Vedanta Excel in Mathematics Book 6 Final (2078)

Properties of Whole Numbers

b) Cube of 20 = 203 I got it !
= 20 × 20 × 20 = 8000 In 600 × 600 × 600, I
should first find
c) Cube of 500 = 5003 6 × 6 × 6 = 216. Then, I
= 500 × 500 × 500 should write six zeros at
=125000000 the end of 216.

Example 3 : Find the square roots of a) 144 b) 3600

Solution :

a) Factorising 144, 2 144
2 72
2 36
2 18
39

3

? 144 = 2 × 2 × 2 × 2 × 3 × 3 Interesting!
= 22 × 22 × 32

? 144 = 2 × 2 × 3 = 12 In 3600 , I should first find
b) 36 = 62 the square root of 36. Then, I
should write half number of
zeros at the end of square root.

? 36 = 6 so, 36 = 6 and 3600= 60

? 3600 = 60

Example 4 : Find the cube root of a) 1728 b) 64000000
Solution :
a) Factorising 1728:

? 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
= 23 × 23 × 33

? 3 1728 = 2 × 2 × 3

= 12

b) 64 = 43 I got it !
In 64000000 , I should first find the
? 3 64 =4 cube root of 64. Then I should write
? 3 64000000 = 400 one-third number of zeros of 64000000
at the end of cube root of 64.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 51 Vedanta Excel in Mathematics - Book 6

Properties of Whole Numbers

Example 5 : 15/15 rose plants are planted along the length and breadth of a
square garden. How many plants are there in the garden ?

Solution :
Here, the number of plants in the garden = 152 = 15 × 15

= 225
Hence, there are 225 rose plants in the garden.

Example 6 : The product of two identical numbers is 2025. Find one of the
numbers.

Solution:
Here, one of the numbers is the square root of 2025.

Now, factorising 2025

? 2025 = 3 × 3 × 3 × 3 × 5 × 5
= 32 × 32 × 52

? 2025 = 3 × 3 × 5
= 45

Hence, the required number is 45.

Example 7: Find the smallest number by which when 18 is multiplied, it

becomes a perfect square.

Solution:

Factorising 18, Interesting !

2 18 When 2 × 32 is multiplied
39 by 2 it becomes 22 × 32
which is a perfect square.
3

? 18 = 2 × 3 × 3 = 2 × 32
Hence, the required smallest number is 2.

Example 8: If the length of a cubical block is 12 cm, find its volume.
Solution:
Length of the cubical block (l) = 12 cm.
Volume of the cubical block = l3 = (12 cm)3

= 12 cm × 12 cm × 12 cm
= 1728 cm3.
Thus, the required volume of the cubical block is 1728 cm3

Vedanta Excel in Mathematics - Book 6 52 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Properties of Whole Numbers

Example 9: If the volume of a cube is 3375 cm3, find its length.
Solution:
Here, the length of the cube is the cube root of 3375.
factorising 3375,

3 3375 ? 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 33 × 53
3 1125 ? 3 3375 = 3 × 5 = 15
3 375
5 125 Hence, the required length of the cube is 15 cm.
5 25

5

EXERCISE 3.4

General Section A – Classwork

1. Let’s study the following multiplication table and find the squares of
the following numbers.

× 1 2 3 4 5 6 7 8 9 10
1 1 2 3 4 5 6 7 8 9 10
2 2 4 6 8 10 12 14 16 18 20
3 3 6 9 12 15 18 21 24 17 30
4 4 8 12 16 20 24 28 32 36 40
5 5 10 15 20 25 30 35 40 45 50
6 6 12 18 24 30 36 42 48 54 60
7 7 14 21 28 35 42 49 56 63 70
8 8 16 24 32 40 48 56 64 72 80
9 9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100

a) 12 = ............ 22 = ............ 32 = ............ 42 = ............

b) 52 = ............ 62 = ............ 72 = ............ 82 = ............

c) 92 = ............ 102 = ............ 02 = ............

2. Let’s investigate the idea from the example and tell and write the squares
of these numbers as quickly as possible.

Example: 22 = 4, 202 = 400, 2002 = 40000, 20002 = 4000000

Here, 20 has one zero and its square (400) has two zeros.
200 has two zeros and it's square (40000) has four zeros.

Similarly, 2000 has three zeros and its square (4000000) has six zeros.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 53 Vedanta Excel in Mathematics - Book 6

Properties of Whole Numbers

a) 32 = ............ 302 = ............ 3002 = ............ 30002 = ............

b) 42 = ............ 402 = ............ 4002 = ............ 40002 = ............

c) 52 = ............ 502 = ............ 5002 = ............ 50002 = ............

d) 62= ............ 602 = ............ 6002 = ............ 60002 = ............

3. Let’s tell and write the square roots of the following numbers as quickly as
possible.

a) 4 = ............ b) 9 = ............ c) 16 = ............ d) 25 = ............

e) 36 = ............ f) 64 = ............ g) 100 = ............ h) 81 = ............

4. Let’s investigate the idea from the example. Then, tell and write the square

roots of these numbers as quickly as possible.

4 = 2, 200 = 20 Interesting!
40000 = 200 400 has two zeros and its square root has one
4000000 = 2000 zero. 40000 has four zeros and its square root
has two zeros.

a) 9 = ........... 900 = ........... 90000 = ........... 9000000 = ...........

b) 16 = ........... 1600 = ........... 160000 = ........... 16000000 = .........

c) 25 = ........... 2500 = ........... 250000 = ........... 25000000 = .........

d) 36 = ........... 3600 = ........... 360000 = ........... 36000000 = .........
5. Let’s tell and write the cubes of these numbers as quickly as possible.

a) 13 = ............ 203 = .............. 2003 = ..................

b) 33 = ........... 303 = .............. 3003 = ..................

c) 43 = ........... 403 = .............. 4003 = ..................

d) 53 = ........... 503 = .............. 5003 = ..................

Creative Section - A

6. Let’s find the squares of each number.

a) 14 b) 25 c) 42 d) 90 e) 115 f) 284

7. Let’s factorise each number. Then identify the perfect square numbers.

a) 9 b) 16 c) 45 d) 144 e) 220 f) 324

Vedanta Excel in Mathematics - Book 6 54 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Properties of Whole Numbers

8. Let's find the square roots of these numbers.

a) 36 b) 81 c) 196 d) 225 e) 324 f) 400
k) 2304 l) 3025
g) 576 h) 729 i) 1225 j) 1764

9. Let’s find the cube of these numbers.

a) 4 b) 7 c) 9 d) 12 e) 25 f) 36

10. Let’s factorise each number. Then, identify whether the
numbers are perfect cube or not.

a) 27 b) 54 c) 64 d) 81 e) 216 f) 512

11. Let's find the cube roots of these numbers.

a) 8 b) 27 c) 125 d) 216 e) 729 f) 1728

Creative Section - B

12. a) 25/25 plants are planted along the length and breadth of a square
garden. How many plants are there in the garden ?

b) Seats are arranged in 32 rows and 32 columns in a film hall. How many
seats are there in the hall altogether ?

c) There are 45 students in class VI. If every student donates the same
amount of money as their number to flood victims, how much money
do they donate altogether ?

13. a) If the product of two identical numbers is 441, find one of the
numbers.

b) In an afforestation programme, every student planted as many plants as
their number. If they planted 576 plants altogether, how many students
took part in the programme ?

14. a) If the product of three identical numbers is 1000, find one of the
numbers.

b) If the volume of cubical dice is 216 cm3, find its length.

15. Find the smallest number by which the following numbers are multiplied
to make them perfect square.

a) 50 b) 75 c) 80 d) 108 e) 175 f) 245

It’s your time - Project Work

Funs of square numbers and square roots
16. Squares of a number with the digit at ones place being 5.

152 = 225 52 = 25, 1 + 1 = 2 and 2 × 1 = 2. So, 152 = 225
252 = 625 52 = 25, 2 + 1 = 3 and 3 × 2 = 6. So, 252 = 625
352 = 1225 52 = 25, 3 + 1 = 4 and 4 × 3 = 12. So, 352 = 1225

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 55 Vedanta Excel in Mathematics - Book 6

Properties of Whole Numbers

Let's investigate the rules from the above illustrations. Tell and write the
squares of these numbers as quickly as possible.

a) 452 = ............ b) 552 = ........... c) 652 = ..............

d) 752 = ........... e) 852 = ........... f) 952 = ..............

17. 22 = 4 = 1 + 2 + 1 32 = 9 = 1 + 2 + 3 + 2 + 1

Investigate the rules from the above illustrations. Then, complete these as
quickly as possible.

a) 42 = 16 = .....................................................................................................

b) 52 = 25 = .....................................................................................................

c) 62 = 36 = .....................................................................................................

d) 72 = 49 = ....................................................................................................

18. Let’s complete the given chart and investigate the fact.

Number Factor No. of factors Remark

11 1 Odd

2 1, 2 2 Even

3 1, 3 2 Even

4 1, 2, 4 3 Odd

5 1, 5 2 Even

6 ................ ................ ................

7 ................ ................ ................

8 ................ ................ ................

9 ................ ................ ................

10 ................ ................ ................

From the above chart, let’s answer the following questions and draw out the
conclusion.
a) List out the numbers which have exactly two factors.
b) List out the numbers which have even number of factors but more than

two factors.
c) List out the numbers which have odd number of factors.
d) List out the square numbers from the chart.
e) Draw out your conclusion about the factors of a square numbers.

Vedanta Excel in Mathematics - Book 6 56 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Unit Integers

4

4.1 Integers - Introduction

Let’s use a positive or negative number to represent the following
conditions.

a) a profit of Rs 20. …................

b) a loss of Rs 5. …................
c) 18° C above 0° C. …................
d) 2° C below 0° C. …................
e) Taking 5 steps forward …................
f) Taking 3 steps backwards …................
g) a drop of Rs 25 in the price of L P gas. …................
h) a deposit of Rs 5000 in the bank account. …................

