Math5

Fraction

Introduction:

Equal parts of a whole are fractions:
1, 2, 5, etc. are fractions.
236
2 means two parts out of three equal parts.
3
In a fraction 2

3
2 is numerator and 3 is denominator.

Diagram Fraction Meaning Numerator Denominator

1 One part out of 1 4
4 4 equal parts

Equivalent fractions:

1 is shaded
2

2 is shaded
4

3 is shaded Each fraction represents the same
6 part but the notation is different.

1, 2, 3, etc. are equivalent fractions.
246
Two or more fractions representing the same part of the whole are equivalent
fractions.

To find the equivalent fraction of the given fraction

Method I:

To find the equivalent fractions of given fraction, multiply both numerator and
denominator by the same number (except 0)

1 = 1×2 = 2
2 2×2 4

1 = 1 × 3 = 3
2 2 × 3 6
1 × 4
1 = 2 × 4 = 4 1, 2, 3, 4, etc are equivalent fractions.
2 8 2468

Oasis School Mathematics Book-5 101

Method II:
To find the equivalent fractions of the given fraction, divide both the numerator
and the denominator by the same number.

5 = 5÷5 = 1 1 and 5 are equivalent fractions
10 10 ÷ 5 2 2 10

3 = 3÷3 = 1 1 and 1 are equivalent fractions
6 6÷2 3 3 6

Like and unlike fractions:

1, 2, 5, 6, etc. are like fractions.
8888
\ Fractions having same denominator are like fractions.
2, 1, 5, 3, etc. are unlike fractions.
5368
\ Fractions having different denominators are unlike fractions.

Proper and improper fractions:

3, 2, 1, 2, etc. are proper fractions.
7567
Fraction in which numerator is less than the denominator is called proper fraction.

6, 7, 8, 5, etc. are improper fractions.
4554
Fraction in which numerator is greater than the denominator is called improper
fraction.

Mixed number 12 .

A fraction like 7 can be expressed as 3
2

7 halves

1 (2 halves) 1 (2 halves) 1 (2 halves) 1
2
7 halves i.e. 3 full and one half

\ 7 = 321
2

\ Mixed number is the combination of a whole number and a proper fraction.

102 Oasis School Mathematics Book-5

Conversion of a mixed number into an improper fraction and an improper
fraction into mixed number

= 1
4

1 1 =1 1 = 4 × 1 + 1= 5
4 4 4 4

\ Improper fraction = whole number × denominator + numerator
denominator

Again,

Mixed number = quotient remainder
denominator

Example :

Convert into mixed number.
17 1
4 = 4 4 4) 17 (4 Quotient
- 16 Remainder
1

Exercise 6.1 c.

1. Write the fraction of the shaded part.
a. b.

2. What fraction of these circles are shaded?
a. b. c.

3. Write each of the divisions in fraction.

a. 11 ÷ 7 b. 9 ÷ 4 c. 3 ÷ 4 d. 11 ÷ 13 e. 8 ÷ 3 f. 3 ÷ 13

4. a. What fraction of an hour is 45 minutes?

b. What fraction of a day is 6 hours?

c. What fraction of a year is 25 days?

5. Write any three equivalent fractions of the following:

a. 1 b. 1 c. 4 d. 2 e. 1
2 4 5 7 5

Oasis School Mathematics Book-5 103

6. Write any two equivalent fractions, dividing both numerator and

denominator by a common number.

a. 10 b. 18 c. 20

20 24 24

7. Write the correct number in the space:

a. 2 = b. 3 = 12 c. 1 = d. 3 = 18
5 15 4 4 5
40

8. Identify whether the given fractions are like or unlike:

a. 2 , 3, 7, 1 b. 4 , 2, 131, 6
5 555 7 5 9

9. Separate the like fractions from the given set:
2, 1 , 3 , 4, 2, 1, 5, 3, 2 , 3
5 10 9 7 6 7 7 7 11 13

10. Identify whether the given fractions are proper or improper:

a. 4 b. 9 c. 3 d. 4 e. 16 f. 2 g. 3 h. 12
7 4 8 13 11 9 11 7

11. Convert the following mixed numbers into improper fractions:

a. 21 b. 7 53 c.2 3 d. 1 1 e. 2 3 f. 5 6
4 11 5 7 7

g.10 2 h.7111 i. 3 3
5 8

12. Convert the following improper fractions into mixed numbers:

a. 12 b.13 c.18 d. 19 e. 22 f. 32 g. 29 h. 53 i. 65
755 3 9 9 11 13 15

13. Answer the following questions:

a. What are the fractions called if they have same denominator?

b. What are the fractions called if they have different denominators?

c. What is a fraction called if its numerator is less than its denominator?

d. What is a fraction called if its numerator is greater than its

denominator?

e. Are 2 and 4 equivalent fractions?
5 10

f. Are 3 and 3 like fractions?
54

Answers: Consult your teacher

104 Oasis School Mathematics Book-5

Conversion of unlike fractions into like fractions

It is easy to compare, add and subtract the like fractions. So we must learn to
convert unlike fractions into like fractions.

Let's see an example and get an idea how we can convert unlike fractions into like

fractions.

Example :

Convert 34, 1 and 5 into like fractions.
Solution: 5 8

L.C.M. = 4, 5, 8 is 2 × 2 × 1 × 5 × 2 = 40

Now, 3 3 10 30 Steps:
4 4 10 40 • Find the L.C.M of denominators
= × =
• Convert all the fractions into their
1 = 1 × 8 = 8 equivalent fractions having L.C.M of
5 5 8 40 denominator.
5 5 5 25
8 = 8 × 5 = 40

3400 , 8 and 25 are the like fractions representing unlike fractions 43, 1 and 58.
40 40 5

Class Assignment

Conversion of unlike fraction into like fraction:

1. Convert the following unlike fractions into like fractions:

4 and 2 12, 2 and 1 32, 4 and 3
5 3 3 5 5 4

Comparison of fractions 3 1
1 4 4
Let's take two fractions 4 and
3
From the figure, it is clear that 3 > 1 4
4 4

If the fractions are like, the fraction having
greater numerator will be the greater fraction.

Again, 1 4 1
3 6 3
Take two fractions and 4
6
From the figure it is clear that 4 > 1
6 3

Oasis School Mathematics Book-5 105

It is difficult to make figure in every case to compare the fractions.

So we have to convert unlike fractions into like fractions to compare them.
1 4
3 and 6

L.C.M. of 3 and 6 is 3 × 1 × 2 = 6

\ 1 = 1×2 = 2
3 3×2 6

4 = 4 × 1 = 4
6 6 × 1 6
If the fractions are unlike, first convert them into like
Since 4 > 2 fractions and compare them.
6 6

Example :

Arrange 1 , 5 and 3 in descending order.
46 3

Solution:
Given fractions 1, 5, 3

463
These are unlike fractions,

L.C.M. of 4, 6 and 3 is 2 × 3 × 2 = 12 I know ! Ascending order

1 = 1×3 = 3 means increasing order and
4 4×3 12 descending order means

5 = 5 × 2 = 10 decreasing order.
6 6 × 2 12

2 = 2 × 4 8
3 3 × 4 = 12

Since, 10 > 8 > 3 3 5 2 1
10 8 12 6 3 4
12 > 12 > \ > >

Class Assignment

Arrange 52, 4 and 1 in ascending Arrange 23, 4 and 1 in descending
order: 7 7 order: 5 2

106 Oasis School Mathematics Book-5

Exercise 6.2

1. Observe the fractions and write>, < or = in the box.

a. 2 1 b. 4 3 c. 12 13 d. 11 11
3 3 7 15 15 17 17
7 7 3
e. 5 13 f. 2 3 g. 8 17
13 15 17
15

2. Make the like fractions and compare them:

a. 2 and 3 b. 1 and 2 c. 3 and 5 d. 1 and 1
5 4 2 3 5 6 4 5

e. 5 and 4 f. 3 and 1 g. 3 and 4
12 9 8 12 9 12

3. Arrange the following fractions in ascending order:

a. 2, 1, 5 b. 7 , 2 , 9 c. 2, 3, 5 d. 1, 3, 7
777 11 11 11 346 2 4 12

4. Arrange the following fractions in descending order:

a. 3, 2, 7 b. 131 , 1, 6 c. 1, 2, 3 d. 3, 1, 5
878 11 11 234 846

Answers: Consult your teacher

Fraction in its lowest term

A fraction is said to be in its lowest term if only the common factor between
the numerator and denominator is 1. We can convert a fraction into its lowest
term by cancelling the common factors of both numerator and denominator.

There are different ways of addition and subtraction of fractions depending
upon its type.

Let's be clear with the help of given examples.

