 cm2
SURFACE AREA OF A RIGHT CIRCULAR CONE
18. Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface
area.
(a) 160 cm2 (b) 162 cm2
(c) 164 cm2 (d) 165 cm2
Directions (Questions 19-20): Curved surface area of a cone is 308 cm2 and its slant height is 14 cm.
Find
19. radius of the base
(a) 4 cm (b) 5 cm
(c) 6 cm (d) 7 cm
20. total surface area of the cone
(a) 461 cm2 (b) 462 cm2
(c) 463 cm2 (d) 464 cm2
21. What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base
radius 6 m? Assume that the extra length of material that will be required for stitching margins and
wastage in cutting is approximately 20 cm
(Use =3.14).
(a) 61 m (b) 62 m
(c) 63 m (d) 64 m
22. The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost
of white-washing its curved surface at the rate of Rs. 210 per 100 m2.
(a) Rs. 1155 (b) Rs. 1266
(c) Rs. 1300 (d) Rs. 1366
23. A joker’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the
area of the sheet required to make 10 such caps.
(a) 4500 cm2 (b) 5500 cm2
(c) 6500 cm2 (d) 7500 cm2
SURFACE AREA OF A SPHERE
24. The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find
the ratio of surface areas of the balloon in the two cases.
(a) 1 : 2 (b) 1 : 3
(c) 1 : 4 (d) 1 : 5
25. The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio
of their surface areas.
(a) 1 : 12 (b) 1 : 13
(c) 1 : 15 (d) 1 : 16
Directions (Questions 26-28): A right circular cylinder just encloses a sphere of radius r (See Fig.). Find:
26. surface area of the sphere
(a)
2
4r
(b)
2
5r
(c)
2
6r
(d)
2
7r
27. curved surface area of the cylinder
(a)
2
4r
(b)
2
5r
(c)
2
6r
(d)
2
7r
28. ratio of the areas obtained in 26. and 27.
(a) 1 : 2 (b) 1 : 1
(c) 2 : 2 (d) 2 : 1
VOLUME OF A CUBOID
29. A matchbox measures 4 cm × 2.5 cm × 1.5 cm. What will be the volume of a packet containing 12
such boxes?
(a) 80 cm3 (b) 140 cm3
(c) 180 cm3 (d) 280 cm3
30. A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of water can it
hold? (1 m3 = 1000 l)
(a) 132000 litres (b) 134000 litres
(c) 135000 litres (d) 136000 litres
31. A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres
of a liquid?
(a) 4.75 m (b) 5.75 m
(c) 6.75 m (d) 7.75 m
32. The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank, if its length
and depth are respectively 2.5 m and 10 m.
(a) 1 m (b) 2 m
(c) 3 m (d) 4 m
33. A godown measures 40 m × 25 m × 10 m. Find the maximum number of wooden crates each
measuring 1.5 m × 1.25 m × 0.5 m that can be stored in the godown.
(a) 12000 (b) 13000
(c) 14000 (d) 16000
34. A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the
new cube? Also, find the ratio between their surface areas.
(a) 6 cm, 4 : 1 (b) 7 cm, 2 : 1
(c) 8 cm, 4 : 1 (d) 9 cm, 2 : 1
VOLUME OF A CYLINDER
35. The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. How many
litres of water can it hold? (1000 cm3 = 1l)
(a) 32.65 litres (b) 33.65 litres
(c) 34.65 litres (d) 35.65 litres
36. The inner diameter of a cylindrical wooden pipe is 24 cm and its outer diameter is 28 cm. The
length of the pipe is 35 cm. Find the mass of the pipe, if 1 cm3 of wood has a mass of 0.6 g.
(a) 2.432 kg (b) 3.432 kg
(c) 4.432 kg (d) 5.432 kg
37. A soft drink is available in two packs—(i) a tin can with a rectangular base of length 5 cm and
width 4 cm, having a height of 15 cm and (ii) a plastic cylinder with circular base of diameter 7 cm
and height 10 cm. Which container has greater capacity and by how much?
(a) The cylinder has the greater capacity by 85 cm3
(b) The cylinder has the greater capacity by 92 cm3
(c) The cylinder has the greater capacity by 105 cm3
(d) The cylinder has the greater capacity by 125 cm3
Directions (Questions 38-39): If the lateral surface of a cylinder is 94.2 cm2 and its height is 5 cm, then
find
38. radius of its base
(a) 1 cm (b) 2 cm
(c) 3 cm (d) 4 cm
39. its volume
(Use =3.14)
(a) 140.3 cm3 (b) 141.3 cm3
(c) 142.3 cm3 (d) 143.3 cm3
40. The capacity of a closed cylindrical vessel of height 1 m is 15.4 litres. How many square metres of
metal sheet would be needed to make it?