We have already learned about the following sets of numbers.
N = {1, 2, 3, 4, 5, …} is the set of natural numbers.
W = {0, 1, 2, 3, 4, 5, …} is the set of whole numbers.

Now, let’s consider any two whole numbers 4 and 9.

Here, 4 + 9 = 13 (13 is a member of the set of whole numbers)

9 – 4 = 5 (5 is a member of the set of whole numbers)
9 u 4 = 36 (36 is a member of the set of whole numbers)

But, now let’s try to subtract 9 from 4. Is it possible? Look at the following number line.

From the number line, 4 – 9 = – 5

Here, – 5 is not a member of the set of whole numbers. 9
Then, in which set does – 5 belong? 4

– 5 belongs to the set of Integers. 5

Thus, the set of all numbers both positive and negative including zero (0) is

called the set of integers.

The set of integers is denoted by the letter ‘Z’.

The number line given below represents the integers.

Z = {…, – 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5, …} is the set of integers.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 57 Vedanta Excel in Mathematics - Book 6

Integer

On the number line, the numbers towards right form 0 (zero) are positive
integers. Thus Z+ = {+ 1, +2, +3, +4, +5, ...} is the set of positive integers. The
positive integers are always greater than 0.

On the other hand, the numbers towards left from 0 (zero) are negative integers.
Thus, Z– = {–1, –2, –3, –4, –5, …} is the set of negative integers. The negative
integers are always less than 0. So, –1, –2, –3, etc. are less than 0.

4.2 Operations on integers

It is easier to understand the fundamental operations (addition, subtraction,
multiplication and division) of integers on number lines. Let’s study and learn
from the following illustrations.

1. Addition rule

(i) The positive integers are always added. The sum holds the positive (+)
sign.

For example, I investigated the rule
(a) (+ 2) + (+ 5) = 2 + 5 = 7 (+) + (+) Add = + sum
(+ 3) + (+ 4) = + 9
(+2) + (+5)

+2 +5

(b) (+ 3) + (+ 4) + (+ 2) = 3 + 4 + 2 = 9

(+3) + (4) + (2)

+3 +4 +2

(ii) The negative integers are always added. I also investigated!
(–) + (–) Add = – sum
The sum holds the negative (–) sign. (– 2) + (– 6) = – 8
For example,
(a) (– 4) + (– 5)= – 4 – 5 = – 9

Vedanta Excel in Mathematics - Book 6 58 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Integer

(b) (– 3) + (– 2) + (– 6) = – 3 – 2 – 6 = – 11

(iii) The positive and negative integers are always subtracted.
The difference holds the sign of the bigger integer.

For example, + 8 is bigger. So, the
(a) (+ 8) + (– 3) = 8 – 3 = 5 difference 5 holds
positive (+) sign.

+8
–3

(b) (– 9) + (+ 5) = – 9 + 5 = – 4 –9 is bigger. So, the
difference 4 holds
–9 negative (–) sign.
+5

2. Multiplication and division rules
(i) The product or quotient of two positive integers is always positive.

For example, I can remember now!
(a) (+ 3) u (+ 2) = 3 u 2 = 6 (+) × (+) = + product
(+) ÷ (+) = + quotient

(b) (+ 8) ÷ (+ 4) = 8 ÷ 4 = 2 (+8) ÷ (+4)

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 59 1 time

Vedanta Excel in Mathematics - Book 6

Integers

(ii) The product or quotient of a positive and a negative integers is always
negative.

For example, I can remember!
(+ 4) u (– 2) = (– 4) u (+ 2) = – 8 (+) × (–) = (–) × (+) = – product
(+ 15) ÷ (– 3) = (– 15) ÷ (+ 3) = – 5 (+) ÷ (–) = (–) ÷ (+) = – quotient

(iii) The product or quotient of two negative integers is always positive.

For example, It's interesting!
(– 5) u (– 3) = 15 (–) × (–) = + product
(– 12) ÷ (– 4) = 3 (–) ÷ (–) = + quotient

Worked-out examples

Example 1: Simplify a) (+ 5) + (– 2) + (+ 8) + (– 6)

b) (– 3) u (+ 9) u (– 2)

Solution:

a) (+ 5) + (– 2) + (+ 8) + (– 6) b) (– 3) u (+ 9) u (– 2)
= (+ 13) + (– 8) = + 5 = 5 = (– 27) u (– 2) = + 54 = 54

Example 2: Evaluate (a) (– 3)2 (b) (– 2)5
Solution:
a) (– 3)2 = (– 3) u (– 3) = + 9 = 9

b) (– 2)5 = (– 2) u (– 2) u (– 2) u (– 2) u (– 2)
= (+ 4) u (+ 4) u (– 2) = (+ 16) u (– 2) = – 32

Vedanta Excel in Mathematics - Book 6 60 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Integers, Rational and Irrational Numbers Integers

EXERCISE 4.1
General Section – Classwork

1. Let’s tell and write the correct answers in the blank spaces.
a) The positive integers lie towards ........................... from 0 (zero) in a
number line.
b) The ........................... integers lie towards left from 0 (zero) in a number
line.
c) Integers between – 3 and + 3 are .............................................................
d) Integers greater than – 2 and less than +1 are ........................................
e) The ascending order of the integers 2, –3, 0, –1, 4 is
....................................................................................................................
f) The descending order of the integers 0, –2, 3, –1, 2, –3, 1 is
....................................................................................................................

2. A loss of Rs 40 is written as – 40. Let’s tell and write the appropriate
integers as quickly as possible.
a) a loss of Rs 15 is .....................
b) a gain of Rs 100 is .....................
c) a drop of Rs 10 in a movie ticket is .....................
d) a rise of Rs 20 in a sack of rice is .....................
e) 16°C above 0° C is .....................
f) 3°C below 0°C is .....................
g) 5 steps forward from a fixed point .....................
h) 2 steps backward from a fixed point .....................

3. Let’s tell and insert appropriate sign > or < between two integers.

a) + 4 +9 b) + 6 +1 c) +2 –3

d) –7 +5 e) –5 –3 f) –2 0

4. Let’s tell and write True of False for the following statements.

a) 0 is less than every positive integer. ........................

b) 0 is a positive integers. ........................

c) 0 is the smallest integer ........................

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 61 Vedanta Excel in Mathematics - Book 6

Integers

d) – 1 is the greatest negative integer ........................

e) The opposite of – 4 is 4 ........................

f) The opposite of +6 is +9 ........................

g) – 5 is greater than 0 ........................

h) The sum of an integer and its opposite is always zero .........................

i) The sum of two negative integers is positive ........................

j) The product of two negative integers in negative ........................

5. Let’s tell and write the correct answers as quickly as possible.

a) 3 more than – 4 = .................. b) – 3 more than – 4 = ..................

c) 3 less than – 4 = .................. d) – 3 less than – 4 = ..................

e) 8 – 5 = ..............., 5 – 8 = ............., – 5 – 8 = ..............

f) – 2 × 5 = ..............., 2 × (– 5) = ............., – 2 × (– 5) = ..............

g) – 16 ÷ 8 = ..............., 16÷ (– 8) = ............... , – 16 ÷ (– 8) = ..............

Creative Section
6. Make mathematical expressions from these number lines, then simplify.

Hint: (– 4) + (– 3) = – 7

Hint: 2 × (– 3) = – 6

Vedanta Excel in Mathematics - Book 6 62 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Integers

7. Simplify. b) (+ 7) + (– 9) c) (– 6) + (– 8)
a) (+ 7) + (– 4) e) (– 9) + (+ 4) + (– 3) f) 2 – 8 – 6 – 11 + 12
d) (+8) + (+ 6) + (– 5) h) (– 4) × (– 6) i) (– 2) ×(– 7) × (– 3)
g) (+3) × (– 5) k) (– 72) ÷ (+ 9) l) (– 63) ÷ (– 7)
j) (+ 18) ÷ (– 6)

8. Evaluate. c) (– 1)4 d) (– 2)3 e) (–2)4 f) (–2)5
a) (– 1)2 b) (– 1)3 i) (– 4)2 j) (– 4)3 k) (–5)2 l) (–5)3
g) (– 3)2 h (– 3)3

9. a) A man runs 10 km due East and A 10 km B
B
then 4 km due West . Find. His

position with respect to his 4 km
starting point.
6 km

Hint: 10 km – 4 km = 6 km East from the starting point.

b) Mrs. Yadhav walks 18 km due North and then 12 km due South. Find
her position with respect to his starting point.

c) Mr. Chamling travels 20 km due East and then 25 km due west. Find
his position with respect to his starting point.

d) Sunayana walks 8 km due East, then turns round and goes 4 km west of
the starting point. Then, she again turns back and returns to the starting
point. What is the total distance travelled by her?

10. a) The temperature of a body first rises by 20° C and then falls by 24° C.
Find the final temperature of the body, if its initial temperature is 10° C.

Hint : Final temperature = 10° + (+ 20°) + (– 24°) = 6° C.

b) The temperature of a body first rise by 25° C and then falls by 30° C.
Find the final temperature of the body, if its initial temperature is

(i) 8°C. (ii) 2°C (iii) 0°C. (iv) – 3° C

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 63 Vedanta Excel in Mathematics - Book 6

Integers

c) The temperature of a body first falls by 15° C and then rises by 20°C.
Find the final temperature of the body, if its initial temperature was :

(i) 28° C (ii) 12° C (iii) 5° C. (iv) – 2°C

It's your time - Game time!
11. Play with one of your friends.

Let’s fill in the addition and multiplication tables separately. Then, compare
your completed tables with your competitor. Winner will be declared if
anyone completes both the tasks as quickly as possible with correct sums.