Example : Example :

Express: 8 into its lowest term: Express: 15 8 into lowest term:
12 Solution: 56
Solution: 8
15586
12 2 8 2 12 2×2×2 I know ! how
2×2×2 2×2×2×7 to factorize
22 26
= 2 × 2 × 3 23 = 15 +
2
= 15 + 1
=3
8 2 7
\ 12 = 3 1
\ 15 7

Oasis School Mathematics Book-5 107

Addition and subtraction of fractions
Addition and subtraction of like fractions

Sum of like fractions = Sum of the numerators
common denominator

Look at these examples properly and get the idea of addition and subtraction

of like fractions.

2+3
77
2 + 3 Add numerator, keep
= 7 2 3 denominator same.
75 7
5
= 7 7

Difference of like fractions = Difference of the numerators
common denominator

Example :

Subtract: 8 – 3
99
= 8 – 3 Subtract numerator,
9 8 keep denominator same.
5 93
= 9
59
9

Look at one more example:

Add: 2 1 + 5 2 Steps:
33

Solution: 1 2 • Convert the mixed number into the improper fraction.
3 3 • Keeping denominator same, add numerator.
2 + 5

= 7 + 17 Alternative method: Steps:
3 3
1 2
= 7 + 17 2 3 + 5 3 • Separate the mixed number
3 into whole number and proper
24 1 2 fraction.
= 3 = 2 + 5 + 3 + 3
Add proper fractions.
=8 •
= 7 + 1 + 2 • Add the whole number with the
3 sum of proper fractions.
3
= 7 + 3

=7+1

=8

108 Oasis School Mathematics Book-5

Example : Example : If the resultant fraction is
improper, we have to convert
Subtract: 5 1 – 3 3 Add: 2 3 + 3 1 + 4 4 it into mixed number.
4 4 5 5 5

Solution: Solution:

5 1 – 3 3 2 3 + 3 1 + 4 4
4 4 5 5 5
13 16 24
= 21 – 15 = 5 + 5 + 5
4 4
13 + 16 + 24
21 – 15 =
= 4 5

6 = 53
4 5
= 3
= 10 5
3
= 2

Exercise 6.3

1. Express the following fractions into their lowest term:

a. 6 b. 9 c. 15 d. 14 e. 27 f. 15
8 12 20 49 81 60

g. 84 h. 96 i. 7 j. 20 k. 50
108 144 77 100 400

2. Express the following mixed fraction into improper fraction and
convert them into their lowest term:

a. 4 6 b. 12 3 c. 18297 d. 30120
12 6

e. 25250 f. 30 6 g. 35 8 h. 50 20
24 32 80

3. Add the following and reduce them into lowest term if necessary:

a. 1 + 1 b. 3 + 1 c. 4 + 2 d. 5 + 4 e. 213 + 332
4 4 7 7 11 11 13 13

f. 414 + 314 g. 237 + 3 3 h. 238 + 3 1 i. 1 5 + 4 5
7 8 12 12

4. Subtract the following and reduce them into lowest term if necessary:

a. 5 – 1 b. 3 – 1 c. 6 – 2 d. 5 – 1 e. 6 – 2
6 6 4 4 7 7 11 11 13 13

f. 2 4 – 215 g. 432 – 1 1 h. 435 – 2 1 i. 273 – 1 1 j. 7121 – 3 5
5 3 5 7 11

Oasis School Mathematics Book-5 109

5. Add the following and reduce them into lowest term if necessary:

a. 1 + 2 + 3 b. 5 + 2 + 1 c. 2 + 3 + 1 d. 2 2 + 513 + 331
7 7 7 11 11 11 9 9 9 3

e. 527 + 2 1 + 337 f. 752 + 215 + 353 g. 385 + 483 + 181 h. 4 5 + 281 + 5 7
7 8 8

Answers : 1
11
1. a) 3 b) 3 c) 3 d) 2 e) 1 f) 1 g) 7 h) 2 i) j) 1 k) 1
4 4 4 7 3 4 9 3 5 8
9 151 101 121 141 151 1 4
2. a) 2 b) 25 c) 55 d) 5 e) 4 f) 4 g) 5 h) 5 3. a) 2 b) 7
23
6 9 1 6 h) 521 i) 556 2 1 4 4 4
c) 11 d) 13 e) 6 f) 7 2 g) 5 7 4. a) 3 b) 2 c) 7 d) 11 e) 13

f) 3 g) 3 1 h) 2 2 i)1 2 j) 3 8 5. a) 6 b) 8 c) 2 d) 11 1 e) 10 6 f ) 13 1
5 3 5 7 11 7 11 3 3 7 5

g) 918 h) 1258

Addition and subtraction of unlike fractions

To add or subtract unlike fractions they have to be first converted into like
fractions. Let's see an example and get the idea about addition and subtraction
of unlike fractions.

Example :

Add: 2 + 3 - Convert all the fractions into like
34 fractions.

L.C.M. of 3 and 4 is 3 × 4 = 12 - Add the like fractions.

2 × 4 + 3 × 3 L.C.M. of 3 and 4 = 12
3 4 4 3 12 ÷ 3 = 4
\2×4
= 8 + 9 12 ÷ 4 = 3
12 12 \3×3

= 8+9 Alternative method:
12
Add: 2 + 3
= 17 34
12 2×4+3×3
= 12
= 15
12 = 8+9
12

= 17 = 1 5
12 12

110 Oasis School Mathematics Book-5

Example :

Add: 5 – 1 Alternative method:
64
Add: 5 – 1 L.C.M. of 6 and 4 = 12
L.C.M. of 6 and 4 is 2 × 3 × 2 = 12 64 12 ÷ 6 = 2
\5×2
Now, = 5×2-1×3
12 12 ÷ 4 = 3
5–1 \1×3
64 = 10 - 3
5×2 12
= 6×2 – 1×3 7
4×3 = 12

= 10 – 3
12 12

= 10 - 3 = 7
12 12

Example :

Add: 213 + 434 + 165

Solution:

L.C.M. of 3, 4 and 6 is 12

231 + 443 + 156 L.C.M. of 3, 4, 6
2 × 3 × 1 × 2 × 1 = 12
= 7 + 19 + 11
3 4 6

= 7 × 4 + 19 × 3 + 11 × 2
3 4 4 3 6 2

= 28 + 57 + 22 = 28 + 57 + 22
12 12 12 12

= 107 = 811
12 12

Example :

Subtract: 5 2 – 3 2
7 5
Solution:

5 2 – 3 2
5 5

= 37 – 17 7
75 7
5
= 37 × 5 – 17 ×
7 5

Oasis School Mathematics Book-5 111

= 185 – 119 Alternative method:
35 35
Subtract: 5 5 – 3 1
= 185 - 119 64
35 Solution:

= 66 5 2 – 3 2
35 7 5
= 37 – 17
= 131 75
35 = 37 × 5 – 17 × 7
35

= 185 - 119

Exercise 6.4 = 66 35
35

= 131
35

1. Add the following and reduce them into the lowest term if necessary:

a. 1 + 3 b. 3 + 1 c. 5 + 7 d. 2 + 2 e. 9 + 3
2 8 8 4 8 12 3 9 10 5

f. 5 + 2 g. 7 + 5 h. 7 + 1 i. 154 + 1 1 j. 278 + 1 1
6 3 4 6 9 6 10 4

2. Subtract the following and reduce them into the lowest term if necessary:

a. 1 – 3 b. 3 – 1 c. 5 – 7 d. 2 – 2 e. 9 – 3
2 8 8 4 6 12 3 9 10 5

f. 5 – 2 g. 7 – 5 h. 7 – 3 i. 154 1 j. 287 – 1 1
6 3 4 6 9 5 – 110 4

3. Simplify:

a. 1 + 3 – 5 b. 5 – 3 + 5 c. 5 – 1 + 3
2 4 6 9 6 12 8 4 2

d. 381 – 143 + 214 e. 514 + 6 1 – 323 f. 653 – 174 – 352
2

g. 512 + 634 + 256 h. 634 – 3 1 – 114 i. 521 – 332 + 243
8

Answers : 5
7 8 8 1 1
1. a) 8 , b) , c) 1 5 , d) 9 , e) 1 2 , f ) 1 2 , g) 2 7 , h) 17 , i) 2 9 , j) 4 1 ,
24 12 18 10 8

2. a) 1 , b) 1 , c) 3 , d) 4 , e) 3 , f) 1 , g) 11, h) 8 , i) 7 , j) 1 5 ,
8 8 12 9 10 6 12 45 10 8

3. a) 152, b) 1376, c) 1 7 , d) 3 5 , e) 8112, f) 2315, g) 15112, h) 2 3 , i) 4172
8 8 8

112 Oasis School Mathematics Book-5

Addition and subtraction of a whole number and a fraction
Let's take a whole number 5 and a fraction 13.