(a) 0.4708 m2 (b) 1.4 m2
(c) 2.2 m2 (d) 0.3708 m2
41. A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior.
The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of the
pencil is 14 cm, find the volume of the wood and that of the graphite.
(a) Volume of wood = 4.28 cm3, Volume of graphite = 0.1 cm3
(b) Volume of wood = 5.28 cm3, Volume of graphite = 0.11 cm3
(c) Volume of wood = 6.28 cm3, Volume of graphite = 0.2 cm3
(d) Volume of wood = 7.28 cm3, Volume of graphite = 0.1 cm3
VOLUME OF A RIGHT CIRCULAR CONE
Directions (Questions 42-43): Find the volume of the right circular cone with
42. radius 6 cm, height 7 cm
(a) 264 cm3 (b) 364 cm3
(c) 464 cm3 (d) 564 cm3
43. radius 3.5 cm, height 12 cm
(a) 152 cm3 (b) 153 cm3
(c) 154 cm3 (d) 155 cm3
Directions (Questions 44-45): Find the capacity in litres of a conical vessel with
44. radius 7 cm, slant height 25 cm
(a) 1.232 l (b) 1.432 l
(c) 1.562 l (d) 1.622 l
45. height 12 cm, slant height 13 cm
(a)
10
35l
(b)
11
35l
(c)
9
34 l
(d)
12
35l
46. If the volume of a right circular cone of height 9 cm is
3
48 cm ,
find the diameter of its base.
(a) 6 cm (b) 7 cm
(c) 8 cm (d) 9 cm
Directions (Questions 47-49): The volume of a right circular cone is 9856 cm3. If the diameter of the
base is 28 cm, find
47. height of the cone
(a) 46 cm (b) 47 cm
(c) 48 cm (d) 49 cm
48. slant height of the cone
(a) 48 cm (b) 50 cm
(c) 52 cm (d) 54 cm
49. curved surface area of the cone
(a) 2200 cm2 (b) 3300 cm2
(c) 4400 cm2 (d) 5500 cm2
50. A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the
volume of the solid so obtained.
(a)
3
100 cm
(b)
3
200 cm
(c)
3
300 cm
(d)
3
400 cm
51. A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its
volume. The heap is to be covered by can as to protect it from rain. Find the area of the canvas
required.
(a) 76.625 m3, 89.825 m2 (b) 86.625 m3, 99.825 m2
(c) 96.625 m3, 109.825 m2 (d) 106.625 m3, 109.825 m2
VOLUME OF A SPHERE
52. Find the volume of a sphere of radius 11.2 cm.
(a) 5887.32 cm3 (b) 6887.32 cm3
(c) 7887.32 cm3 (d) 8887.32 cm3
53. A shot-putt is a metallic sphere of radius 4.9 cm. If the density of the metal is 7.8 g per cm3, find
the mass of the shot-putt.
(a) 3.75 kg (nearly) (b) 3.85 kg (nearly)
(c) 4.75 kg (nearly) (d) 4.85 kg (nearly)
54. A hemispherical bowl has a radius of 3.5 cm. What would be the volume of water it would
contain?
(a) 69.8 cm3 (b) 70.8 cm3
(c) 80.8 cm3 (d) 89.8 cm3
Directions (Questions 55-56): Find the volume of a sphere whose radius is
55. 7 cm
(a) 1437
1
3
cm3 (b) 1637
1
3
cm3
(c) 1837
1
3
cm3 (d) 1937
1
3
cm3
56. 0.63 m
(a) 0.05 m3 (approx.) (b) 1.05 m3 (approx.)
(c) 2.05 m3 (approx.) (d) 3.05 m3 (approx.)
Directions (Questions 57-58): Find the amount of water displaced by a solid spherical ball of diameter
57. 28 cm
(a) 11498
2
3
cm3 (b) 12498
2
3
cm3
(c) 13498
2
3
cm3 (d) 14498
2
3
cm3
58. 0.21 m
(a) 0.002751 m3 (b) 0.003851 m3
(c) 0.004251 m3 (d) 0.004851 m3
59. The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction
of the volume of the earth is the volume of the moon?