Addition Table
+ +5 +4 +3 +2 +1 0 –1 –2 –3 –4 –5
+5
+4
+3
+2
+1
0
–1
–2
–3
–4
–5

Multiplication Table
× +5 +4 +3 +2 +1 0 –1 –2 –3 –4 –5
+5
+4
+3
+2
+1
0
–1
–2
–3
–4
–5

Vedanta Excel in Mathematics - Book 6 64 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Unit Fraction and Decimal

5

5.1 Fraction - Looking back
Classwork - Exercise

1. Let’s chose the correct answer and fill in the blanks.

a) 2 , 5 and 8 are ......................... fractions (like / unlike)
9 9 9

b) 1 , 2 and 6 are ........................ fractions (like / unlike)
5 3 7

c) 2 is a .................... fraction. (proper / improper)
5

d) 7 is a ................... fraction. (proper / improper)
4

e) The fraction equivalent to 2 is ................... ( 6 , 8 ).
3 9 15

2. Let’s tell and write the correct answers as quickly as possible.

a) 1 + 2 = ............ b) 5 – 2 = ............ c) 1 × 3 = ............
7 7 9 9 2 5

d) 3 ÷ 1 = ............ e) 1 of Rs 20 = ............ f) 1 of Rs 36 = ............
2 2 4

5.2 Equivalent fractions

Let’s study the fractions of the shaded parts and investigate the idea of equivalent

fractions. 1
2
The fraction of the shaded part is

The fraction of the shaded part is 2
4

The fraction of the shaded part is 3
6

Here, the fractions 1 , 2 and 3 represent the equal portion of the whole rectangle.
2 4 6

So, these fractions are called the equivalent fractions.

If two or more than two fractions represent the equal portion of the whole then
they are said to be equivalent fractions.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 65 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

5.3 To find the fractions equivalent to the given fraction

Let’s consider a fraction 2 .
3
2 2× 2 4
Here, 3 = 3× 2 = 6

2 =23 × 3 = 6
3 × 3 9
2 =32 × 4 8
3 × 4 = 12

Thus, 4 , 6 , 8 , etc. are the fractions equivalent to the given fraction 2 .
6 9 12 3
12
Also, consider a fraction 18 .

Here, 12 = 12 ÷ 2 = 6
18 18 ÷ 2 9
12 12 ÷ 3 4
18 = 18 ÷ 3 = 6

12 = 12 ÷ 6 = 2
18 18 ÷ 6 3

Thus, 2 , 4 , 6 are the fractions equivalent to the given fraction 12 .
3 6 9 18

Thus, the fractions equivalent to the given fraction are obtained by multiplying

or dividing the numerator and the denominator of the given fraction by the same

natural number.

Test of equivalent fractions

Let’s consider any two fractions 3 and 15 .
4 20
To find the cross–product of numerators and denominators:

3 15 I investigated the idea!
4 20 If the cross-products of numerators
and denominators of two fractions
3 u 20 = 4 u 15 are equal, the fractions are
equivalent.
60 = 60

So, 3 and 15 are equivalent fractions.
4 20

5.4 Like and unlike fractions

Let’s look at the denominators of the following groups of fractions and
investigate the idea of like and unlike fractions.

1 , 2 , 4 , 7 , etc. are the like fractions. It's my investigation!
9 9 9 9 Like fractions have the same

2 , 4 , 5 , 2 , etc. are the unlike fractions. denominators, but unlike fractions
3 7 9 11 have different denominators.

Vedanta Excel in Mathematics - Book 6 66 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

5.5 To convert unlike fractions into like fractions

Let’s consider any two unlike fractions 2 and 3 .
3 4
To convert unlike fractions
The L.C.M. of the denominators 3 and 4 is 12 into like fractions.
– Find the L.C.M. of
Here, 12 ÷ 3 = 4 and 2 = 2×4 = 8
3 3×4 12 denominators.
– Divide the L.C.M. by each
And 12 ÷ 4 =3 and 3 = 3×3 = 9
4 4×3 12 denominator.
– Multiply the numerator
Thus, 8 and 9 are the like fractions.
12 12 and denominator of each
fraction by the quotient.

5.6 Comparison of fractions

(a) Comparison of like fractions
Let’s observe the following example and investigate the idea of comparing
the like fraction.
1
5

3
5

2

5

4 I investigated the idea!
5 To compare like fractions, I should
just compare the numerators
Here, 1 < 2 < 3 < 4 . because denominators of like
5 5 5 5 fractions are same.

Similarly, 4 < 5 , 8 > 2 , etc.
7 7 9 9

Thus, the greater fraction has larger numerator if the fractions are like.

(b) Comparison of Unlike fractions
Let’s observe the following example and make the idea to compare unlike
fractions.

28
3 12

39
4 12

From the above illustration, it is clear that 8 < 9 . ? 2 < 3 .
12 12 3 4

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 67 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

Thus, for the comparison of unlike fractions, they should at first be converted

into like fractions. Then we can compare their numerators. Let’s consider any
5 3
two unlike fractions 6 and 8 .

Here, the L.C.M. of 6 and 8 is 24.

? 5 = 5×4 = 20 , 3 = 3×3 = 9
6 6×4 24 8 8×3 24

Here, 20 > 9 . So, 5 > 3 .
24 24 6 8

Worked-out examples

Example 1: Out of 28 students of class VI, 25 passed in Maths with 'A' grade.

Likewise, out of 30 students of class VII, 23 passed in Maths with

'A' grade. What fractions of the students are passed in 'A' grade

and other than 'A' grade in each class?

Solution:

(a) For class VI 25
28
Fraction of the students who obtained 'A' grade =

Fraction of the students who obtained other grades = 1 – 25 = 28 – 25 = 3
28 28 28

For section VII 23
30
Fraction of the students who obtained 'A' grade =

Fraction of the students who obtained other grades = 1 – 23 = 30 – 23 = 7
30 30 30

Example 2: Write the first three fractions equivalent to 3 .
5
Solution:

Here, 3 = 3×2 = 6 , 3×3 = 9 and 3×4 = 12
5 5×2 10 5×3 15 5×4 20

So, 6 , 9 and 12 are the first three fractions equivalent to 3 .
10 15 20 5

Example 3: Are the fractions 5 and 20 equivalent?
Solution: 7 21

Here, 5 20
7 21
5 u 21 = 105 and 7 u 20 = 140

5 u 21 is not equal to 7 u 20

So, 5 and 20 are not equivalent fractions.
7 21

Vedanta Excel in Mathematics - Book 6 68 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

Example 4: Convert 3 and 5 into the like fractions.
4 6

Solution:

Here, the L.C.M. of the denominators 4 and 6 is 12.

Now, 3 = 3×3 = 9 , 5 = 5×2 = 10
4 4×3 12 6 6×2 12

So, 9 and 10 are the like fractions.
12 12

Example 5: Compare the following fractions. (a) 4 and 2 (b) 5 and 7
Solution: 9 9 6 8

(a) Comparing the numerators of 4 and 2 , 4 > 2.
9 9
4 2
? 9 > 9

(b) The L.C.M. of the denominators 6 and 8 is 24.

Now, 5 = 5×4 = 20 , 7 = 7×3 = 21
6 6×4 24 8 8×3 24

Comparing the numerators of 20 and 21 , 20 < 21.
24 24

? 20 < 21 or 5 < 7 .
24 24 6 8

Example 6: Arrange the fractions 2 , 5 and 4 in ascending order.
Solution: 3 6 9

Here, the L. C. M. of the denominators 3, 6 and 9 = 18

Now, 2 = 2 × 6 = 12 , 5 = 5 × 3 = 15 , 4 = 4 × 2 = 8
3 3 × 6 18 6 6 × 3 18 9 9 × 2 18

Since, 8 < 12 < 15 . So, 4 < 2 < 5 .
18 18 18 9 3 6

EXERCISE 5.1
General Section – Classwork

1. Let’s tell and write the correct numerals in the blanks.

a) 1 = 1 × ..... = ........... = 1 × ..... = ........... = 1 × 4 = ...........
2 2 ×2 ........... 2 × 3 ........... 2 × ..... ...........

b) 3 = 3 × ..... = ........... = 3× 3 = ........... = 3 × ..... = 12
4 4×2 ........... 4× ..... ........... 4 × .....
.......

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 69 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

2. Let’s tell and write the first three fractions equivalent to these fractions.

a) 1 = ........ = .......... = .......... b) 2 = .......... = ......... = ..........
3 5

3. Let’s tell and write ‘Yes’ if equivalent or ‘No’ if not equivalent these pairs
of fractions.

a) 1 and 3 ............. b) 2 and 10 ........... c) 3 and 9 ...............
2 6 3 12 4 12

4. Let’s look at the figures and fill the proper number in the blank spaces.

1 1 1 1 a) 1 = ........... b) ........... = 4
4 4 4 4 4 8 4 8

11 11 11 11 c) 3 = 6 d) 2 = 4
88 88 88 88 ........... 8 4 ...........

5. Let’s tell and write the appropriate sign (< or >) in the blanks.

a) 4 .......... 5 b) 7 .......... 2 c) 1 ........ 1 d) 3 .......... 3
7 7 9 9 8 5 8 10

Creative Section - A

6. a) In class - VI of a school, there are 15 girls and 20 boys. 9 students are

wearing spectacles, 11 girls are wearing ribbons and 7 boys are wearing

caps.

(i) What fraction of the students are girls and boys?

(ii) What fraction of the students are wearing spectacles?

(iii) What fractions of the girls are wearing ribbons and boys are wearing cap?

b) Mrs. Maharjan bought one dozen of bananas and ate 5 of them

(i) what fraction of the bananas was eaten?

ii) What fraction of the bananas was left?