Let's add them Quick method
1
3 + 3 5 + 1 = 5 1
2 2
= 5 + 1
1 3 6 + 2 = 6 2
5 × 3 + 1 × 1 L.C.M. of 1 and 3 = 3 3 3
= 3 I understand !
1 1
= 15 + 1 7 + 2 = 7 2
3
16 I just have to
= 5331 3) 1 6 (5 remove + sign.
= – 15

1

Class Assignment

Add quickly: 3 + 23 =

5 + 41 = 6 + 32 =
10 + 14 =
15 + 71 = 20 + 3 =
10

Take a whole number 6 and a fraction 1 .
Let's 1 4
subtract 4 from 6.
1 L.C.M. of 1 and 4 = 4
6 – 4

= 6 – 1 4) 2 3 (5
1 4 – 20
= 6 × 4 – 1 × 1
4 3

= 24 – 1
4
23
= 4

= 534

Oasis School Mathematics Book-5 113

Exercise 6.5

1. Add: b. 6 + 4 c. 8 + 1 d. 7 + 92 e. 8 + 1
a. 2 + 35 7 3 3

2. Subtract:

a.5 – 12 b. 6 – 1 c. 4 – 1 d. 5 – 83 e. 8 – 1
3 7 6

3. a. A milkman mixes 12l of milk with 18l of water. How much mixture does he get?
b. kg of Manasuli rice and 215 kg of Basmati rice. How
Salim bought h3e54
much rice did buy altogether?

c. Three bags weigh 2 1 kg, 3 1 kg and 412kg respectively. What is the total
2 2
weight of the bags?
2 3
4. a. Bishwash and Bismita ate 8 and 10 part of a cake. Who ate more cake and
by how much?

b. In a garden, 1 part of the garden is covered by mango trees and 2 part of the
3 5
garden is covered by oranges trees. Find the remaining part of the garden.

c. Ansu was given 312 hours to do a test. She finished the test in 231 hours.
How earlier did she finish the test?

Answers :

1. a) 2 3 b) 6 4 c) 8 1 d) 7 2 e) 8 1 2. a) 4 1 b) 5 2 c) 3 6 d) 4 5
57 3 3 3 237 8

e) 7 5 3. a) 5 l b) 6 kg c) 10 1 kg 4. a) Bismita ate 1 part more b) 4 , c) 1 1
6 8 2 20 15 6

Multiplication by fraction

Multiplication of a fraction by a whole number:

3 × 1 = 1 + 1 + 1 p= 3 Multiplication is the repeated addition.
8 o8 8 8 8

whole number × numerator
\ Whole number × fraction = denominator of the fraction

While multiplying a fraction by a whole number, multiply the numerator by

whole number and leave the denominator same.

\ 5 × 1 = 53, 6 × 3 = 18, 3 × 1 = 34, etc.
3 5 5 4

114 Oasis School Mathematics Book-5

Multiplication of a fraction by a fraction

3 part is shaded 1 of 3 part is shaded
4 2 4

1 3 Double shaded = 3
2 4 part represents 8

3 of We have to multiply numerator
4 and denominator separately.
To multiply two fractions:

Multiply the numerators 1 × 3 =3

Multiply the denominators 2 × 4 = 8
1 3 3
\ 2 × 4 = 8 Note: If the fraction is a mixed number, then change it into improper

fraction and then multiply and reduce the product into its lowest term.

Example :

Multiply: 215 × 5 2
3
Solution: 215 × 523

= 11 × 17
5 3
11 × 17 = 187
= 187 = 12 7 5 × 3 = 15
15 15

Value of a given fraction of a whole quantity:

Consider the following example,
1
Find the value of 4 of 12

There are 12 circles.
41circles are shaded. Number of shaded circles = 3.
1
\ 4 of 12 = 3

Example :

Find the value of 1 of 21.
3
Solution:

1 of 21 = 1 × 21
3 3
= 1 ×321
= 7

Oasis School Mathematics Book-5 115

Exercise 6.6

1. Multiply the following:

a. 1 × 5 b. 1 × 7 c. 1 ×8 d. 1 ×6 e. 1 × 10
3 5 3 7 3

f. 1 × 10 g. 1 × 12 h. 1 × 18 i. 1 × 21
5 6 3 7

2. Multiply:

a. 2 × 6 b. 1 × 5 c. 3 ×7 d. 3 × 4 e. 4 × 3
6 7 4 8 9

f. 3 × 16 g. 12 × 12 h. 5 × 3 i. 6 × 15 j. 16 × 3
8 15 5 16 8

k. 112 × 4 l. 345 × 15 m. 413 × 8 n. 3 1 × 11
2

3. Multiply the given fractions and reduce into the lowest terms if

necessary:

a. 1 × 3 b. 2 × 4 c. 2 × 1 d. 2 × 3 e. 4 × 3
2 4 3 8 5 7 5 7 5 4

4. Find: 1 3
3 4
a. 1 of 2 b. 1 of 2 c. of d. 4 of 3 e. 2 of 1
4 3 2 3 5 4 5 3

f. 2 of 331 g. 2 of 312 h. 4 of 423
5 7 9

5. Find the product and convert it into lowest term:

a. 1 × 1 × 1 b. 1 × 1 × 2 c. 2 1 × 1 × 2 d. 3 × 331 × 7
2 3 3 3 4 3 3 6 3 3 12

e. 431 × 253 × 212 f. 221 × 134 × 331 g. 3 × 313 × 5
8 12

6. Find the value of:

a. 1 of 20 b. 1 of 36 c. 1 of 28 d. 1 of 50
2 3 4 5

e. 2 of 25 f. 3 of 55 g. 5 of 54 h. 3 of 100
5 5 6 4

i. 3 of 480 j. 3 of 240 k. 3 of 420
8 4 7

Answers: Consult your teacher

116 Oasis School Mathematics Book-5

Reciprocal of a fraction

Two numbers are said to be reciprocal to each other if their product is 1.
a b
b × a = 1

\ a and b are reciprocal to each other.
b a

Reciprocal of a number is 1 divided Reciprocal of a number can be
obtained simply by interchanging
by the number. Reciprocal of 5 = 1 numerator and denominator.
5

Example :

Find the reciprocal of.

a. 2 b. 56 c. 1
3

Solution:

a. Given fraction = 2 b. Given fraction = 5 c. Given fraction 31==313
Its reciprocal = 1 Its reciprocal = 6 Its reciprocal =
1 6
2 5

Class Assignment

Find the reciprocal of the following: Reciprocal of 5 =
Reciprocal of 2 = 7

Reciprocal of 7 = Reciprocal of 1 =
4

Reciprocal of 1 = Reciprocal of 3 =
5 7

Reciprocal of 2 = Reciprocal of 572 =
3

Reciprocal of 631 =

Division of whole number by a fraction 1
4
A man has 2 litres of milk. He has to divide l of milk in each bottles. How
many bottles will he need?

Oasis School Mathematics Book-5 117

Picture tells that he needs 8 such bottles. 1
4
Mathematically, it is obtained dividing 2 by
1 Steps:
i.e. 2 ÷ 4

= 2 × 4 • Write a whole number into a fraction.
1 1 • Multiply the dividend with the reciprocal of the divisor.

=8 • Reduce it into the lowest term if necessary.

Class Assignment

Divide:

5 ÷ 2 10 ÷ 1 9 ÷ 3 16 ÷ 1
3 5 5 3

Division of a fraction by a whole number

5 bottles of milk is to be poured equally in 4 glasses. How much milk will be
6
there in each glass?
5
To get it, we have to divide 6 by 4.
i.e.
5 ÷4 1
6 4
= 5 × Steps:
65 • Find the reciprocal
= of divisor.
24 • Multiply dividend with reciprocal of divisor.

• Reduce it into lowest term if necessary.

Class Assignment

Divide:

1 ÷ 5 2 ÷ 3 3 ÷ 2 3 ÷ 4
5 3 7 8

118 Oasis School Mathematics Book-5

Dividing a fraction by a fraction

AhiT.eaovb gioneytg=i34htle,a43n÷wsg×e3481t18hhmaovfoe18ftmoc••oS.d ltHoievupoirdwsMFe:edinmu34dlpatmiatnphpyleybepryrte.hice81Heicpdmeersiov.wcciadaalnenonthfdsdebtiomvyictsahoukerte.r?ietciinprtoocsaol mofedipviiesocer.s each
=6
• Reduce it into the lowest term if necessary.