(a)
1
24
(b)
1
34
(c)
1
64
(d)
1
84
60. Find the volume of a sphere whose surface area is 154 cm2.
(a)
2
179 3
cm3 (b)
2
279 3
cm3
(c) 254 cm3 (d) 256 cm3
Directions (Questions 61-62): Twenty seven solid iron spheres, each of radius r and surface area S are
melted to form a sphere with surface area
S.
Find the
61. radius
r
of the new sphere
(a) 1 r (b) 2 r
(c) 3 r (d) 4 r
62. ratio of S and
S.
(a) 1 : 7 (b) 1 : 8
(c) 1 : 9 (d) 1 : 10
ANSWERS
1. (a) 2. (d) 3. (c) 4. (c) 5. (a) 6. (d)
7. (a) 8. (c) 9. (a) 10. (b) 11. (a) 12. (a)
13. (b) 14. (a) 15. (c) 16. (d) 17. (a) 18. (d)
19. (d) 20. (b) 21. (c) 22. (a) 23. (b) 24. (c)
25. (d) 26. (a) 27. (a) 28. (b) 29. (c) 30. (c)
31. (a) 32. (b) 33. (d) 34. (a) 35. (c) 36. (b)
37. (a) 38. (c) 39. (b) 40. (a) 41. (b) 42. (a)
43. (c) 44. (a) 45. (b) 46. (c) 47. (c) 48. (b)
49. (a) 50. (a) 51. (b) 52. (a) 53. (b) 54. (d)
55. (a) 56. (b) 57. (a) 58. (d) 59. (c) 60. (a)
61. (c) 62. (c)
AREAS RELATED TO CIRCLES PRACTICE SET-5
PERIMETER AND AREA OF A CIRCLE
Mark (√) against the correct answer in each of the following:
1. The cost of fencing a circular field at the rate of Rs. 24 per metre is Rs. 5280. The field is to be
ploughed at the rate of Rs. 0.50 per m2. Find the cost of ploughing the field
22
Take = .
7
(a) Rs. 1925 (b) Rs. 2026
(c) Rs. 2927 (d) Rs. 3925
2. The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has
circumference equal to the sum of the circumferences of the two circles.
(a) 27 cm (b) 28 cm
(c) 29 cm (d) 30 cm
3. The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area
equal to the sum of the areas of the two circles.
(a) 7 cm (b) 8 cm
(c) 9 cm (d) 10 cm
4. The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel
make in 10 minutes when the car is traveling at a speed of 66 km per hour?
(a) 4175 (b) 4275
(c) 4375 (d) 4475
AREAS OF SECTOR AND SEGMENT OF A CIRCLE
5. Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.
(a)
132
7
cm2 (b)
133
8
cm2
(c)
134
9
cm2 (d)
135
10
cm2
6. Find the area of a quadrant of a circle whose circumference is 22 cm.
(a)
77
7
cm2 (b)
77
8
cm2
(c)
78
9
cm2 (d)
79
10
cm2
7. A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the
corresponding segment of the circle.
(Use =3.14 and 3 1.73)
(a) 22.32 cm2 (b) 44.44 cm2
(c) 66.22 cm2 (d) 88.44 cm2
Directions (Questions 8-9): A horse is tied to a peg at one corner of a square shaped grass field of side 15
cm by means of a 5 m long rope (see Fig.). Find:
8. the area of that part of the field in which the horse can graze.
(a) 19.625 m2 (b) 29.625 m2
(c) 40.625 m2 (d) 50.625 m2
9. the increase in the grazing area if the rope were 10 m long instead of 5 m.
(Use =3.14)
(a) 55.875 cm2 (b) 56.875 cm2
(c) 57.875 cm2 (d) 58.875 cm2
Directions (Questions 10-11): A brooch is made with silver wire in the form of a circle with diameter 35
mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in
Fig. Find:
10. the total length of the silver wire required.
(a) 285 mm (b) 286 mm
(c) 385 mm (d) 386 mm
11. the area of each sector of the brooch.
(a)
185
4
mm2 (b)
285
4
mm2
(c)
385
4
mm2 (d)
485
4
mm2
12. An umbrella has 8 ribs which are equally spaced (see Fig.). A summing umbrella to be a flat circle
of radius 45 cm, find the area between the two consecutive ribs of the umbrella.
(a)
22275
28
cm2 (b)
33275
27
cm2
(c)
44475
27
cm2 (d)
55575
28
cm2
13. A round table cover has six equal designs as shown in Fig. If the radius of
the cover is 28 cm, find the cost of making the designs at the rate of
Rs. 0.35 per cm2.