7. Find the first three fractions equivalent to the following fractions by

multiplication process.

a) 1 b) 2 c) 2 d) 3 e) 4 f) 8
2 3 5 4 7 9

8. Find the first three fractions equivalent to the following fractions by
division process.

a) 6 b) 8 c) 12 d) 18 e) 20 f) 24
12 16 18 24 30 36

9. Test whether the following pairs of fractions are equivalent.

a) 1 and 2 b) 2 and 3 c) 3 and 12 d) 7 and 21
3 6 5 7 5 15 8 24

10. Convert the following pairs of fractions into like fractions.

a) 1 and 2 b) 3 and 3 c) 2 and 5 d) 3 and 7
2 3 4 5 3 6 4 10

Vedanta Excel in Mathematics - Book 6 70 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

11. Compare these pairs of fractions by using ‘<’ or ‘>’ sign.
5 7 8 3 7 2 3
a) 9 and 9 b) 11 and 6 c) 4 and 8 d) 5 and 10
11

12. Arrange the following fractions in ascending order. 2
1 3 5 2 5 7 1 3 5 1 5 3
a) 2 , 4 and 8 b) 3 , 6 and 9 c) 3 , 4 and 6 d) 4 , and 10 .

13. Arrange the following fraction in descending order.
3 4 9 5 2 7 3 5 3 7 5 11
a) 4 , 5 and 10 b) 6 , 9 and 12 c) 8 , 6 and 4 d) 12 , 16 , 24

Creative Section - B 3 1
4 8
14. a) Father spends parts of his monthly income on food and parts in

your education. On which heading does he spend more money?
3 5
b) Sunita travelled 7 parts of distance by bus and 14 parts by taxi when

she was going to airport. By which vehicle did she travel more?
2 3
c) Dhaniya can do 3 parts of a work in 1 day and Bhuniya can do 4 parts

of the work in 1 day. Who can do more work in 1 day ?
2 3
d) A man gave 5 parts of his property to his wife, 10 parts to his son and

1 parts to his daughter. To whom did he give more property? Find.
6

It’s your time - Project Work

15. a) Let's write any three pairs of equivalent fractions.

Draw three pairs of rectangles of the same size. Divide each pair of
rectangles into as many equal parts as your fractions. Then shade the
parts to show your each pair of equivalent fractions.

b) Let's take three rectangular sheets of paper. Fold each of them to show

an equivalent fraction of each of 1 , 1 and 1 .
2 3 4

c) Let's draw each pair of these shaded rectangles in sheets of paper.

Then draw dotted line to convert unlike fractions into the like frac-

tions. The first one is done for you.

(i) (ii)

1 = 4 1 = 3 1 = 1 =
3 12 4 12 2 3

(iii) (iv)

1 = 1 = 1 = 1 =
2 4 2 5

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 71 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

5.7 Proper, improper fractions and mixed numbers
Proper fractions
\

1 represents only the fractional parts 3 represents only the fractional parts
2 4
1 3
2 is a proper fraction. 4 is a proper fraction.

A fraction whose numerator is less than denominator is called a proper fraction.

4 , 2 are also the proper fractions. 4 2
5 3 5 3

Improper fractions

3 represents a whole and half of the whole.
2
It does not represent only the fractional parts. 2 1 3
2 + 2 = 2
3
2 is an improper fraction.

A fraction whose numerator is greater than denominator is called a improper

fraction. 5
3
5 , 9 are also the improper fractions.
3 4 9
4

Mixed numbers 3 3 112
2 2
In a proper fraction , it has 1 whole and a half parts. =

1 1 is a mixed number. 1 1
2 2

A combination of a whole number and a proper fraction is called a mixed number.

1 2 , 2 3 are also mixed numbers. 132 234
3 4

Note that, a mixed number is obtained by adding a whole number with a proper

fraction. For example:
1 1 3 3 4 4
1 + 2 = 1 2 , 2+ 4 = 2 4 , 5+ 7 = 5 7 , and so on.

Every improper fractions can be converted 7 = 7 ÷ 3
into mixed numbers or vice versa. 3

3 1 1 17 2 It gives 2 quotient and 1 remainder.
2 2 3 5 5
= 1 , 7 = 2 , = 3 , and so on. ∴ 7 = 213 = quotient remainder
3 3 divisor

Vedanta Excel in Mathematics - Book 6 72 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

5.8 Reducing fractions to their lowest terms

We can reduce the fractions to their lowest terms by dividing the numerator and
denominator by the same number.

For example : 12 = 12 62 = 2 12 = 12 ÷ 2 = 6 ÷ 3 = 2
18 18 93 3 18 18 ÷ 2 9 ÷ 3 3

AHgearein, ,atthfeirnstu,mtheerantourmaenrdatdoernaonmdindaetnoormofin69atiosrdoivfid11e28dibs yd3ivtiodegdetb23yw2htiochgeist 6 .
9

the

fraction with lowest terms. Alternatively, the numerator and denominator can

directly be divided by their H.C.F. to reduce the given fraction into its lowest

terms. For example : 2 H.C.F. of 12 and 18 is 6. So, the
3 numerator and denominator
12 = 12 ÷ 6 = 2 or, 12 = 12 = 2 are divided by 6.
18 18 ÷ 6 3 18 18 3

Worked-out examples

Example 1 : Convert a) 11 into mixed number b) 2 1 into improper fraction.
4 4

Solution : I got the rule !
Mixed number
a) 11 = 11 ÷ 4 4 11 2 = quotient remainder
4 –8 divisor
3
? 141 = 243 241

241 2 × 4 + 1 =8 +1 9 9 means 4 parts are taken 2 times and
4 4 4 4 more part is taken =2×
b) = = 1 4 +1 = 9
4 4

Thus, improper fraction = divisor × quotient + remainder
divisor

Example 2 : Reduce a) 21 b) 45 c) 1800 into the lowest terms.
Solution: 35 60 24000

a) 213 = 3 H.C.F. of 21 and 35 is 7. So, 21 ÷ 7 = 3 and 35 ÷ 7 = 5
355 5

b) 45 3 = 3 H.C.F. of 45 and 60 is 15. So, 45 ÷ 15 = 3 and 60 ÷ 15 = 4
60 4 4

183 At first, cancel the equal number of zeros from
240 1800
c) 1800 = = 3 the numerator and denominator of 24000 .
24000 40 Then, reduce 21840into its lowest
40 terms.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 73 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

EXERCISE 5.2
General Section A – Classwork
1. Let's say and write the fractions of the shaded parts. Mention whether the

fractions are proper or improper.

a) b)

......... is ......................... fraction ......... is ......................... fraction

c) d)

......... is ......................... fraction ......... is ......................... fraction

2. Let's say and write the improper fractions and the mixed numbers
represented by the shaded diagrams.

a) b) c)

............ = ............ ............ = ............ ............ = ............

3. Let’s tell and fill in the blanks with appropriate words.
a) A fraction with numerator less than the denominator is called

a/an ................................. fraction

b) A fraction whose numerator is greater than the denominator is called

a/an ................................. fraction.

c) 1 is a/an ................................. fraction.
2

d) 132 is a ................................. number.

4. Let’s tell and write the improper fractions as quickly as possible.

a) 3 1 = .......... b) 4 2 = .......... c) 5 3 = .......... d) 10 4 = ..........
2 3 4 7

Vedanta Excel in Mathematics - Book 6 74 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

5. Let’s tell and write the mixed numbers as quickly as possible.

a) 3 = ............ b) 7 = ............ c) 15 = ............
2 3 4

e) 1 + 1 = ............ f) 2+ 3 = ............ g) 3 + 4 = ............
3 4 7

Creative Section

6. Change these improper fractions into mixed numbers.

a) 7 b) 13 c) 19 d) 32 e) 43 f) 67
4 6 5 7 8 10

7. Change these mixed numbers into improper fractions.

a) 234 b) 372 c) 458 d) 794 e) 1083 f) 6 9
10

8. Reduce these fractions to their lowest terms.

a) 4 b) 10 c) 6 d) 12 e) 18 f) 32
6 12 9 16 24 40

g) 45 h) 20 i) 120 j) 2400 k) 3600 l) 4200
54 40 150 2800 27000 63000

9. 20% means 20 = 12015= 1 . Express these percents into fractions and
100 5

reduce them to their lowest terms.

a) 10% b) 15% c) 25% d) 40% e) 60% f) 75%

It’s your time - Project Work!

10. a) How is the meaning of 2 and 3 not same? Explain by shading diagrams
3 2

and present in your class.

b) How is the meaning of 5 and 114 same? Explain by shading diagrams
4

and present in your class.

c) Is it possible to eat 3 equal pieces of pizzas from 2 whole pizzas? How?
Write the answer with shaded figures and present in your class.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 75 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

5.9 Addition and subtraction of fractions

(i) Addition and subtraction of like fractions

Like fractions can be added or subtracted simply by adding or subtracting their
numerators. For example:

Add: 2 + 3 = 2 + 3 = 5 += =
7 7 7 7

Subtract: 4 – 1 = 4 –1 = 3 –==
5 5 5 5

(ii) Addition and subtraction of unlike fractions

Let’s study the following illustrations and investigate the idea of addition and
subtraction of unlike fractions.

Add : 1 + 3 = 1 × 2 + 3
4 8 4 × 2 8

= 2 + 3 = 2 + 3
8 8 8

= 5
8

From the above illustrations, its clear that the unlike fractions 1 and 3 are
converted into like fractions and they are added. 4 8

Again, let’s take another illustration of addition.

Add : 1 2 + 2 1 = (1 + 2) + 2 + 1
3 6 3 6

=3+ 2 × 2 + 1
3 × 2 6

=3+ 4 + 1
6 6

= 3 + 5 = 3 5 3
6 6

Subtraction of unlike fractions is also performed in the similar ways.

Vedanta Excel in Mathematics - Book 6 76 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

Worked-out examples

Example 1 : Add 2 + 5
3 7

Solution : I got it !