Class Assignment

Divide:

2 ÷ 1 5 ÷ 1 2 ÷ 1 1 ÷ 2
3 2 6 3 5 6 3 6

Exercise 6.7

1. Divide:
a. 5 ÷ 12 1 c. 8 ÷ 75 d. 4 ÷ 32 5
b. 7 ÷ 3 e. 7 ÷ 6

f. 2 ÷ 13 g. 12 ÷ 3 h. 15 ÷ 3 i. 12 by 1 j. 15 by15
4 5 2

2. Divide:

a. 3 ÷ 2 b. 5 ÷ 2 c. 3 ÷ 5 d. 1 ÷ 4 e. 1 ÷7
4 12 7 2 3
2 3
f. 5 ÷ 4 g. 7 by 2 h. 5 by 2 i. 3 by 2
12 11

3. Divide:

a. 1 ÷ 2 b. 2 ÷ 5 c. 1 ÷ 41 d. 2 ÷3 e. 8 ÷ 3
3 5 3 7 7 3 10 17 24

f. 5 ÷ 2 g. 248 ÷ 6 h. 158 ÷ 261 i. 921 ÷ 531 j. 213 by 523
7 21 5

k. 351 by 253

Oasis School Mathematics Book-5 119

4. Solve the following:
1 1 5 6 3 4 442 1 1
a. 4 ÷ 2 ÷ 3 b. 7 ÷ 8 ÷ 9 c. ÷ 5 ÷ 3 2

d. 241 ÷ 132 ÷ 214 e. 2 1 ÷ 2 1 ÷ 5
4 5 2

Answers : 1 2
5 5
1. a) 10 b) 21 c) 11 d) 6 e) 8 f) 6 g) 16 h) 25 i) 24 j) 75

2. a) 3 b) 5 c) 3 d) 1 e) 1 f) 1 g) 3 h) 5 i) 3
8 24 35 8 21 10 14 24 22

3. a) 5 b) 14 c) 4 d) 2 2 e) 11 5 f) 7 1 g) 2 1 h) 4 3 i) 1 25
6 15 7 9 17 2 12 4 32

j) 7 k) 1 3 l) 13 4. a) 3 b) 1 5 c) 6 3 d) 3 e) 9
17 13 19 10 7 7 5 22

Verbal problems on division of a fraction

In our daily life we come across many situations where we need to divide a

whole number by a fraction,a fraction by a fraction and a fraction by a whole

number. Let's see some examples.

Divide 5 litre of juice equally among 4 children.
8

Here, 5 ÷ 4
8

= 5 × 1 This is the division of a
8 4 fraction by a whole number.

= 5
32
5
\ Each child gets 32 litre of juice.

Exercise 6.8

1. a. A man has 5l of juice. He has to divide 1 l of juice in each cup. How many
5
cups will he need?
3
b. A man has 20 litres of milk. He has to divide 4 litres in each bottle. How
many bottles does he need?

2. ba.. 543cbhoiltdtlreenoafrme silhkariisnpgo32uorfedpizezqau.aWllhyaitnftroac4tiognlaosfspesiz.zWa whailtl each student get? is
fraction of milk

there in each glass?

120 Oasis School Mathematics Book-5

3. a. A girl has 2 m long ribbon. How many pieces of ribbons each having the
3
1
length of 6 m can be made from it ?

b. A man has 3 m of cloth. He has to cut it into pieces of 1 m each. How
4 12
many pieces he can make?

Answers : 2. a) 2 b) 3 3. a) 4 b) 9
1. a) 25 b) 15 15 16

Simplification of fractions

The order of operations in the simplification of fractions also follows the rule
of BODMAS.

Where, ‘B’ stands for brackets [{( )}] in order

‘O’ stands for of (×)

‘D’ stands for division (÷)

‘M’ stands for multiplication (×)

‘A’ stands for addition (+)

Examp‘lSe’ s:tands for subtraction (-)

Simplify: 5 + q 1 ÷ s 1 + 1 – 1 ptr = 5 1 ÷ s21+01tr
6 20 5 o5 10 6 + q20

Solution: 5 + 1 ÷ 1 + 1 – 1 ptr = 5 1 3
6 q20 s5 o5 10 6 + q20 ÷ 10 r

= 5 + 1 ÷ 1 +o2–1 ptr = 5 1 10
6 q20 s5 10 6 + q20 ×3r

= 5 1 ÷s 1 1 = 5 + 1 = 5+1
6 + q20 5 + 10 tr 6 6 6

Exercise 6.9 = 6 =1
6

1. Multiply the following:

a. 2 + 1 – 1 b. 2 + 1 × 3 c. 2 – 1 + 4 × 5
3 4 2 5 2 5 3 2 5 8

d. 3 ÷ 5 × 10 + 1 – 1 e. 5 × 1 ÷ 1 + 2 – 1 f. 3 + 5 ÷ 1 – 1 × 2
5 8 9 3 2 6 2 4 3 2 5 6 6 2 3

g. 4 ÷ 7 of 8 h. 4 ÷ 2 of 3 i. 9 1 ÷ 3 of 7 × 4
5 15 9 3 3 4 3 5 9 5

Oasis School Mathematics Book-5 121

2. Simplify:

a. 4 1 × 1 + 1 p
2 o3 6

b. 1 – 1 ÷ 1
o33 2p 2

c. 2 + 1 + 1 + 1 – 1
3 s2 3 o2 6pt

d. 4 + s21 – 3 – 1 ÷ 1
3 o4 2pt 4

e. 2 ÷ 1 ÷ 3 × 4 + 1
5 o2 2 5p 2

f. 5 – 3 – 1 + 1 + 4
8 s5 o2 4pt 5

g. 1 – 1012 ÷ 5 t of 3
s12 3 6 5

h. 1 – 3 – 1 + 1 – 1
2 q4 s2 o4 8 ptr

i. o–2 – 1 of 1 p ÷o721 ÷ 212 p
2 3

Answers : 7 2 9 5 4 2
5 10 3 10 6 15 13 3
1. a) 12 b) c) d) e) 1 f) 5 g) 1 14 h) 2 i) 16

2. a) 2 1 b) 5 2 c) 1 5 d) 2 1 e) 2 f ) 1 23 g) -4 h) 3 i) – 13
4 3 6 3 40 24 8 18

122 Oasis School Mathematics Book-5

Worksheet

Comparison of unlike fractions.

Divide each strip of one whole into 2, 3, 4, 5, 6, 7and 8 equal parts as shown in

the figure.
1. Three equivalent fractions of 21are ............, ............ and .............
1
An equivalent fraction of 4 is .............

An equivalent fraction of 1 is .............
3

An equivalent fraction of 2 is .............
3

An equivalent fraction of 3 is .............
4
2. Compare the fractions:



3. Write the fractions of 1 , 1 , 2 , 3 and 1 in ascending order.
4 3 5 7 8
............................................................................................................

4. Write the fractions of 5 , 3 , 3 , 1 and 7 in descending order.
6 7 5 2 8
.............................................................................................................

Oasis School Mathematics Book-5 123

Decimals 1

10
A piece of paper is divided into ten

equal parts. One part represents
1
10 i.e. 0.1. 0.1

1 whole is shaded in the figure.

4 is shaded. i.e. 0.4 is shaded. 1.4
10

The shaded part represents 1.4.

Remember: If there is no whole number,
we have to write 0 in the whole
1.4 number position before decimal.

Whole number part decimal point decimal part

Here, one whole is divided into 100 Note:
1
equal parts, one part represents 100 i.e. 1 = 0.01
10
0.01. 1
100
= 0.01

2 = 0.2
10
3
100 = 0.03

1 whole is shaded 7 parts out of 100 are shaded

1.07
How to read the decimal number?

Write Read
1.6 one point six
1.07 one point zero seven
13.38 thirteen point three eight

124 Oasis School Mathematics Book-5

Write the following fractions in decimal.

1 = 0.1 2 = 5 =
10 10 10

7 = 1 = 2
10 100 100 =

15 = 25 = 65 =
100 100 100

Place value of decimal numbers

The fractions having denominators 10 are called tenths.

2 , 5 , 6 , etc. are tenths.
10 10 10

The fractions having denominators 100 are called hundredths.

3 , 5 , 6 , etc. are hundredths.
100 100 100

The fractions having denominators 1000 are called thousandths.

2 , 3 , 6 , 12 , etc. are thousandths.
1000 1000 1000 1000

1 = one tenth = 0.1 tenths place
10
3
1000 = three - hundredths = 0.03 hundredths place

15 tenths place
1000
= fifteen thousandths = 0.015 thousandths place

hundredths place

tenths place

Let's see the place value chart of 432.675.