(Use 3 1.7)
(a) Rs. 160.68 (b) Rs. 161.58
(c) Rs. 162.68 (d) Rs. 163.58
14. Area of a sector of angle p (in degrees) of a circle with radius R is
(a)
2R
180
p
(b)
2
R
180
p
(c)
2R
360
p
(d)
2
2R
720
p
AREAS OF COMBINATIONS OF PLANE FIGURES
15. Find the area of the shaded region in Fig., where ABCD is a square of side
14 cm.
(a) 40 cm2 (b) 41 cm2
(c) 42 cm2 (d) 43 cm2
16. Find the area of the shaded design in Fig., where ABCD is a square of side
10 cm and semicircles are drawn with each side of the square as diameter.
(Use =3.14)
(a) 54 cm2 (b) 55 cm2
(c) 56 cm2 (d) 57 cm2
17. Find the area of the shaded region in Fig., if PQ = 24 cm, PR = 7 cm and O is the centre of the
circle.
(a)
4523
28
cm2 (b)
4823
28
cm2
(c)
5023
28
cm2 (d)
5223
28
cm2
18. Find the area of the shaded region in Fig., if radii of the two concentric circles with centre O are 7
cm and 14 cm respectively and
AOC = 40°.
(a)
134
3
cm2 (b)
144
3
cm2
(c)
154
3
cm2 (d)
164
3
cm2
19. Find the area of the shaded region in Fig., where a circular arc of radius 6 cm has been drawn with
vertex O of an equilateral triangle OAB of side 12 cm as centre.
(a)
660 36 3
7
cm2 (b)
760 36 3
8
cm2
(c)
850 26 3
8
cm2 (d)
950 28 3
9
cm2
20. In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in
the middle as shown in Fig. Find the area of the design.
(a)
11428 568 3
7
cm2 (b)
22528 768 3
7
cm2
(c)
30428 548 3
7
cm2 (d)
42428 548 3
7
cm2
21. In Fig., ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such
that each circle touch externally two of the remaining three circles. Find the area of the shaded
region.
(a) 40 cm2 (b) 41 cm2
(c) 42 cm2 (d) 43 cm2
Directions (Questions 22-23): Fig. depicts a racing track whose left and right ends are semicircular. The
distance between the two inner parallel line segments is 60 m and they are each 106 m long. If the track is
10 m wide, find:
22. the distance around the track along its inner edge
(a)
2804
7
m (b)
2924
7
m
(c)
3224
7
m (d)
4024
7
m
23. the area of the track
(a) 4110 m2 (b) 4320 m2
(c) 4500 m2 (d) 4610 m2
24. The area of an equilateral triangle ABC is 17320.5 cm2. With each vertex of the triangle as centre,
a circle is drawn with radius equal to half the length of the side of the triangle (see Fig.). Find the
area of the shaded region.
(Use = 3.14 and 3 1.73205)
(a) 1620.5 cm2 (b) 2720.5 cm2
(c) 2810.5 cm2 (d) 2900.5 cm2
25. On a square handkerchief, nine circular designs each of radius 7 cm are made (see Fig.). Find the
area of the remaining portion of the handkerchief.
(a) 175 cm2 (b) 276 cm2
(c) 377 cm2 (d) 378 cm2
26. In Fig., a square OABC is inscribed in a quadrant OPBQ. If OA = 20 cm, find the area of the
shaded region.
(Use = 3.14)
(a) 228 cm2 (b) 230 cm2
(c) 331 cm2 (d) 334 cm2
27. In Fig., ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as
diameter. Find the area of the shaded region.
(a) 95 cm2 (b) 96 cm2
(c) 97 cm2 (d) 98 cm2
28. Calculate the area of the designed region in Fig. common between the two quadrants of circles of
radius 8 cm each.
(a)
254
7
cm2 (b)
256
7
cm2
(c)
258
7
cm2 (d)
260
7
cm2
SURFACE AREAS AND VOLUMES
SURFACE AREA OF A COMBINATION OF SOLIDS
29. The decorative block shown in Fig. is made of two solids—a cube and a hemisphere. The base of
the block is a cube with edge 5 cm, and the hemisphere fixed on the top has a diameter of 4.2 cm.
Find the total surface area of the block.
22
Take = 7
(a) 163.86 cm2 (b) 170 cm2
(c) 180.46 cm2 (d) 280 cm2
30. Mayank made a bird-bath for his garden in the shape of a cylinder with a hemispherical depression
at one end (see Fig.). The height of the cylinder is 1.45 m and its radius is 30 cm. Find the total
surface area of the bird-bath.
22
Take = 7
(a) 1.1 m2 (b) 2.2 m2
(c) 3.3 m2 (d) 4.1 m2
31. 2 cubes each of volume 64 cm3 are joined end to end. Find the surface area of the resulting cuboid.
(a) 160 cm2 (b) 170 cm2
(c) 180 cm2 (d) 190 cm2
32. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the
hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the
vessel.
(a) 472 cm2 (b) 572 cm2
(c) 672 cm2 (d) 772 cm2
33. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the
hemisphere can have? Find the surface area of the solid.
(a) Greatest diameter = 7 cm, surface area = 332.5 cm2
(b) Greatest diameter = 8 cm, surface area = 452.5 cm2
(c) Greatest diameter = 10 cm, surface area = 632.5 cm2
(d) Greatest diameter = 12 cm, surface area = 832.5 cm2
34. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends
(see Fig.). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find
its surface area.
(a) 210 m2 (b) 220 m2
(c) 230 m2 (d) 240 m2
35. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as
shown in Fig. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total
surface area of the article.
(a) 332 cm2 (b) 352 cm2
(c) 374 cm2 (d) 395 cm2
VOLUME OF A COMBINATION OF SOLIDS
36. A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm
and the height of the cone is equal to its radius. Find the volume of the solid in terms of
.
(a)
cm3 (b) 3 cm3
(c) 4 cm3 (d) 5 cm3
37. A gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how
much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two
hemispherical ends with length 5 cm and diameter 2.8 cm (see Fig.)
(a) 335 cm3 (b) 336 cm3
(c) 337 cm3 (d) 338 cm3
38. A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter, the diameter of the
spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its volume to be
345 cm3. Check whether she is correct, taking the above as the inside measurements, and
3.14.
(a) Not correct. Correct answer is 346.51 cm3
(b) Correct
(c) Cannot be determined
(d) None of these
CONVERSION OF SOLID FROM ONE SHAPE TO ANOTHER
39. A cone of height 24 cm and radius of base 6 cm is made up of modeling clay. A child reshapes it
in the form of a sphere. Find the radius of the sphere.
(a) 2 cm (b) 4 cm
(c) 6 cm (d) 8 cm
40. A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform
thickness. Find the thickness of the wire.
(a) 0.47 mm (approx.) (b) 0.5 mm (approx.)
(c) 0.67 mm (approx.) (d) 0.7 mm (approx.)
41. A hemispherical tank full of water is emptied by a pipe at the rate of
4
37
litres per second. How much time will it take to empty half the tank, if it is 3 m in diameter?
22
Take = 7
(a) 12.5 minutes (b) 16.5 minutes
(c) 20.5 minutes (d) 22.5 minutes
42. A metallic sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of radius 6 cm.
Find the height of the cylinder.
(a) 2.74 cm (b) 3.76 cm
(c) 4.80 cm (d) 5.82 cm
43. Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single solid
sphere. Find the radius of the resulting sphere.
(a) 10 cm (b) 11 cm
(c) 12 cm (d) 13 cm
44. A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a
platform 22 m by 14 m. Find the height of the platform.
(a) 2.5 m (b) 4.7 m
(c) 6.2 m (d) 8.5 m
45. How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to form a
cuboid of dimensions 5.5 cm × 10 cm × 3.5 cm?
(a) 100 (b) 200
(c) 300 (d) 400
46. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in her
field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/h, in
how much time will the tank be filled?
(a) 100 minutes (b) 200 minutes
(c) 300 minutes (d) 400 minutes
ANSWERS
1. (a) 2. (b) 3. (d) 4. (c) 5. (a) 6. (b)
7. (d) 8. (a) 9. (d) 10. (a) 11. (c) 12. (a)
13. (c) 14. (d) 15. (c) 16. (d) 17. (a) 18. (c)
19. (a) 20. (b) 21. (c) 22. (a) 23. (b) 24. (a)
25. (d) 26. (a) 27. (d) 28. (b) 29. (a) 30. (c)
31. (a) 32. (b) 33. (a) 34. (b) 35. (c) 36. (a)
37. (d) 38. (a) 39. (c) 40. (c) 41. (b) 42. (a)
43. (c) 44. (a) 45. (d) 46. (a)