The L.C.M. of 3 and 7 is 21. 5
2 7
2 + 5 = 2 × 7 + 5 × 3 Now, I should change 3 and into like
3 7 3 × 7 7 × 3
14 15 fraction making the denominator 21.
21 21
= +

= 14 + 15 = 29 =1281
21 21

Direct process: L.C.M. of 3 and 7 is 21.

2 + 5 =2 ×7+5×3 21 is divided by each denominator
3 7 21 and the quotient is multiplied by
the corresponding numerator.
=14 + 15 29 =1281
21 = 21

5 – 14 . Alternative process
6 9
Example 2: Subtract 4 5 4 5 4
6 9 6 9
Solution : 4 – 1 = (4 – 1) + ( – )

5 4 29 13 = 3 + (3 × 5–2 × 4)
6 9 6 9 18
4 – 1 = –
15 – 8
3 × 29 – 2 × 13 = 3 + 18
= 18

= 87 – 26 = 3 + 7 = 3178
18 18

= 61 = 3178 Alternative process
18
5 3 – 2 5 + 1 7
Example 3 : Simplify 5 3 – 2 5 + 1 7 . 4 6 + 8
4 6 8 3 5 7
= (5 – 2 1) + ( 4 – 6 + 8 )

Solution : 6 × 3 – 4 ×5 + 3 × 7
24
3 5 7 23 17 15 = 4 +
4 6 8 4 6 8
5 – 2 + 1 = – + 18 – 20 + 21
12
6 × 23 – 4 × 17 + 3 × 15 = 4 +
24
= 19 42194
24
138 – 68 + 45 183 – 68 115 42149 = 4 + =
24 24 24
= = = =

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 77 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

Example 4: In 1 Mdary. ,JhMar.cSahnedropa112capnardtoof14 part, Mr. Adhikari can do 1 part
and a work. 6

a) How much work would they do in 1 day working together ?

b) How much work is remained to complete?

Solution : 1 1 1
4 6 12
a) Here, the total amount of work done by three person in 1 day is + +

1 + 1 +112 = 3 +2+ 1 = 6 = 1 + +
4 6 12 12 2
11 1
So, they would do 1 parts of the work working together. 46 12
2
1 1 2 – 1 1 ++
2 2 2 2
b) Remaining work=whole work – = 1 – = = 32 1
12 12 12
1
So, 2 parts of the work is remained to complete.

Example 5: Sunayana took a novel book from her school library. She
1 3
completed reading 6 parts of the book in the first day, 8 parts in

the second day and 5 parts in the third day. If she completed
12
the novel in the fourth day, what parts of the novel was left for

the last day?

Solution : 1 3 5
6 8 12
Here, the total parts of the novel read in first three days = + +

1 + 3 + 5 = 4 × 1 +3× 3 + 2 × 5
6 8 12 24

= 4 + 9 + 10 = 23
24 24

Again, remaining part of the novel left for fourth day = 1 – 23 = 1
1 for the last day. 24 24
Thus, 24 part of the novel was left

EXERCISE 5.3
General Section – Classwork
1. Let’s say and find the total of the shaded parts with different colours.

a) b)

........... + ........... = ........... ........... + ........... = ...........

Vedanta Excel in Mathematics - Book 6 78 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

2. Let’s subtract the fraction of the crossed parts from the fraction of the
shaded part.
a) b)

........... – ........... = ........... ........... – ........... = ...........

3. Let’s add the fraction of the shaded parts. Then add the unlike fractions.

a) + =+ b) + = +

1 + 1 = 3 + 2 =.......... ........ + ........ = ........ + ........ = ........
2 3 6 6

c) + = + d) + =+

........ + ........ = ........ + ........ = ........ ........ + ........ = ........ + ........ = ........

4. Let’s subtract the fraction of the crossed parts from the fraction of the
shaded parts.

a) b)

1 – 1 = ........ ........ – ........ = ........
2 4

c) d)

........ – ........ = ........ ........ – ........ = ........

5. Let’s write the mixed numbers from the shaded parts and add the mixed
numbers.

a) + b) +

1 1 + 1 1 = ........ ............ + ............ = ............
2 4

c) + d) +

............ + ............ = ............ ............ + ............ = ............

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 79 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

6. Let’s say and write the sums or differences as quickly as possible.

a) 1 + 1 = ........... b) 1 + 2 = ........... c) 2 + 3 = ...........
4 4 5 5 7 7

d) 7 – 5 = ........... e) 10 – 6 = ........... f) 8 – 2 = ...........
9 9 11 11 15 15

g) 213 + 1 1 = ........... h) 3 3 + 2 15 = ........... i) 4 59 – 1 2 = ...........
3 5 9

7. 1 + 3 = 7 Add 3 + 4 and write the sum in numerator.
4 4 Subtract 7 – 2 and write the difference in numerator.

1 – 2 = 5
7 7

Let's apply the tricky ways of calculation, tell and write the answers as
quickly as possible.

a) 1 + 2 = ............... b) 1 + 3 = .............. c) 1 + 2 = ..............
3 4 7

d) 1 – 3 = .............. e) 1 – 2 = ............... f) 1 – 4 = ................
4 5 9

8. Let’s tell and write the answers as quickly as possible.

a) A bread is divided into 4 equal parts. Mr. Chand eats 1 parts and
Mr. Rawal eats 2 parts.

(i) Fraction of total parts eaten by them = ...................

(ii) Fraction of remaining parts = ...................

b) An income is divided into 6 equal parts. 2 parts are spent on food and
3 parts are spent on education.

(i) Fraction of total parts which are spent = .....................

(ii) Fraction of remaining parts of income = .....................

Vedanta Excel in Mathematics - Book 6 80 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

Creative Section - A
9. Let's add:

a) 1 + 2 b) 1 + 2 c) 3 + 5 d) 8 + 2
2 3 3 5 4 8 9 3

e) 3 + 11 f) 2 + 7 g) 323 + 156 h) 2 2 + 3170
5 15 3 12 5

i) 4 5 + 3134 j) 4130 + 5145 k) 2145 + 5230 l) 1254 + 2376
7

10. Let's subtract:

a) 1 – 1 b) 3 – 2 c) 5 – 2 d) 4 – 3
2 3 4 3 6 3 5 10

e) 6 – 3 f) 5 – 3 g) 7 – 5 h) 3 2 – 1 1
7 14 6 8 8 12 3 2

i) 5 3 – 2 5 j) 4 45 – 19 k) 7 65 – 3 2 l) 538 – 2172
4 8 10 9

11. Let's simplify

a) 1 + 1 + 1 b) 2 + 1 + 5 c) 3 + 5 – 7 d) 8 – 2 + 1
2 3 4 3 6 9 4 6 8 9 3 6

e) 3 – 7 – 1 f) 11 – 3 – 1 g) 1 1 + 21 + 3 1 h) 2 1 + 3 2 + 1 1
5 15 10 12 8 4 2 4 8 2 3 6

i) 3 1 + 4 1 + 6 1 j) 3 2 + 2 5 – 4 4 k) 2 3 – 3 1 + 1112 l) 5 1 – 2 74 – 1154
3 4 6 3 6 9 4 8 2

Creative Section - B 4 3 170 .
5
12. a) Samriddhi bought a pencil for Rs 5 and an eraser for Rs What is

the total cost of both the articles? Find.
1 5
b) Kamal travelled 3 4 km by tempo and 6 6 km by bus to reach his uncle’s

home. What distance did he travel altogether?
1
c) The weight of Manisha is 25 2 kg and that of her school bag with
stationery is 1 kg. What the total weight while she is carrying
2 4 will be

her school bag? 3 41
of
13. a) Lakhan needs m of cloth for a pants and a shirt of his new school’s
dress. If 3 cloth is required for the pants, how much cloth is
1 4 m

required for a shirt?

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 81 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

b) The weight of empty gas cylinder is 16 4 kg and that of the cylinder filled
with gas is 31 7 5 of the gas filled in the cylinder?
15 kg. What is the weight

14. a) taATanhxmdei,ad165ni18spgtaaikrvnmtescsetbo83byaepatcwabhruetasesrnioatfyntw.tdhWoethhppealratroceppemaesrraittsisyn1otinof5gthkhdmiesi.spstAroaonnpm,cee1ar5n2thyetpraawarrveateslllekltfeeotddwh.8iHist41hodwkahmuimmgbha?ytneyar
b)

kilometre did he walk ?

It’s your time - Project Work!

15. a) Let's write any three pairs of unlike fractions of your own choice. Add
each pair of fraction by using shaded diagrams.

b) Let's write any three pairs of mixed numbers of your own choice. Add
each pair by using shaded diagrams.

5.10 Multiplication and division of fractions
(i) Multiplication of a fraction by a whole number

Let’s study the following illustration and investigate 2u 1 means 1 + 1
the idea of multiplication of a fraction by a whole 3 3 3

number. 1 11 2
3 33 3
Multiply: 2u means

2 u 1 = 2×1
3 3
2
= 3

Thus, (whole number u fraction) = whole number u numerator
denominator

(ii) Multiplication of a fraction by another fraction

Let’s study the following illustration and investigate the idea of multiplication

of a fraction by another fraction. 1 3 means one half of 3
2 4 4
13 u
Multiply: 2 u 4
3

13 1×3 3 4
2u4= 2×4 8
= 1 of 3 = 3
2 4 8

Vedanta Excel in Mathematics - Book 6 82 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

(iii) Reciprocal of a number or a fraction

Let’s investigate the idea of the reciprocal of a given number.

Reciprocal of 2 = 1 , Reciprocal of 3 = 1 , Reciprocal of 7 = 1 , and so on.
2 3 7
Thus, the reciprocal of any number is 1 divided by the number.

In the case of a fraction, its reciprocal is obtained just by interchanging its

numerator and denominator. For example,

Reciprocal of 2 = 3 , Reciprocal of 4 is 5 , and so on.
3 2 5 4

(iv) Division of a whole number by a fraction

Let’s study the following illustration and investigate the idea of division of a
whole number by a fraction.