Hundreds Tens Ones Decimal Tenths Hundredths Thousandths
(H) (T) (O)
100 10 1 (t) (h) (h)

4 3 2 1 2 2
10 100 1000

.6 7 5

Oasis School Mathematics Book-5 125

Place value of 4 = 4 × 100 = 400 I know,

Place value of 3 = 3 × 10 = 30 110 = 0.1, 1 = 0.01
100
Place value of 2 = 2 × 1 = 2 1 0.001, 150
1000 = = 0.5

Place value of 6 = 6 × 1 = 6 = 0.6 6 = 0.6
10 10 10

Place value of 7 = 7 × 1 = 7 = 0.07
100 100

Place value of 5 = 5 × 1 = 5 = 0.005
1000 1000

Expanded form and standard form of a decimal number

23.134 = 2 × 10 + 3 × 1 + 1 × 1 1 +4× 1
10 + 3 × 100 1000

= 2 × 10 + 3 × 1 + 1 × 0.1 + 3 × 0.01 + 4 × 0.001

Standard form Expanded form

Exercise 6.10 b.

1. Write in decimal:
a.

c. d.

2. In the number 372.845, write the digit in the:

a. tens place b. ones place c. hundreds place

d. tenths place e. hundredths place f. thousandths place

3. Form decimal number with:
a. 6 in ones place and 2 in tenths place.
b. 7 in tens place, 5 in ones place, 3 in tenths place, 5 in hundredths place.
c. 2 in tens place, 3 in ones place, 5 in tenths place, 3 in hundredths
place and 4 is thousandths place.

4. Show the number 219.453 in the place value chart and write the place
value of each digit.

126 Oasis School Mathematics Book-5

5. Write these numbers in figure:
a. Seven point two three b. Twelve point zero nine
c. Twenty eight point one six d. Thirty seven point six three

6. Write the given numbers in word:

a. 6.35 b. 19.64 c. 134.69 d. 235.92 e. 0.537

7. Write the following fractions in decimals:

a. 2 b. 3 c. 4 d. 6 e. 8
10 100 1000 10 100

f. 12 g. 154 h. 624 i. 7 j. 82
100 1000 1000 1000 100

8. Write the given decimal numbers in expanded form.

a. 0.84 b. 12.63 c. 54.652 d. 37.185

9. Write the given decimal numbers in standard form.
a. 2 × 10 + 5 × 1 + 6 × 0.1 + 5 × 0.01
b. 3 × 100 + 2 × 10 + 1 × 1 + 3 × 0.1 + 7 × 0.01 + 8 × 0.001
c. 4 × 1000 + 2 × 100 + 2 × 10 f+ 5 × 1 + 3 × 0.1 + 2 × 0.01 + 1 × 0.001

Answers: Consult your teacher

Conversion of decimals into fractions
Let's see an example and get the idea of conversion of decimal into fraction.

0.85 85 17 = 17 Steps:
= 100 20 20
• Count the number of decimal places in the
2 digits after decimal decimal.
means I have to write
two zeros after 1 in • Ignore the decimal point and write
all the digits on the numerator of the
denominator. fraction.

• Write as many zeros after 1 in the
denominator as there were decimal
places in the fraction.

• Reduce the fraction into its lowest
term.

Oasis School Mathematics Book-5 127

Example :

Convert 2.05 into fraction. Alternative method:

Solution: 2.05 = 2 + 0.05

2.05 = 205 51 1 = 2 + 5 = 2 + 1 = 2 1
100 = 2100 20 = 2 20 100 20 20

Class Assignment 2.015

Convert the following decimals into fractions:
1.25 5.6 0.25

Conversion of fractions into decimal:

We already know:

i. When the denominator is 10, 100, 1000 etc.

1 = 0.2
20
3 I already know it
100
= 0.03

7
1000 = 0.005

i. When the denominator is another number:

Convert, 25 into decimal. I have to convert
Here, 5 20 0.4 denominator into 10,

100, 1000, etc.

- 20
2 ×
5 = 0.4 Divide numerator by denominator.

Alternative method: Multiply both numerator
and denominator by
2 = 2 × 2 = 4 = 0.4 2 to make 10 in the
5 5 2 10 denominator.

128 Oasis School Mathematics Book-5

Class Assignment

Convert the following fractions into decimals:

4 3 1
10 = 2 4

1 = 3
100 4

2 = 1
100

3 = 2
100

6 =
100

Comparison of decimal:

Let's take any two numbers 23.84 and 15.23

Here, Compare whole numbers
23 > 15 first, then compare tenths,
\ 23.84 > 15.23 hundredths, thousandths etc.

Example : I understand ! We have to
compare whole the number first
Compare 28.63 and 28.82
Here, compare whole numbers, 28 = 28 then tenths, hundredths.
Compare tenths 6 < 8
\ 28.63 < 28.82

Exercise 6.11

1. Convert the following decimals into fractions:

a. 0.8 b. 0.3 c. 0.12 d. 0.45 e. 2.5 f. 6.3
l. 19.82
g. 8.5 h. 12.8 i. 5.25 j. 6.75 k. 12.95

m. 0.314 n. 0.0516 o. 150.14 p. 7.214 q. 25.306

Oasis School Mathematics Book-5 129

2. Convert the following fractions into decimals:

a. 2 b. 150 c. 1300 d. 4 e. 1040 f. 25
10 1000 100

g. 25 h. 35 i. 21 j. 45 k. 43 l. 121

m. 134 n. 721 o. 83 p. 85 q. 3 1
5

3. Compare the given decimal numbers: c. 75.94 and 63.38
a. 12.84 and 13.35 b. 163.28 and 47.12 f. 34.27 and 34.23
d. 18.61 and 18.54 e. 19.64 and 19.31 i. 0.842 and 0.841
g. 63.18 and 63.17 h. 15.312 and 15.314

4. Arrange the following numbers in ascending and descending order:

a. 3.14, 4.75, 2.69 b. 5.23, 5.27, 5.18

c. 6.572, 6.275, 6.752 d. 4.83, 4.75, 4.86

Answers: Consult your teacher.

Like and unlike decimal:
2.3, 3.57 and 8.164 are unlike decimals
5.12, 6.35, 2.50 are like decimals.

If the number of digits If the number of digits
after decimal point is after decimal points is not
same, such decimals are same, such decimals are
like decimals. unlike decimals.

Conversion of unlike decimals into like decimals

2.3, 2.52 and 3.614 are unlike decimals 2.3 and 2.300 are same
2.300, 2.520 and 3.614 are like decimals 2.52 and 2.520 are same

Remember !
• Writing or removing zeros at the end of a decimal number does not change its

value.

130 Oasis School Mathematics Book-5

Addition and subtraction of decimals

Look at these examples properly and get the idea about addition and subtraction
of decimals.

Example :

Add: 15.24 + 8.6

15.24 15.24 • Convert the decimals into like decimals.
8.6 +8.60
23.84 • Arrange the digits according to the place value so that
the decimal points are exactly one below the other.

• Start to add from hundredths. Carry over if needed.

Example : • Convert the decimals into like decimals.
Subtract: 27.18 - 15.7
• Arrange the digits according to the place value so that
27.18 27.18 the decimal points are exactly one below the other.
15.70 - 15.70
11.48 • Start to subtract from hundredths, borrow if necessary.

Exercise 6.12

1. Add:

a. 24.16 + 5.8 b. 16.35 + 12.1 c. 46.25 + 6.4
f. 18.97 + 15.84
d. 4.57 + 13.94 e. 13.63 + 15.95 i. 216.87 + 87.764
l. 0.874 + 13.26 + 5.7
g. 0.637 + 12.834 h. 13.742 + 18.217

j. 45.996 + 18.84 k. 12.64 + 7.5 + 18.372

m. 18.64 + 15.736 + 5.9 n. 25.76 + 0.81 + 219.6

2. Subtract:

a. 6.78 - 3.32 b. 8.69 - 7.42 c. 9.32 - 4.12 d. 14.6 - 3.84
g. 45.1 - 36.111
e. 34.17 - 32.717 f. 28.654 - 13.82 j. 12 - 9.324
m. 361.2 - 65.286
h. 536.3 - 442 .18 i. 8 - 5.376

k. 200.26 - 97.865 l. 67.86 - 12.375

3. Simplify:

a. 5.4 + 7 - 6.2 b. 4.7 + 2.6 - 3.8 c. 5.2 - 3.3 + 4.2

d. 4.63 + 2.32 - 3.9 e. 5.46 - 2.84 + 1.45 f. 6.74 + 5.31 - 2.378

g. 48.93 + 50.05 + 20.007 h. 53.358 + 26.732 - 37.4 i. 105.38 + 36.79 - 46.372

j. 118.32 - 15.632 - 12.54 Answers: Consult your teacher.

Oasis School Mathematics Book-5 131

Word problems on decimals

There are many situations in our daily life when we have to add and subtract
the decimals. Let's see an example and get an idea of addition and subtraction.

• Read the questions properly
• Decide what you have to do?

Example :

Ankit had Rs 25.65. His mother gave him Rs 23.80. How much money does he have

altogether?

Solution:

Money that Ankit has = Rs 25.65

Money given by his mother = + Rs 23.80

Total money he has = Rs 49.45

Exercise 6.13

1. a. What is the difference between 5.732 and 4.67?
b. What should be added to 5.672 to make 12?
c. What should be subtracted from 13 to make 8.432?
d. From which number 15.37 should be subtracted to make 32.437?

2. a. Sandesh bought a book for Rs 35.85, a pen at Rs 17.95 and a pencil at Rs
8.45. What is the total amount he spent?

b. A milkman has 45.35l of milk. He sold 25.395l milk during the day. How
much milk is left with him?

c. Santosh bought a plate of Mo: Mo: worth Rs 68.75 and a cold drink worth
Rs 15.35. He paid Rs 100 note. How much change did he get back?

3. Find the perimeter of the given figure:
a. b. 8.62 cm

5.62 cm 3.25 cm
5 cm

7.81 cm

5.62 cm 9.1 cm

Answers: Consult your teacher.
132 Oasis School Mathematics Book-5

Multiplication of decimal by 10, 100, 1000, etc:
Let's learn the multiplication of a decimal number by 10, 100, 1000, etc.

3.174 × 10 = 31.74 Shift decimal point one step right Try
4.374 × 100 = 437.4 Shift decimal point two steps right 0.268 × 10 = ....................
1.205 × 10 = ....................
5.624 × 1000 = 5624 Shift decimal point three steps right 12.63 × 100 = ....................
256.2 × 100 = ....................
Example : I have to shift 0.026 × 10 = ....................
decimal point two 5.125 × 100 = ....................
Multiply: 0.374 × 100 0.016 × 100 = ....................
Solution: steps right. 0.0125 × 1000 = ....................
0.374 × 100 7.1723 × 1000 = ....................
= 37.4 72.65 × 10 = ....................

Multiplication of decimals by a whole numbers

Let's learn multiplication of a decimal number by a whole number.
Example :

7.432 • Perform the multiplication as if we are multiplying
× 6 the two whole numbers.
44.592
• Put the decimal point in the product to get as many
decimal places in the multiplicand.

Example : As there are 3 digits after the
Multiply: 3.034 × 6 decimal point in multiplicand, I
have to put decimal after 3 digits
9.034 counting from the right in the
× 6
product.

54.204

Multiplication of decimal by another decimals

Let's learn the multiplication of two decimal numbers with the help of given
example.

Multiply: 9.85 × 4.6 • Perform the multiplication as if we are multiplying
Solution: 9.85 two whole numbers.

× 4.6 • Put the decimal point in the product to get as
5910 many decimal places in the product as there are in
39400 multiplicand and multiplier altogether.
45.310

Oasis School Mathematics Book-5 133

Example:
One kg of apples costs Rs 45.25. What is the cost of 9 kg of apples?

Solution:
Cost of 1 kg apples = Rs 45.25
Cost of 9 kg apples = Rs 45.25 × 9
Now,

45.25
× 9
407.25

\ Cost of 9 kg of apples = Rs 407.25.

Exercise 6.14

1. Multiply: b. 0.314 × 10 c. 0.014 × 10
a. 0.36 × 10 e. 6.37 × 10 f. 3.074 × 10
d. 5.12 × 10 h. 0.1082 × 100 i. 5.172 × 100
g. 0.367 × 100 k. 17.853 × 100 l. 0.0842 × 1000
j. 16.872 × 100 n. 5.34 × 1000 o. 6.37 × 1000
m. 5.3164 × 1000
d. 3.5 × 4
2. Multiply: h. 3.824 × 8
l. 8.16 × 18
a. 0.7 × 5 b. 0.8 × 4 c. 2.6 × 3 p. 62.854 × 27
g. 1.372 × 7
e. 4.7 × 5 f. 0.821 × 6 k. 6.52 × 12
o. 3.378 × 36
i. 4.721 × 9 j. 14.82 × 7

m. 12.34 × 16 n. 0.83 × 25

3 . Multiply: b. 1.2 × 0.4 c. 2.4 × 0.6 d. 1.8 × 0.8
a. 0.2 × 0.5 f. 3.4 × 0.25 g. 5.8 × 0.16 h. 3.957 × 0.9
e. 5.6 × 0.3 j. 0.0004 × 0.03 k. 4.371 × 0.5 l. 8.215 × 2.4
i. 75.9 × 2.3

4. a. If 1 kg of sugar costs Rs 55.35, what is the cost of 10 kg of sugar?
b. If a bag contains 45.63 kg of rice, how much rice is contained in 100 such bags?

5. a. The cost of 1 m cloth is Rs 150.65. What is the cost of 9 m cloth?
b. The cost of 1 kg of rice is Rs. 45.55. What is the cost of 0.34 kg of rice?

134 Oasis School Mathematics Book-5

6. Find the area of given rectangles:

a. 5.6 cm b. 4.5 cm

3 cm 2.5 cm

Division of decimals: Answers: Consult your teacher.

Division of decimals by 10, 100 and 1000:

We already know that

2 ÷ 10 = 2 = 0.2 (two - tenths) Shift decimal point one step left.
10
3 Shift decimal point two steps left.
3 ÷ 10 = 10 = 0.3 (three- tenths)

5 ÷ 100 = 5 = 0.05 (five - hundredths)
100
12
12 ÷ 100 = 100 = 0.12 (twelve - hundredths)

Example :

4.543 ÷ 10 = 0.4543 While dividing a number by 10, shift decimal
36.18 ÷ 10 = 3.618 point one step left.
23.16 ÷ 100 = 0.2316
5614.34 ÷ 1000 = 5.61434 While dividing a number by 100, shift decimal
point two steps left.

While dividing a number by 1000, shift
decimal point three steps left.

Division of decimals by a whole number:

Let's learn the division of a decimal number by a whole number,

Divide 12.645 by 5

Solution:

5 ) 12.645 ( 2.529 Division of the decimal number by a
- 10 whole number is same as the division
26 of a whole number by a whole number.
- 25 Just we keep decimal after dividing the
14 whole number.
- 10
45
- 45
0

\ 12.645 ÷ 5 = 2.529

Oasis School Mathematics Book-5 135

Division of a decimal number by another decimal number:

Let's learn to divide a decimal number by another decimal number.
Example :

Dividing 5.22 ÷ 0.9

Here, 5.22 = 5.22 × 10 = 52.2
0.9 0.9 × 10 9

Now, 9 ) 52.2 ( 5.8 • Make divisor a whole number multiplying
both numerator and denominator by
- 45 appropriate number.

72 • Divide as in the division of decimal number
by a whole number.
- 72

0

\ 52.2 ÷ 9 = 5.8

Exercise 6.15

1. Divide:

a. 16 ÷ 10 b. 523 ÷ 10 c. 23.16 ÷ 10 d. 628.24 ÷ 10
g. 726 ÷ 100 h. 57.68 ÷ 100
e. 738.06 ÷10 f. 18 ÷ 100 k. 7.62 ÷ 100 l. 5.632 ÷ 100
o. 15.3 ÷ 1000 p. 17.82 ÷ 1000
i. 723.84 ÷ 100 j.882.57 ÷ 100 s. 235.894 ÷ 1000 t. 573.87 ÷ 1000

m. 0.8234 ÷ 100 n. 28 ÷ 1000

q. 1.6543 ÷ 1000 r. 0.25 ÷ 1000

2. Divide:

a. 0.42 ÷ 2 b. 0.45 ÷ 9 c. 0.56 ÷ 7 d. 1.26 ÷ 2

e. 8.42 ÷ 2 f. 1.255 ÷ 5 g. 0.728 ÷ 8 h. 13.2 ÷ 12

i. 5.304 ÷ 12 j. 51.2 ÷ 16

3. Divide:

a. 5.1 ÷ 0.3 b. 1.25 ÷ 2.5 c. 0.216 ÷ 0.6 d. 8.64 ÷ 0.24

e. 3.35 ÷ 0.05 f. 7.385 ÷ 3.5 g. 57.6 ÷ 1.5 h. 0.765 ÷ 0.17

i. 4.84 ÷ 0.11

4. a. A rope of 3.92 m is cut into 4 equal parts. What is the length of each part?

b. The cost of 5 pens is Rs 62.75. What is the cost of a pen?

c. How many pieces of wire each of 3.5 cm can be cut from a wire of
17.5 cm?

136 Oasis School Mathematics Book-5

Answers :
1. Consult your teacher 2. a) 0.21 b) 0.05 c) 0.08 d) 0.63 e) 4.21 f) 0.251

g) 0.091 h) 1.1 i) 0.442 j) 3.2 3. a) 17 b) 0.5 c) 0.36 d) 36 e) 67
f) 2.11 g) 38.4 h) 4.5 i) 44 4. a) 0.98 b) 12.55 c. 5

Rounding off decimal numbers
Rounding off means replacement of a number by another convenient number
which is easy to understand and is close to the original number.

We round off the decimal numbers nearest to ones, tenths, hundredths and
so on.

Rounding off to the nearest ones (whole number)
For rounding off to the nearest ones, see the digit in the tenths place. If it is less
than 5 (1, 2, 3, 4), the digit in the ones place remains same and the digit in the
tenths place is replaced by 0.
If the digit in the tenths place is 5 or more than 5 (5, 6, 7, 8, 9), 1 is to be added
to the ones place and the digit in the tenths place is replaced by zero.

Look at the given examples and get the idea of it. My focus should be
Round off to the nearest ones: on tenths place.
a. 13.6 b. 5.2 c. 133.5
Solution:

a. 13.6
The digit in the tenths place = 6, which is greater than 5. So we have to

add 1 in ones place and replace the tenths place by 0.
\ 13.6 to the nearest ones = 14.
b. 5.2
Digit in the tenths place = 2, which is less than 5. Now ones digit remains

same and tenths digit is replaced by ‘0’.
\ 5.2 to the nearest ones = 5
c. 133.5
Digit in the tenths place = 5
Now, we have to add 1 in ones place and replace tenths place by ‘0’.
\ 133. 5 to the nearest ones = 134.

Oasis School Mathematics Book-5 137

Class Assignment

Round off to the nearest ones:

15.7 = 12.1 = 16.5 =
23.1 =
18.3 = 11.7 =

28.6 = 15.8 =

Rounding off to the nearest tenths (one decimal place)
For rounding off to the nearest tenths, see the digit in the hundredths place.

• If the digit in the hundredths place is less than 5 (1, 2, 3, 4), tenths place remains
same and hundredths place is replaced by ‘0’.

• If the digit in the hundredths place is 5 or more than 5, 1 is to be added in the
tenths place and hundredths place is replaced by ‘0’.

Example :

Round off to the nearest tenths (one decimal place).
a. 4.62 b. 7.48 c. 13.85

Solution:

a. In 4.62, the digit in the hundredths place = 2
which is less than 5
So, we have to keep tenths place same, replace hundredths place by zero.
\ 4.62 to the nearest tenths = 4.6
b. In 7.48, the digit in hundredths place = 8
which is greater than 5.
So we have to add 1 in tenths place and replace hundredths place by zero.
\ 7.48 to the nearest hundredths = 7.5
c. In 13.85, the digit in hundredths place = 5.
So, we have to add 1 in tenths place and replace hundredths place by zero.
\ 13.85 to the nearest tenths = 13.9

Class Assignment

Round off to the nearest tenths:

3.64 = 56.21 = 28.67 =
87.18 =
63.62 = 84.26 =

88.88 = 57.32 =

138 Oasis School Mathematics Book-5

Rounding off to the nearest hundredths (2 decimal place)
For rounding off to the nearest hundredths, we have to see the digit in
thousandths place.

Example :

Round off to the nearest hundredths (2 decimal places).
a. 15.846 b. 19.812 c. 15.565
Solution:
a. In 15.846, digit in thousandths place = 6.
Which is greater than 5.
\ 15.846 to the nearest hundredths = 15.85
b. In 19.812, digit in thousandths place = 2
Which is less than 5.
\19.812 to the nearest hundredths = 19.81
c. In 15.565, digit in thousandths place = 5
\ 15.565 to the nearest thousandths = 15.57

Class Assignment

Round off to the nearest hundredths:

17.864 = 93.914 = 28.612 =

289.127 = 37.378 = 315.315 =

876.129 = 6.218 =

Exercise 6.16

1. Round off each of the following to the nearest ones (whole
number):

a. 13.3 b. 18.6 c. 132.5 d. 17.8 e. 26.3

f. 15.21 g. 18.57 h. 198.324

2. Round off each of the following to the nearest tenths (one decimal
place):

a. 19.42 b. 27.86 c. 65.45 d. 17.87 e. 13.93

f. 121.814 g. 352.942 h. 0.573

Oasis School Mathematics Book-5 139

3. Round off each of the following to the nearest hundredths (two decimal
place):

a. 5.632 b. 18.657 c. 19.865 d. 0.846 e. 132.972

f. 51.6827 g. 74.3759 h. 213.8634

4. Round off each of the following to the nearest decimal places given in
the brackets:

a. 0.846 (whole number) b. 7.84 (1) c. 13.76 (1)

d. 153.816 (2) e. 0.9763 (2) f. 57.8461 (2)

Answers: Consult your teacher.

Objective Questions

Colour the correct alternatives:

1. Which one of the following is an improper fraction?

1 1 2 9
3 5 7

2. The sum of 5 1 and 732 is 13
3

12 13 2
3

3. Value of 3 2 × 4 1 is equal to
5 6

12 2 14 5 14
30 30

4. 4 of 3 is equal to
5 4

3 16 3
5 15 16

1 2 3
5. 6 ÷ 3 is equal to 6

18

140 Oasis School Mathematics Book-5

6. 1 ÷ 5 is equal to
2

5 2 1
2 5 10

31 1 6 3
7. 5 ÷ 2 of 5 is equal to 25 50

6 0.3 0.03

8. 3 is equal to 2.30 2.003
100
1.506 1506
3

9. 2.3 is equal to
2.03

10. 15.06 × 10 is equal to
150.6

11. 3.125 ÷ 100 is equal to

0.3125 0.03125 312.5

12. Rounding off 2.6 to its nearest whole number is
3 2.7 2.64

Number of correct answer

Oasis School Mathematics Book-5 141

Unit Test Full marks: 34

1. Compare the given fractions: 2
2
a. 1 and 49 b. 3 and 5 2
2 5 6 3
3
2. Convert the given improper fractions into the mixed number: 4
a. 12 b. 11
56

3. Add or subtract and reduce them into lowest term if necessary:

a. 5 + 32 b. 518 – 2 3
6 4

4. Multiply:

a. 3 × 16 b. 1 × 3 c. 14 × 1
9 2 4 7

5. Divide:

a. 5 ÷ 1 b. 12 ÷ 2 c. 2 ÷ 5
2 5 3 7

6. Simplify:

a. o3 1 – 2 1 ÷ 3 1 b. 1 – 3 – 1 + 1 – 1
2 3p 2 2 q4 s2 s4 8tr

7. Show the number 37.82 in place value chart: 2

8. Write the number 63.76 in the expanded form: 2
1.5 × 2 = 3
9. Add or subtract:
a.18.64 + 15.736 + 5.9 b. 200.26 - 97.865

10. Multiply: 4×1=4
a.0.468 × 10 b. 3.149 × 100 c. 1.372 × 7 d. 3.95 × 0.9

11. Divide: 4×1=4
a.18 ÷ 10 b. 0.56 ÷ 7 c. 0.216 ÷ 0.6 d. 15.3 ÷ 100

12. Round off the following to the nearest decimal places: 3×1=3
a. 0.837 (whole number) b. 13.76 (1) c. 57.8461 (2)

Marks
142 Oasis School Mathematics Book-5

UNIT Percentage, Unitary

7 Method, Ratio, Simple
Interest, Profit and Loss

12 Estimated Teaching Hours: 10
93

6

Contents • Percentage
– Relation of percentage with decimal and

fraction
– Value of given percent of given quantity
• Unitary method
• Simple interest
• Profit and loss
• Ratio

Expected Learning Outcomes

Upon completion of this unit, students will be able
to develop the following competencies:

• To convert decimal and fraction into percentage and convert
percentage into decimal and fraction

• To find the value of given percent of a given quantity
• To find the unit value from the value of many items and to

find the value of many items from the unit value
• To find the simple interest of the given sum using unitary

method
• To find profit or loss if cost price (CP) and selling price (SP)

are given
• To find the ratios of two quantities

Materials Required : Graph sheet, glue, etc.

Oasis School Mathematics Book-5 143

Percent Per cent

Look and learn.

For every Hundred

Percent is composed of two words per and cent. Per means for every and cent
means out of hundred.

So percent means out of hundred.

I got 90% marks I got 85% marks

Anasuya got 90 marks out of 100 Nova got 85 marks out of 100 full
full marks. marks.

The symbol of percent is % or p.c.

Relation of percent and fraction:

In the given figure, a square is divided into 100

equal parts. 18

Fraction represented by the shaded parts = 100
Percent of shaded part = 18%.
82
Fraction represented by non shaded part = 100

Percent of non-shaded part = 82%.

82 = 82%, 25 = 25%, 60 = 25%,
100 100 100

Class Assignment

1. Find the fraction and percentage of shaded part of the given figures
having 100 squares:

a. b.

Fraction =

Fraction = Percentage
=
Percentage
=

144 Oasis School Mathematics Book-5

2. a. 65 students out of 100 are boys. What are there percentage of boys in the class?
b. Pooja scored 85 out of 100 full marks. What percent did she score?

3. Express the following fractions in percentage:

a. 3 = b. 15 = c. 60 = d. 75 = e. 86 =
100 100 100 100 100

Conversion of percentage into fraction:

To convert percent into fraction, we follow the
following steps.

20% = 1 20 = 1 • Remove the symbol %.
100 5 5 • Divide it by 100.
• Simplify the fraction into its lowest term.
55% = 11 55 = 11
100 20 20

Conversion of percentage into decimal:
To convert percentage into decimal, follow the following steps.

Example :

50% = 50 = 0.5 • Remove the symbol %.
100 • Divide it by 100.
• Convert the fraction into decimal.
85% = 85 = .85
100

Conversion of fraction into percentage:
To convert fraction into percentage, follow the following steps:

Example :

2 = 2 × 100% = 40% • Multiply the fraction by 100.
5 5 • Put the symbol %.
3 3
10 = 10 × 100% = 30%

Conversion of decimal into percentage:
To convert decimal into percentage, follow the following steps:
Example :

0.23 = 0.23 × 100% = 23% • Multiply the decimal by 100.
0.45 = 0.45 × 100% = 45% • Put the symbol %.

Oasis School Mathematics Book-5 145

Example :

Himal scored 20 out of 25 full marks. Find the percent of his marks.

Solution:

Marks obtained by Himal = 20 out of 25
4 20 20
His percentage = 255 = 4 × 20%
× 100% = 80%

\ He scored 80% marks.

Exercise 7.1

1. Express the following percentage in fraction:

a. 20% b. 25% c. 30% d. 50% e. 35%

f. 65% g. 40% h. 55% i. 75%

2. Express the following percentage in decimal:

a. 25% b. 5% c. 50% d. 56% e. 72%

3. Express the following fractions in percentage:

a. 3 b. 2 c. 3 d. 4 e. 3
10 5 4 20 50

f. 7 g. 19 h. 18 i. 7
10 20 50 20

4. Express the following decimal in percentage:

a. 0.03 b. 0.26 c. 0.85 d. 0.87 e. 0.92

5 . Write the fraction of the given shaded figures and convert them into
percentage:

a. b. c.

6. Marks obtained by 5 students of class V out of full marks 25 are given
below. Express their marks in percentage.

Sumi : 18

Ruby : 22

Ranjan : 15

Asmin : 17 Answers: Consult your teacher
Pratik : 13

146 Oasis School Mathematics Book-5

To find the value of percentage of the given quantity:

To find the value of the percentage of given quantity, we follow the following steps.

Example : Steps:
Find the value of 15% of 300. • Express the given percentage in fraction.
Solution:
15% of 300 • Multiply the fraction by given quantity.

= 15 × 300
100

= 15 × 3

= 45

Example :

Out of 120 students in a class, 45% students are boys. Find the number of girls.

Solution:

Total students = 120

Percentage of boys = 45%

Number of boys = 45 9 × 1206

100 20

= 54

Number. of girls = 120 - 54

= 66

Exercise 7.2

1. Find the value of:

a. 20% of 180 b. 5% of 120 c. 15% of 600 d. 30% of 300

e. 40% of 240 f. 25% of 700 g. 60% of 200 h. 45% of 400

2. a. In a class, there are 40 students. If 10% students are absent, find the
number of absent students.

b. Out of Rs. 120, a boy had, spent 40%. How much money did he spend?

c. There are 900 students in a school. If 45% of them are girls, find the number
of girls.

d. A man earns Rs 4000 in a month and he spends 20% of his income on food.
Find how much money does he spend on food.

Answers : c) 90 d) 90 e) 96 f) 175 g) 120 h) 180
1. a) 36 b) 6 c) 405 d) 800
2. a) 4 b) 48

Oasis School Mathematics Book-5 147

Unitary Method

Unit means one. The word unitary is derived from the word unit. In this
chapter, we will learn the process of finding the cost of a unit from the cost of
many items and vice versa.

To find the values of many items when the value of unit is given:

Cost of 1 apple = Rs 7

Cost of 3 apples = Rs 7 + Rs 7 + Rs 7 Value of many items = unit value x
= Rs 3 x 7 number of items.

= Rs 21
Example :

Cost of a pen is Rs 25. What is the cost of Remember ! = Rs 6
8 such pens? = Rs 2x6 = Rs 12
• Cost of 1 copy = Rs 3x6 = Rs 18
Solution: • Cost of 2 copies = Rs 4x6 = Rs 24
• Cost of 3 copies
Cost of a pen = Rs 25 • Cost of 4 copies

Cost of 8 pens = Rs 8 × 25

= Rs 200

Class Assignment

Complete the given table.

Articles Unit cost Cost of 2 units Cost of 3 units
Pencil Rs 5 Rs 2 x 5 = Rs. 10 Rs. 3 x 5 = Rs 15
Apple Rs 6
Orange Rs 7
Copy Rs 12

To find the value of one item, if the value of many items is given:

Cost of 4 pens is Rs 40

Cost of 1 pen is Rs 40 ÷ 4 Value of one item = value of many
= Rs 10 items ÷ number of item.

Example :

Cost of 15 oranges is Rs 75. What is the cost of one orange?

Solution:

Cost of 15 oranges = Rs 75

Cost of 1 orange = Rs (75 ÷ 15)

= Rs 5

148 Oasis School Mathematics Book-5

Class Assignment Total cost Unit cost
Rs. 360 Rs 360 ÷ 3 = Rs 120
Complete the given table. Rs 540
Rs 512
Number of articles Rs 732
3
5
8
12

Further problems related to unitary method:

In our daily life we also face different the situations together, where we have

to find unit value as well as the value of many items together.
Example :

The cost of 6 copies is Rs 90. Find the cost of 8 copies.

Solution:

Cost of 6 copies = Rs 90 6)90(15 Remember !
-6 15 • First we have to find the
Cost of 1 copy = Rs 90 ÷ 6 30 × 8 unit value.
–30 120 • Then multiply the unit
= Rs 15 0
value by the number of
Cost of 8 copies = Rs 8 × 15 items.

= Rs 120

Class Assignment

Complete the given table.

Number of Total cost Unit cost Cost of 3 Cost of 5
articles 48÷ 2 = 24 articles articles
2 48
4 80 24 × 3 = 72 24 x 5=120
6 72
8 144

Exercise 7.3

1. a. If the cost of an article is Rs 20, what is the cost of 12 articles?
b. If the cost of a pencil is Rs 10, what is the cost of a dozen pencils?
c. A story book has 124 pages. How many pages are there in 6 such books?
d. A bus runs 35 km in one hour. How far will it run in 12 hours?

Oasis School Mathematics Book-5 149

2. a. If the cost of 5 articles is Rs 250, what is the cost of an article?
b. The cost of 8 balls is Rs 192. What is the cost of a ball?
c. A bus covers 384 km in 12 hours. How many kilometers does it cover in

one hour?
d. 16m of clothes costs Rs 480. Find the cost of 1 m cloth.

3. a. If 3 dozen bananas cost Rs 150, what is the cost of 5 dozen bananas?
b. If 15kg of rice costs Rs 480, what is the cost of 18kg of rice?
c. A man earns Rs 1250 in 5 days. How much does he earn in 12 days?
d. A pipe can fill 320l of water in 40 minutes. How much water does it fill in

1 hour?

Answers :
1. a) Rs. 240 b) Rs. 120 c) Rs. 744 d) 420 km 2. a) Rs. 50 b) Rs. 24
c) 32 km d) Rs. 30 3. a) Rs. 250 b) Rs. 576 c) Rs. 3000 d) 480 litre

Simple Interest Sunayana kept Rs. 2000 in a bank

Aayush kept Rs. 3000 in a bank

After 1 year Sunayana gets Rs 2120 from the bank and and Aayush gets Rs
3180 from the bank.

How much extra money does Sunayana get? Rs 120

How much extra money does Aayush get? Rs 180

The money deposited by them is the principal.

The extra money they get is the interest.

\ For Sunayana For Auyush

Principal = Rs 2000 Principal = Rs 3000

Interest = Rs 120 Interest = Rs 180

Here, Interest of Rs 2000 in 1 year is Rs 120 I understand ! Rs 180 is the
interest of Rs 3000 in 1 year
Rs 120
Interest of Re 1 in 1 year is 2000

Interest of Rs 100 in year is Rs 120 × 100
2000

= Rs 6

Since the interest of Rs 100 in 1 year = Rs 6, rate of interest (R) = 6%.

150 Oasis School Mathematics Book-5