Divide: 3 ÷ 1 3 ÷ 1 means, how many halves are there in 3?
2 2

3÷ 1 =3u 2 11 11 11
2 1 22 22 22

= 6 2 halves + 2 halves + 2 halves = 6 halves
1

=6

Thus, to divide a whole number by a fraction, we should multiply the whole
number by the reciprocal of the fraction.

The above division can also be shown by a number line.

3 ÷ 1
2
2
3 × 1 =6

(v) Division of a fraction by a whole number

Let’s study the following illustration and investigate 1
3
the idea of division of a fraction by a whole number.
1
Divide: 1 ÷ 2. It means divide 1 part again into two parts. 6
3 3

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 83 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

From the diagram,

1 ÷ 2 = 1 × 1 = 1 × 1 = 1
3 3 2 2 × 3 6

Thus, to divide a fraction by a whole number, we should multiply the fraction
by the reciprocal of the whole number.

(vi) Division of a fraction by a fraction

Study the following illustration and investigate the idea of division of a
fraction by a fraction.

Divide: 3 ÷ 3 3
2 4 4

It means how many are 3
3 4
there in 2 ? (There are two . )

From the diagram,

3 ÷ 3 = 2 3 In 3 ÷ 3 , there are two 3 .
2 4 2 2 4 4

Here, 3 is multiplied by the reciprocal of 3 .
2 4

3 ÷ 3 = 3 4 =2
?2 4 2 u3

Worked-out examples

Example 1 : Multiply a) 6 × 4 b) 3 3 × 1 1
Solution: 9 7 6

6 × 4 = 26× 4 b) 3 3 × 1 1 = 27414× 7
9 93 7 6 61
8 2 4 × 1
= 3 = 2 3 = 1 × 1 = 4

Example 2 : Divide a) 14 ÷ 7 b) 5 ÷ 10 c) 2 2 ÷ 3 1
8 9 9 3

Solution:

a) 14 ÷ 7 = 2 × 8 = 2 × 8 = 16
8 71 1
14

b) 5 ÷ 10 = 5 1 1 = 1
9 9 10 18
÷ 2

c) 2 2 ÷ 3 1 = 20 ÷ 10 = 202 × 13011= 2 × 1 = 2
9 3 9 3 93 3 × 1 3

Vedanta Excel in Mathematics - Book 6 84 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

Example 3 : Simplify a) 2 × 9 ÷ 21 b) 2 2
3 10 25 3 c) 3

4 4

Solution: 3211× 190351×1

a) 2 × 9 ÷ 21 = 255 = 1 × 5 = 5
3 10 25 21 7 1 × 7 7

b) 2 = 2 ÷ 4 = 21 × 1 2= 1
3 3 3 4 6
4

c) 2 = 2 ÷ 3 =2× 4 = 8 = 2 2
3 4 3 3 3

4

Example 4 : Find a) 5 of Rs 160 b) 2 ÷ 1 of 2
8 3 3 5
Solution:

a) 5 of Rs 160 = 5 × Rs 160 = 5 × Rs 20 = Rs 100
8 8

b) 2 ÷ 1 of 2 = 2 ÷ 1 × 2 It is the wrong process
3 3 5 3 3 × 5

= 2 ÷ 2 2 ÷ 1 of 2 = 2 ÷ 1 × 2
3 15 3 3 5 3 3 5

= 2 1× 15 5 = 2 × 3 × 2
3 21 3 1 5
1

=5 = 4
5
Which is wrong answer.

Example 5: Mr. Pandey can do 1 part of a work in 1 day. How much work
Solution: 15
does he do in 5 days?

Here, in 1 day, Mr. Pandey does 1 part of the work.
15

So, in 5 days, he does 1 ×5 parts of the work.
15

= 1 part of the work.
3

So, he does 1 part of the work in 5 days.
3

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 85 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

Example 6: How many jars each of capacity 5 21 litres are needed to empty
55 litres of water?

Solution:

Here, the required number of jar = 55 ÷ 5 1 = 55 ÷ 11 = 55 × 2 = 10
2 2 11
Hence, 10 jars are needed.

Example 7: A man earns Rs 9,600 in a month. He spends 3 parts of his
earning to run his family. 4

a) Find his monthly expenditure.

b) Find his monthly saving.

Solution :

a) Here, monthly income = Rs 9,600

Monthly expenditure = 3 of Rs 9,600
4 ×
=134 2400
3 × Rs 2,400 = Rs 7,200
Rs 9600 =

Hence, his monthly expenditure is Rs 7,200.

b) Again monthly saving = Rs 9,600 – Rs 7,200 = Rs 2,400

Hence, his monthly saving is Rs 2,400.

Example 8: The distance between Chakraghatti and Biratnagar is 60 km.
3
Mr. Sunuwar travelled 5 part of the distance by a bicycle and the

remaining distance by a bus.

a) How many kilometres did he travel by bicycle ?

b) How many kilometres did he travel by bus ?

Solution :
a) Here, the distance between the two places = 60 km.

Distance travelled by a bicycle = 3 of 60 km
5
Hence, he travelled 36 km by a=bic35y1×cle6.102 km = 36 km

b) Again, the remaining distance = 60 km – 36 km
= 24 km

Hence, he travelled 24 km by a bus.

Vedanta Excel in Mathematics - Book 6 86 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

EXERCISE 5.4
General Section – Classwork

1. Let's complete multiplication from the given shaded diagrams.

a) + = = 2 × 1 = 2 = 1
2 2

b) + + = = ......... × ......... = ......... = .........

c) + + = = ......... × ......... = ......... = .........

d) =.........×.........=.........= .........
++++=

2. Let's complete multiplication from the given shaded diagrams.

a) = 1 of 1 = 1 × 1 = 1
2 2 2 2 4

b) = 1 of ......... = ........................ = .........
3

c) = ......... of ......... = ........................ = .........

3. Let's say and write how many halves, thirds, and quarters?

a) How many halves are there in 4 breads?

4 ÷ 1 = 4 × 2 = .........
2 1

b) How many thirds are there in 3 breads?
......... ÷ ......... = ......... × ......... = .........

c) How many quarters are there in 5 breads?
......... ÷ ......... = ......... × ......... = .........

d) How many halves are there in a half bread?

1 ÷ 1 = ......... × ......... = .........
2 2

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 87 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

e) How many thirds are there in a half circle?

1 ÷ 1 = ......... × ......... = ...3...... = 1 1
2 3 2 2

f) How many quarters are there in a half circle?

......... ÷ ......... = ......... × ......... = ......... = .........

4. Let's complete division from the shaded diagrams.
a) Divide a half bread into 2 equal parts.

1 ÷ 2 = 1 × 1 = ........
2 2 2

b) Divide a half bread into 3 equal parts.

........ ÷ ........ = ........ × ........ = ........

c) Divide two-thirds of a bread into 2 equal parts.
........ ÷ ........ = ........ × ........ = ........

5. Let's circle half, thirds, and quarters of the given numbers. Say and write
the answer.

a) Half of 4 pizzas. 1 of 4 = 1 × 4 = 2 pizzas
b) Half of 6 pizzas. 2 2

........ of ........ = ........ × ........ = ........ pizzas

c) One-third of 6 pizzas.

........ of ........ = ........ × ........ = ........ pizzas

d) Two-thirds of 6 pizzas.

........ of ........ = ........ × ........ = ........ pizzas

e) Three-quarters of 8 pizzas.

........ of ........ = ........ × ........ = ........ pizzas

Vedanta Excel in Mathematics - Book 6 88 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

6. Let’s say and write each repeated addition as a multiplication as shown in
example. Then find the product.

Example: 1 + 1 + 1 = 3 × 1 = 3 a) 1 + 1 + 1 = ............
5 5 5 5 5 4 4 4

b) 1 + 1 + 1 + 1 = ............ c) 2 + 2 + 2 + 2 + 2 = ............
7 7 7 7 9 9 9 9 9

7. Let’s say and write the answers as quickly as possible.

a) 3210××665 = ............... b) 1 × 9 = ............... c) 10 × 3 = ...............
d) = ................ e) 13 × = ............... 5
2 6 1 9
7 f) 3 × 10 = ...............

g) 1 of 20 = ............... h) 1 of 15 = ............... i) 1 of Rs 44 = .............
2 3 4

j) reciprocal of 1 = ..... k) reciprocal of 6 = ..... l) reciprocal of 4 = ....
2 5
1 1 1
m) 2 ÷ 3 = ......... n) 3 ÷ 4 = ......... o) 2 ÷ 2 = .............

8. Let’s say and write the answers as quickly as possible.

11

a) 2 = ......... b) 2 = ......... c) 1 = ......... d) 1 = .........
3 4 2 3

1 1 4 5

e) 3 = ......... f) 4 = ......... g) 2 = ......... h) 1 = .........
2 2 1 5

4 3

Creative Section - A

9. Let's multiply.

a) 3 × 1 b) 8 × 2 c) 5 × 2 d) 2 × 9 e) 5 × 24
5 7 9 3 6

f) 10 × 7 g) 18 × 2 h) 2 × 4 i) 3 × 5 j) 4 × 7
8 15 3 5 4 8 7 8

k) 5 × 8 l) 5 × 8 m) 15 × 12 n) 1 12 × 1 1 o) 4 3 × 1115
8 15 6 15 16 25 3 8

10. Let's divide.

a) 4 ÷ 2 b) 9 ÷ 3 c) 3 ÷ 9 d) 4 ÷ 12 e) 1 ÷ 3
3 5 5 9 2 4

f) 6 ÷ 36 g) 3 3 ÷ 1 1 h) 3 1 ÷ 2 1 i) 9 3 ÷ 3 3 j) 3 3 ÷ 2 1
7 49 4 2 8 2 8 4 8 4

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 89 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

11. Let's simplify.

a) 2 × 9 × 5 b) 4 × 15 × 8 c) 3 × 28 × 22
3 10 6 5 16 9 7 33 24
8 15 6 10 15
d) 3 × 10 ÷ 35 e) 16 × 28 ÷ 7 f) 20 ÷ 9 × 16
5 21 25 27
1252 11134 h) 1210 ÷ 4190 5 2
g) 2 2 ÷ × × 1 9 i) 3 34 × 2 23 ÷ 2 9
11
250 180 1500 1000 2500 500 2700 5000
j) 240 × 200 ÷ 1200 k) 800 ÷ 4000 × 6000 l) 2500 ÷ 1800 × 1400
3500

12. Let's simplify. 2 6 5 9

a) 1 b) 4 c) 3 d) 7 e) 6 f) 10
2 8 4 9 10 18

33 18 25

13. Let's find the value.

a) 2 of Rs 30 b) 4 of Rs 100 c) 3 of 40 pupils
3 5 8
6 3 7
d) 7 of 350 kg e) 5 of 200 m f) 10 of 50 km

g) 4 of 270 men h) 5 of 360 girls i) 3 of 140 girls
9 6 7
251
14. 25% of 40 means 1004 × 4100 = 10. Likewise, Simplify and find the value of
the following.

a) 5% of Rs 200 b) 10% of Rs 60 c) 20 % of Rs 500
d) 25% of 60 boys e) 50% of 450 students f) 75% of Rs 400.
15. Let's simplify.

a) 1 ÷ 3 of 2 b) 2 ÷ 5 of 3 c) 3 of 2 ÷ 8 d) 1 of 2 72 ÷ 3
2 4 5 3 6 10 4 7 21 4 5

16. a) If the cost of 1 kg 34ofkrmiceinis1Rmsin85u12te., find the cost of 10 kg of rice.
b) A car can travel 1 How many kilometres does it
travel

in 12 minutes with the same speed ?

17. a) Teacher cut one-third part of a chart paper and she gave it to her student
to colour half of this part. What fraction of the whole chart paper did
the student colour?

b) Mother gives half of a whole bread to brother. Brother divides this half
bread into three equal parts and he gives one part to his sister.
(i) What fraction of the whole bread does sister get?
(ii) What fraction of the whole bread is left with brother?

Vedanta Excel in Mathematics - Book 6 90 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

18. a) Anamol can do 1 part of a work in 1 day. How much work does he do
in 3 days ? 9

b) Mrs. Maharjan can do 1 part of a work in 1 day. She worked for 2 days
6
and left. The remaining parts of the work is done by Mrs. Rai.

(i) How much work did Mrs. Maharjan do in 2 days ?

(ii) How much work was left to do ?

(iii) How much work was done by Mrs. Rai ?
1
c) Bhurashi can do 20 part of a work in 1 day. She worked for 4 days and

left. If the remaining parts of the work was done by Rita, how much

work did she do ? 1
2
19. a) How many jars each of capacity 2 litres are needed to empty a vessel
of 50 litres of milk?
3 13
b) A rope is 40 m long. How many metre pieces can be cut from the
rope?

20. a) In the Polymerise Chain Reaction (PCR) test among 7,290 suspected
3
people 10 of them were found having Corona positive. Find the number

of people who have Corona positive. 5
9
b) Mr. Shakya earns Rs 18,000 in a month. He spends parts of his
earning to run his family.

(i) How much money does he spend to run his family ?

(ii) How much money does he save in a month ?

c) (Ffoia)otHhdeoarwnedamr25nuscphRarsmts2o0on,ne0y0th0deoineedsauhmceaostnipotenhn.odHfoehnissfpoceohndidld?sr41enp.art of her earning on

(ii) How much money does he spend on education ?

(iii) What is his total expenditure in a month ?

(iv) How much money does he save in a month ?

d) The distance from Dhanghadi to Mahendranagar is 60 km.
1 3
Mr Chaudhari travelled 3 part of the distance by motorbike, 5 parts of

the distance by taxi and the remaining distance he walked. How many

kilomertres did he walk ?

Creative Section - B

21. Divide: 2 2 4 4 1 1 78 7 1
3 3 9 9 4 8 4
a) 6 ÷ and ÷ 6 b) 24 ÷ and ÷ 24 c) 1 ÷ and 1 ÷ 1

Look at the quotients in every pair of division. What do you notice?

Discuss with your friends.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 91 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

d) Perform the operations as given in each pair of numbers.

(i) 4 1 – 3 and 4 1 ÷ 3 (ii) 5 1 – 4 and 5 13 ÷ 4 (iii) 6 1 – 5 and 6 1 ÷ 5
2 2 3 4 4

Are the difference and quotient of numbers in each pair equal? Find.

22. a) Out of 30 students in a class, two-third were successful to get A+ grade in

Maths test. Among these successful students, one-quarter got A+ grade

even in science also.

(i) What fraction of the successful students got A+ grade in science?

(ii) How many students were successful to get A+ grade in science?

(iii) How many students got other than A+ grade in these two subjects?

b) Out of 40 students of class VI, three-fifth of the students participated in
tpoaorkticpiapratnints,h41ightojoukmppa?rt
various games on sport day. Among the in high
jump. (i) What fraction of the students

(ii) How many students took part in high jump?

(iii) How many students did not take part in any games?

It’s your time - Project Work!

23. It's a fun of calculation. Let’s investigate the rules and complete the

remaining sums.

a) 3 = 3 × 2 b) 1 1 2 = 1
2 × 2

3 + 6 = 6 × 3 1 + 1 = 2
2 × × 3
1 2 2 3

3 + 6 + 9 = 9 × 4 1 + 1 + 1 = 3
2 × × × 4
1 2 2 3 3 4

3 + 6 + 9 + 12 = ................. 1 2+2 1 3+3 1 4+4 1 5 = ..........
× × × ×
3 + 6 + 9 + 12 + 15 =............. 1

1 1 2+2 1 3+3 1 4+4 1 5+5 1 6=......
× × × × ×

24. a) Let's fold different sheets of paper into halves, thirds, and quarters. Find
separately the number of halves, thirds and quarters in 1, 2, 3, 4, and 5
sheets of paper.

b) Let's fold a sheet of paper into halves. Then, shade the half of a half
folding.

c) Let's fold a sheet of paper into quarters. Then, shade the half of a quarter.

Vedanta Excel in Mathematics - Book 6 92 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

5.11 Tenths, hundredths, and thousands - Looking back
Let’s say and write the fraction and decimals of the coloured blocks.

Colours Fractions Decimals Colours Fractions Decimals
Red Pink

Green Blue

Colours Fractions Decimals
Green
Blue

Pink

5.12 Decimals

Let’s consider the fractions like 7 , 23 , 259 , etc. The denominators of
10 100 1000
these fractions are 10 or the power of 10. Such fractions are called the decimal

fractions.

Here, 7 = 0.7 (Zero point seven) is a decimal number.
10

23 = 0.23 (Zero point two three) is a decimal number.
100

259 = 0.259 (Zero point two five nine) is a decimal number.
1000

5.13 Conversion of Fractions into decimals

(i) Fractions with denominators 10 or power of 10.

In such fractions, the decimal point is shifted as many number of digits to the
left as there are zeros in power of 10. Let’s learn from these examples.

Example 1: Convert a) 7 b) 3 c) 3 into decimals.
Solution: 10 100 1000

a) 3 = 3.0 = 0.3 10 has one zero. So, decimal point
10 10 is shifted one digit to the left.

10 3 10 3.0 0.3 I got it !
–0
30 3 means 3 tenths which is 0.3.
– 30 10
×

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 93 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

b) 3 = 03.0 = 0.03 100 has two zeros. So, decimal point
100 10 is shifted two digits to the left.

100 3 100 3.00 0.03
–0
30 3 I also understood !
0 100
300 means 3 hundredths which is 0.03 !!
– 300
×

c) 3 = 003.0 = 0.003 1000 has three zeros. So, decimal point
1000 1000 is shifted three digits to the left.

Example 2: Convert a) 16 b) 65 c) 459 d) 253 into decimals.
10 100 1000 10000

Solution:

a) 16 = 16.0 = 1.6 b) 65 = 65.0 = 0.65
10 10 100 100

c) 459 = 459.0 = 0.459 d) 253 = 0253.0 = 0.0253
1000 1000 10000 10000

(ii) Fractions having denominators any other natural numbers

In such fractions, the numerator is directly divided by the denominator to get
decimal number.

Example 3: Convert a) 3 b) 321 c) 2 1 into decimals.
Solution: 4 3

Vedanta Excel in Mathematics - Book 6 94 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

5.14 Conversion of decimals into fractions

To convert decimals into fractions, we should count the number of digits after
the decimal point. If there is only one digit after the decimal point, we should
remove the decimal and write 10 in the denominator of the required fraction.
Similarly, if there are two digits after the decimal point, we write 100, for three
digits we write 1000, and so on in the denominator.

Let’s study the following examples and learn to convert decimals into fractions.

Example 4: Convert a) 0.8 b) 1.75 c) 0.084
Solution:

a) 0.8 = 18045= 4 In 0.8, there is one digit after the decimal
5 point. So, I should write 10 in the
denominator removing the decimal point !

b) 1.75 = 11075047= 7 = 143 In 1.75, there are two digits after decimal
4 point. So, I should write 100 in the
denominator removing the decimal point !

c) 0.084 = 84 21 = 21 In 0.084, there are three digits after the
1000 250 decimal point. So, I should write 1000 in
250 the denominator removing decimal points !

EXERCISE 5.5
General Section – Classwork

1. Let’s tell and write the number names in decimal as quickly as possible.

a) 0.25 ......................................... b) 0.153 .........................................

c) 0.04 ......................................... d) 0.007 .........................................

2. Let’s tell and write the decimals of these fractions as quickly as possible.

a) 7 = .............., 7 = .............., 7 = ..............
10 100 1000

b) 18 = .............., 18 = .............., 18 = ..............
10 100 1000

c) 125 = .............., 125 = .............., 125 = ..............
10 100 1000

d) 3 = .............., 3 = .............., 3 = ..............
1000 10000 100000

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 95 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

3. Let’s tell and write the fractions (not necessary in the lowest terms).

a) 0.4 = .............., 0.04 = .............., 0.004 = ...............

b) 0.9 = .............., 0.09 = .............., 0.009 = ...............

c) 1.5 = .............., 0.15 = .............., 0.015 = ...............

d) 32.7 = .............., 3.27 = .............., 0.327 = ...............
Creative Section

4. Convert the following fractions into decimals.

a) 6 . 6 , 6 b) 4170, 47 , 47 c) 256 , 256 , 256
10 100 1000 100 1000 10 100 1000

d) 1 , 12 , 123 e) 5 , 55 , 555 f) 3 , 34 , 345 , 3456
100 100 100 10000 1000 100 10000 1000 100 10

5. Let’s convert the following fractions into decimals.

a) 1 b) 1 c) 1 d) 3 e) 2 f) 3
2 4 5 4 5 5

g) 5 h) 7 i) 3 j) 9 k) 13 l) 27
8 16 20 25 40 50

m) 112 n) 2 3 o) 4 2 p) 7 2 q) 9 5 r) 10 4
4 5 3 7 9

6. Convert the following decimals into fractions and reduce them to their
lowest terms or express in mixed numbers wherever necessary.

a) 0.2, 0.02, 0.002 b) 0.4, 0.04, 0.004 c) 0.5, 0.05, 0.005

d) 1.5, 0.15, 0.015 e) 12.5, 1.25, 0.125 f) 22.5, 2.25, 0.225

7. 25% means 25 = 0.25. Express these percents into decimals.
100

a) 3% b) 8% c) 25% d) 48% e) 56% f) 97%

It’s your time - Project Work!

8. a) Let's draw 9 rectangular strips each of 10 cm long in a chart paper.
Divide each strip into 10 equal parts. Shade the parts of each strip
separately to show the decimal numbers from 0.1 to 0.9.

b) Let's compare the shaded parts of tenths and hundredths. Discuss with
your friends and answer the questions.

Vedanta Excel in Mathematics - Book 6 96 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

0.1 0.3 0.5
0.10 0.30 0.50

i) Is 0.1 (one-tenth) same as 0.10 (ten-hundredths)?
(ii) Is 0.3 (three-tenths) same as 0.30 (thirty-hundredths)?
(iii) Is 0.5 (five-tenths) same as 0.50 (fifty-hundredths)?

5.15 Place and place values of decimals

Tenths, hundredths, thousandths, etc. are the places of decimal numbers.

Let’s study the following illustrations and learn to find the place values of digits with

decimals. In 0.647
6 is at tenths place. So, it is 0.6.
0.647 6 4 is at hundredths place. So, it is 0.04.
10 7 is at thousandths place. So, it is 0.007.
6 tenths = = 0.6

4 hundredths = 4 = 0.04
100

7 thousandths = 7 = 0.007
1000

Now, in expanded form, 0.647 can be written as: 0.647 = 0.6 + 0.04 + 0.007

5.16 Addition and subtraction of decimals

While adding or subtracting decimals, we should arrange them in such a way
that the digits at the same places should lie in the same column.

Worked-out examples

Example 1 : Simplify a) 0.9 + 0.09 + 0.009 b) 3.45 + 16.7 – 15.324

Solution: 0.900 To make the equal decimal places, we
0.090 can write as many zeros as required
a) 0.9 + 0.009 at the end of the decimal numbers.
0.09 0.999
In 16.7, one zero is written at
+ 0.009 hundredths place to make equal

Again, 20.150 decimal places to 3.45 !
– 15.324
b) 3.45 4.826 In 20.15, one zero is written at
+ 16.70 thousandths place to make equal
20.15 decimal places to 15.324 !!

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 97 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

Example 2 : You bought a bag for Rs 520.75 and an umbrella for Rs 315.60. If
you gave a 1000 rupee note to the shopkeeper, what change did
the shopkeeper return to you?

Solution:

Here, the cost of bag = Rs 520.75 Again, Rs 1000.00

the cost of umbrella = + Rs 315.60 – Rs 836.35
Rs 163.65
Total cost = Rs 836.35

Thus, the shopkeeper returned Rs 163.65.

EXERCISE 5.6
Section A – Classwork

1. Let’s say and write the place value of each digit of decimal numbers.

a) 0.685 b) 0.295 c) 0.743
.......... = ..........
6 = 0.6 .......... = .......... .......... = ..........
10 .......... = .......... .......... = ..........
.......... = ..........
8 = 0.08
100

.......... = ..........

2. Let’s tell and write the place values of the coloured digits.

a) 0.263 ........................ b) 0.085 ........................ c) 0.432 ........................

3. Let’s tell and write the sums or differences as quickly as possible.

a) 0.2 + 0.3 = ......... b) 0.2 + 0.03 = ......... c) 0.02 + 0.03 = .........

d) 0.4 + 0.9 = ......... e) 0.8 + 0.7 = ......... f) 0.3 + 0.8 = .........

g) 0.06 + 0.09 = ......... h) 0.5 – 0.3 = ......... i) 0.6 – 0.2 = .........

j) 0.4 – 0.04 = ......... k) 0.4 – 0.05 = ......... l) 1 – 0.5 = .........

m) 1 – 0.8 = ......... n) 2 – 0.6 = ......... o) 5 – 0.05 = .........
Creative Section - A

4. Write the following decimal numbers in expanded forms.

a) 0.25 b) 0.64 c) 0.714 d) 0.4765

Vedanta Excel in Mathematics - Book 6 98 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fractions and Decimals

5. Add or subtract.

a) 0.15 + 0.47 b) 0.5 + 0.23 c) 0.69 + 0.8 d) 4.56 + 3.98

e) 7.265 + 8.84 f) 15.06 + 18.947 g) 0.42 – 0.24 h) 0.8 – 0.47

i) 0.75 – 0.006 j) 6.04 – 3.56 k) 10.001 – 5.846 l) 5 – 0.875

6. Puzzle time!

Let’s fill in the missing numbers to complete the sums.

a) b) – = c) + 0.1 = 0.9
0.2 + = 0.5
+ + +– – –– + –
= 0.4 0.6 – =0
0.5 + = 0.6 –

= = = = = == = =
+ 0.3 =
+ 0.7 = 0.3 – = 0.1

7. Simplify. b) 0.65 + 0.34 + 0.96 c) 0.7 + 0.9 – 0.125
a) 0.5 + 0.6 + 0.7 e) 3.5 + 0.08 – 1.7 f) 6.24 – 9.5 + 5.004
d) 1.05 – 0.6 – 0.08

8. a) Find the difference between 8.72 and 0.872.

b) What should be added to 64.35 to make 100?

c) What should be subtracted from 55.55 to get 5.555?

9. a) On your birthday, you bought a cake for Rs 250.25, and a packet of
sweets for Rs 145.90. If you gave Rs 500 rupee note to the shopkeeper,
what change did the shopkeeper return to you?

b) The initial temperature of a body is 40.5°C. It’s temperature first rises
by 35.7°C and then falls by 29.8°C. Now, what is the temperature of the
body ?

c) The distance between Itahari and Damak is 42 km. Mr. Dhamala travelled
18.325 km by a bus, 15.675 km by a taxi and the remaining distance by
a motorbike. How many kilometres did he travel by motorbike ?

d) A dairy man has 50.5l of milk. He has to deliver 30.75l and 32.5l of milk
in two marriage ceremonies in the same time. How much more milk
should he manage ?

Creative Section - B

10. a) Find the sum of place values of 3’s and subtract it from the place value
of 2 in the numeral 0.23436.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 99 Vedanta Excel in Mathematics - Book 6

Fractions and Decimals

b) In the numeral 6.89098, subtract the sum of place values of 9’s from the
sum of place values of 8’s.

11. a) From a sack of a quintal of potatoes, a shopkeeper sold 25.65 kg of
potatoes yesterday and 20.48 kg this morning. How many kilograms of
potatoes are left to sell?

b) The distance between Sita’s house and her school is a kilometer. On
the way to the school, her friend Gita’s house is 0.256 km away from
her house and their friend Rita house is 0.375 km far from Gita’s house.
How far is the Rita’s house from school?

12. Let's copy these sums. Then fill in the blanks with appropriate decimals to
make the equations valid.

a) 0.2 + ...................... + 0.002 = 0.222

b) ...................... + 0.07 + 0.007 = 0.777

c) 0.9 + 0.09 + ...................... = 0.999

d) 26.54 + ...................... + 34.65 = 102.96

e) 46.125 + 75.924 – ...................... = 58.365

f) 100.50 – 50.25 – ...................... = 16.75

It’s your time - Project Work!

13. a) Let's cut a few number of rectangular paper strips each of 10 cm long.
Divide each strip into 10 equal parts. Then, shade the parts with
different colour to show the following sums.

(i) 0.2 + 0.3 (ii) 0.4 + 0.5 (iii) 0.6 + 0.7 (iv) 0.8 + 0.9

b) Let's visit to the available website and find today's exchange rates of the
following currencies with Nepali currency. Then, answer the questions.

1 US Dollar 1 Australian Dollar 1 European Euro

(i) By how is 1 US Dollar more expensive than 1 Australian Dollar?

(ii) By how is 1 Euro more expensive than 1 US Dollar?

(iii) Which one is the cheapest currency among these three currencies?

Vedanta Excel in Mathematics - Book 6 100 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur