Vedanta Excel in Mathematics Books 4 Final (2079)

FFuunnddaammeennttaallOOppeerraattioionnss--II

d) A packet of 500 ml of milk gives 17 g of protein. If you drink 18 packets
of milk in a month, how much protein do you get?

e) We get roughly 72 calories from 1 boiled egg. How much calories do we
get from 30 boiled eggs?

f) A water tanker full of water carries 7,500 litres of water in one trip.
How much water does it carry in 20 trips?

g) The distance between your home and your school is 8 km. How many
kilometres do you travel in 15 days?

h) The distance between place A and place B is 32 km. A bus carries
passengers from A to B and B to A 7/7 times everyday. How many
kilometres does the bus travel in a day?

i) A bus can travel 55 km in 1 hour. How many kilometres does it travel in
24 hours?

13. a) There are 7 days in one week. How many days are there in 52 weeks?

b) There are 12 months in one year. How many months are there in
12 years?

c) There are 365 days in one year. How many days are there in 15 years?

14. a) In a school assembly, students are arranged in 16 rows with 25 students
in each row. How many students are there in the assembly?

b) In a hall, chairs are arranged in 15 columns with 18 chairs in each
column. How many chairs are there in the hall?

It's your time - Project work!

15. a) A 8-10 year old child needs to drink roughly 1500 ml of water each day.
Estimate, how many millilitres of water do you drink in 1 day and in 1

week?

b) How many grams (or kilograms) of vegetables do your family (or your

hostel) consume in 1 day? Discuss with your family members (or with

hostel incharge) and estimate the quantities of vegetables consumed

in 30 days.

c) How many grams (or kilograms) of rice do your family (or your hostel)

consume in 1 day? Discuss with your family members (or hostel

incharge) and estimate the quantities of rice consumed in 30 days.

16. a) Let's draw as many circles as the number of rows equally in each row
in a chart paper. Then find the number of circles by multiplication.

b) Let's draw as many circles as the number of columns equally in
each column in a chart paper. Then find the number of circles by

multiplication.

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3.7 Division - Looking back
Let's investigate a few interesting ideas about division from these

illustrations.

When 6 pencils are shared between 2 children, how many pencils does
each child share?

6 ÷ 2 = 3 Each shares 3 pencils.

How many threes are there in 6?

6 ÷ 3 = 2 There are 2 threes in 6.

Classwork - Exercise
1. Let's say and write the correct numbers in the blank spaces.

a) If 3 children share 6 pencils, how many does each get?

Each gets ÷ = pencils.

How many twos are there in 6? ÷ =

b) If 2 boys share 8 apples, how many does each get?

Each gets ÷ = apples.

How many fours are there in 8?

÷ =

c) If 4 girls share 12 guavas, how many does each get?

Each gets ÷ = guavas.

How many threes are there in 12?

÷ =

2. Let's circle and group the marbles. Then complete the division.

a) 9 divided into 3 groups b) 10 divided into 2 groups



÷ = ÷=
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vedanta Excel in Mathematics - Book 4

c) 15 divided into 5 groups FFuunnddaammeennttaallOOppeerraattioionnss--II

d) 16 divided into 4 groups

÷ = ÷=

3.8 Dividend, divisor, quotient, and remainder

Classwork - Exercise

Let's say and write the answer in the blank spaces.

1. a) 18 ÷ 3 = 6 dividend is 18 divisor is 3 quotient is 6

b) 36 ÷ 4 = dividend is divisor is quotient is

c) 56 ÷ 7 = dividend is divisor is quotient is

d) 70 ÷ 10 = dividend is divisor is quotient is

2. a) 9 ÷ 2 = 4 quotient and 1 is remainder

b) 7 ÷ 2 = quotient and is remainder

c) 10 ÷ 3 = quotient and is remainder

d) 11 ÷ 3 = quotient and is remainder

e) 18 ÷ 4 = quotient and is remainder

f) 23 ÷ 5 = quotient and is remainder

3.9 Division as repeated subtraction

Let's investigate the relation between division and subtraction from the
following illustrations.

1. How many times 2 is subtracted from 6 to get 0?

6 – 2 = 4 4–2=2 2–2=0
1 time
2 times 3 times So, 6 ÷ 2 = 3

2. How many times 3 is subtracted from 12 to get 0?

12 – 3 = 9 9 – 3 = 6 6–3=3 3 – 3 = 0
4 times
1 time 2 times 3 times So, 12 ÷ 3 = 4

51Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

FuFnudnadmamenetnatlaOl Opepreartaitoinosns- I- II

Did you investigate the fact of division and subtraction?
Division is a process of repeated subtraction.

Classwork - Exercise
1. Let's write the correct number in the blank spaces.
a) How many times 2 is subtracted from 8 to get 0?

8 – 2 = , 6 – 2 = , 4 – 2 = , 2 – 2 = So, 8 ÷ 2 =

b) How many times 3 is subtracted from 9 to get 0?

9 – 3 = , 6 – 3 = , 3 – 3 = So, 9 ÷ 3 =

c) How many times 4 is subtracted from 16 to get 0?

16 – 4 = , 12 – 4 = , 8 – 4 = , 4 – 4 = So, 16 ÷ 4 =

d) How many times 6 is subtracted from 12 to get 0?

12 – 6 = , 6 – 6 = So, 12 ÷ 6 =

3.10 Relation between multiplication and division

Classwork - Exercise

Let's investigate the relation between multiplication and division from
the given example. Then, say and write the correct numbers in the blank
spaces.

1. a) It is 4 × 3 = 12 It also means 12 ÷ 4 = 3

4 times 3 dots 12 dots are divided into 4 groups.

b) × = and ÷ =

c) × = and ÷ =

d) × = and ÷ =

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2. a) 5 × 3 = 15, So, 15 ÷ 5 = 3 and FFuunnddaammeennttaallOOppeerraattioionnss--II

15 ÷ 3 = 5

b) 2 × 4 = , So, 8 ÷ 2 = and 8 ÷ 4 =

c) 4 × 7 = , So, 28 ÷ 4 = and 28 ÷ 7 =

d) 8 × 6 = , So, ÷ 8 = 6 and ÷6 =8

e) 9 × 10 = , So, ÷ 9 = 10 and ÷ 10 = 9

It's your time!
3. Let's write your numbers. Then, have fun of multiplication and division.

a) × = So, ÷ = and ÷ =

b) × = So, ÷ = and ÷ =

c) × = So, ÷ = and ÷ =

Quiz time!

4. Let's say and write the numbers as quickly as possible.

a) The product of two numbers is 14 and one of them is 2.

The other number is ÷ =

b) The product of two numbers is 18 and one of them is 6.

The other number is ÷ =

c) The quotient of 30 divided by a number is 5.

The number is ÷ =

d) The quotient of 56 divided by a number is 8.

The number is ÷ =

e) Dividend is 13, divisor is 2, quotient = remainder =

f) Dividend is 27, divisor is 5, quotient = remainder =

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Puzzle Time!
5. Let's fill in the missing numbers to complete the sums.

× 4 = 20 90 ÷ =9 ÷ 4=
× ××
××× ÷ ÷÷
÷ =3
×= ÷= = ==

= == = == ÷ 8 =9

15 × = 60 15 ÷ =3

6. Let's recall the multiplication tables of 1, 2, 3, .... 10. Tell and write the
answers as quickly as possible.

a) 24 ÷ 4 = b) 30 ÷ 5 = 5 × 1 = 5, 5 × 2 = 10, 5 × 3 = 15,
c) 42 ÷ 7 = d) 54 ÷ 6 = 5 × 4 = 20, 5 × 5 = 25, 5 × 6 = 30
So, 30 ÷ 5 = 6

e) 72 ÷ 9 = f) 80 ÷ 8 =

g) 45 ÷ 5 = h) 56 ÷ 7 =

3.11 Division by row and column

Let's investigate the fact that how dividing into rows is different from dividing
into columns.

Dividing into rows Dividing into column
6 ÷ 2 = 3 in each row 6 ÷ 3 = 2 in each column

Classwork - Exercise

1. Let's complete these rows and columns division.

a) b) c)

÷ = ÷ = ÷=

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d) e) FFuunnddaammeenntatal lOOppeerraattioionnss--II

f)

÷ = ÷ = ÷=

2. Let's draw dotted lines and divide into rows or in columns. Then, find
the quotient.

a) b) c)

12 ÷ 3 = 10 ÷ 5 = 24 ÷ 4 =

3.12 Dividing tens, hundreds, thousands,... by 10, 20, 300, 4000, ...

Let's investigate the rule of division with these numbers and become
faster than a calculator!

1. Divide 60 by 20. Equal number of zeros from
60 ÷ 20 = 6 ÷ 2 = 3 60 and 20 are cancelled.
Then 6 ÷ 2 = 3!

2. Divide 2800 by 400. Equal number of zeros from
2800 ÷ 400 = 28 ÷ 4 = 7 2800 and 400 are cancelled.
Then 28 ÷ 4 = 7!

3. Divide 1200 by 30. Equal number of zeros from 1200 and
1200 ÷ 30 = 120 ÷ 3 = 40 30 are cancelled. Then divide 12 by 3
and add one zero to the quotient 4.

Classwork - Exercise

1. Let's say and write the quotient as quickly as possible.

a) 80 ÷ 20 = b) 90 ÷ 30 = c) 80 ÷ 40 =

d) 100 ÷ 50 = e) 120 ÷ 40 = f) 140 ÷ 20 =

55Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

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g) 1500 ÷ 300 = h) 1800 ÷ 600 = i) 2400 ÷ 400 =
j) 2700 ÷ 300 = k) 3200 ÷ 800 = l) 4500 ÷ 500 =
m) 800 ÷ 20 = n) 1600 ÷ 80 = o) 2800 ÷ 70 =

3.13 Division facts

Now, let's learn a few important facts about division.

(i) When a number is divided by 1, the quotient is the number itself.

4 ÷ 1 = 4, 7 ÷ 1 = 7, 12 ÷ 1 = 12, and so on.

(ii) When a number is divided by itself, the quotient is always 1.

5 ÷ 5 = 1, 8 ÷ 8 = 1, 15 ÷ 15 = 1, and so on.

(iii) When 0 is divided by any non-zero number, the quotient is always 0.

0 ÷ 6 = 0, 0 ÷ 9 = 0, 0 ÷ 18 = 0, and so on.

(iv) Dividend = Divisor × Quotient + Remainder

In 7 ÷ 2 = 3 is the quotient and 1 is remainder.

So, 7 = 2 × 3 + 1 = 6 + 1 = 7 = Divisor × Quotient + Remainder

3.14 Division of bigger numbers
1. Let's divide 54 by 4.

4 5 4 13 ← Quotient
54 ÷ 4 = = – 4 ↓

14
40 ÷ 4 12÷4 R 1 ten 3 ones Remainder – 1 2
2 ← Remainder

2. Let's divide 118 by 5.

118 ÷ 5 = = 5 1 1 8 23 ← Quotient
= – 10 ↓
100 ÷ 5 15 ÷ 5 R
18
–15

3 ← Remainder

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3. Let's divide 7246 by 7. 4. Let's divide 9018 by 9.

7) 7246 )1035 Quotient 9) 9018 )1002 Quotient
–7 –9
00
02 –0
–0 01
–0
24 18
– 21 – 18
0 Remainder
36
– 35 Here, divisor 25 has two digits.
So, at first try to divide two digits
1 Remainder 27 of the dividend 278.
27 ÷ 25 = 1 time and 2 remainder.
Then continue the process.

5. Let's divide 278 by 25.

25) 278 )11
–25

28
– 25

3
Q = 11 and R = 3

6. Let's divide 5976 by 84. Here, we cannot divide 59 by 84.
So, let's divide 597 by 84.
84) 5976)71 Trick: Let's think 59 ÷ 8. It goes 7 times.
–588 Let's try 84 × 7 = 588.
So, 597 ÷ 84 = 7 times and remainder 9.
96 Then, continue the process.
– 84

12
Q = 71 and R = 12

Exercise - 3.2
Section A - Classwork
1. Let's say and write the dividend and divisor. Then, find the quotient.

a) b) c)

÷ = ÷ = ÷ =

2. Let's draw as many dots as to match the sums. Then say and write the
quotient as quickly as possible.

a) 15 ÷ 5 = =

57Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

FFuunnddaammenentatal Ol Opperearatitoinons s- -I II = =
=
b) 15 ÷ 3 =
c) 24 ÷ 6 =
d) 24 ÷ 4 =

3. Let's write the total number of dots and divide by number of rows or
by number of columns. Then, find the number of dots in each row and
column.

a) b)

÷ = dots in each row ÷ = dots in each column

4. Let's say and write the missing dividend, divisor or quotient.

a) 36 ÷ = 9 b) 42 ÷ = 6 c) 56 ÷ = 7

d) 63 ÷ 7 = e) ÷ 9 = 8 f) ÷ 8 = 10

5. Let's say and write the answer as quickly as possible.

a) The cost of 4 pencils is Rs 32. The cost of 1 pencil = 32 ÷ 4 = Rs

b) The cost of 5 sweets is Rs 35. The cost of 1 sweet = ÷ = Rs

c) 40 cherries are shared between 8 children equally.

Each of them gets ÷= cherries.

d) 45 students are kept in 5 rows equally. The number of students in each row

=÷= students.

e) How many times are twos be subtracted from 10 to get 0?

÷ = times
f) How many times are fives be subtracted from 15 to get 0?

÷ = times 58 Approved by Curriculum Development CentreSanothimi, Bhaktapur

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6. Let's say and write the quotients as quickly as possible.

a) 100 cm = 1m So, 200 cm = 200 ÷ 100 = m

b) 1000 m = 1 km So, 4000 m = 4000 ÷ 1000 = km

c) 1000 g = 1 kg So, 5000 g = 5000 ÷ 1000 = kg

d) 1000 ml = 1 l So, 8000 ml = 8000 ÷ 1000 = l

e) 100 kg = 1 quintal So, 7000 kg = 7000 ÷ 100 = quintal

7. Let's say and write the number of rupees notes.

a) How many Rs 5 notes are there in Rs 100? ÷=

b) How many Rs 10 notes are there in Rs 100? ÷=

c) How many Rs 10 notes are there in Rs 1000? ÷=

d) How many Rs 20 notes are there in Rs 500? ÷=

e) How many Rs 50 notes are there in Rs 1000? ÷=

f) How many Rs 100 notes are there in Rs 1000? ÷ =

8. Let's complete division by using number lines as shown.
a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

12 ÷ 4 =

b)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

÷=

c)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

÷=

59Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

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Section B

9. Let's rewrite the sums and find the quotients in your exercise book.

a) 80 ÷ 20 b) 800 ÷ 20 c) 8000 ÷ 20 d) 8000 ÷ 200

e) 60 ÷ 30 f) 600 ÷ 30 g) 6000 ÷ 30 h) 6000 ÷ 300

i) 120 ÷ 40 j) 250 ÷ 50 k) 360 ÷ 90 l) 4200 ÷ 60

m) 5600 ÷ 70 n) 32000 ÷ 400 o) 45000 ÷ 500 p) 72000 ÷ 800

Let's divide and find the quotient and remainder.

10. a) 58 ÷ 5 b) 74 ÷ 6 c) 65 ÷ 4 d) 84 ÷ 7 e) 96 ÷ 8

f) 213 ÷ 2 g) 326 ÷ 3 h) 416 ÷ 4 i) 530 ÷ 5 j) 927 ÷ 9

k) 429 ÷ 2 l) 638 ÷ 3 m) 853 ÷ 4 n) 767 ÷ 5 o) 852 ÷ 6

11. a) 4602 ÷ 4 b) 5963 ÷ 5 c) 6792 ÷ 6 d) 7856 ÷ 7 e) 3276 ÷ 3

f) 7156 ÷ 7 g) 8024 ÷ 8 h) 1348 ÷ 3 i) 2352 ÷ 4 j) 5257 ÷ 7

12. a) 168 ÷ 14 b) 185 ÷ 15 c) 216 ÷ 21 d) 360 ÷ 24 e) 550 ÷ 32

f) 152 ÷ 16 g) 210 ÷ 26 h) 1470 ÷ 35 i) 3358 ÷ 54 j) 5699÷ 78

Let's read these problems carefully and solve them.

13. a) The cost of 5 kg of rice is Rs 425. Find the cost of 1 kg of rice.

b) If the cost of 1 kg of rice is Rs 85, how many kilograms of rice can be
bought for Rs 425?

c) The cost of 8 l of milk is Rs 768. Find the cost of 1 l of milk.

d) If the rate of cost of milk is Rs 96 per litre, how many litres of milk can
be purchased for Rs 768?

14. a) In 1 dozen of pencils there are 12 pencils. How many dozens of pencils
are there in 180 pencils?

b) 1 crate eggs contains 30 eggs. How many crates of eggs are there in
750 eggs?

c) 5 dozens of cricket balls can be kept in a box. How many boxes are
needed to keep 1440 balls?

15. a) If 12 quintals of weight is equal to 1200 kg, how many kilograms are
there in 1 quintal?

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b) If 9 metric tons weight is equal to 9000 kg, how many kilograms are
there in 1 metric ton?

c) 15 bottles of equal capacity can hold 7500 ml of liquid. Find the capacity
of each bottle in millilitres.

16. a) 154 children of 14 football teams of different schools are playing
football in an inter-school football match. How many players are there
in each team?

b) There are 11 players in a football team. 154 children of different school
teams are playing football in an inter-school football match. How many
teams are playing the match?

17. a) 540 students are arranged in 18 rows with the equal number of students
in each row. How many students are there in 1 row?

b) 480 students are arranged in some columns with 32 students in each
column. How many columns of students are there?

18. a) There are 7 days in one week. How many weeks are there in 364 days?

b) There are 12 months in one year. How many years are there in 300
months?

c) There are 365 days in one year. How many years are there in 4380
days?

It's your time - Project work!

19. a) Let's write any three 2-digit numbers. Divide them separately by any
three 1-digit divisor. Then, show that

Dividend = Divisor × Quotient + Remainder

b) Let's write any three 3-digit numbers. Divide them separately by any
three 1-digit divisor. Then, show that

Dividend = Divisor × Quotient + Remainder
Let's stick your findings on the wall-magazine of your school!

20. Let's draw 12 circles in 1 row, 2 rows, 3 rows, 4 rows, 6 rows and 12 rows

with equal number of circles in each row in a chart paper. Find the number
of circles in each row by using division process. Stick your findings on the
school's wall-magazine!

61Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

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3.15 Divisibility Test
Let's discuss about the answer of these questions.
a) Are 2, 4, 6, 8 and 10 exactly divisible by 2?
b) Are 3, 5, 7 and 9 exactly divisible by 2?
c) Are 3, 6, 9, 12, 15 and 18 exactly divisible by 3?
d) Are 4, 5, 7, 8, 10 and 11 exactly divisible by 3?
When a dividend is divisible by a divisor with no remainder, the dividend

is called exactly divisible by the divisor.
10 ÷ 2 = 5 quotient with no remainder. 10 is exactly divisible by 2.
13 ÷ 2 = 6 quotient with 1 remainder. 13 is not exactly divisible by 2.
Now, let's learn a few rules of divisibility test.

Exactly Rules of divisibility test
divisible by

The digit at ones place of any number is 0 or even number

2 (2, 4, 6, 8). So, 90, 132, 754, 3616, 5978, ... are exactly

divisible by 2.

The sum of the digits of any number is exactly divisible by

3 3. In 225, 2 + 2 + 5 = 9 and 9 is exactly divisible by 3. So,

225 is exactly divisible by 3.

The number formed by last two digits of any number is

4 exactly divisible by 4. So, 92, 308, 924, 2564, ... are exactly

divisible by 4.

5 The digit at ones place is 0 or 5. So, 80, 170, 245, 4195, ...
are exactly divisible by 5.

6 Any even number exactly divisible by 3 are also exactly
divisible by 6. So, 84, 198, 2580, ... are exactly divisible by 6.

The sum of the digits of any number is exactly divisible by

9 9. In 486, 4 + 8 + 6 = 18 and 18 is exactly divisible by 9.

So, 486 is exactly divisible by 9.

10 The digit at ones place is 0. So, 70, 350, 830, 4120, ... are
exactly divisible by 10.

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Classwork - Exercise
1. Let's circle the numbers which are exactly divisible by the given

numbers.
a) by 2 → 50 92 117 354 700 945 1681 4218
b) by 3 → 62 87 102 460 540 893 2154 5307
c) by 4 → 48 94 108 282 628 714 3236 8560
d) by 5 → 56 70 145 378 500 903 4785 7129
e) by 6 → 72 93 234 416 738 825 5160 9204
f) by 9 → 93 108 319 558 6300 7210 2457 6980
g) by 10 → 60 95 130 305 550 1002 3700 5670
2. Let's use the rule of divisibility test to find whether these numbers are

exactly divisible by 3 or 9.

a) Is 174 exactly divisible by 3? 1 + 7 + 4 = 12 Yes.

b) Is 289 exactly divisible by 9? 2 + 8 + 9 = 19 No.

c) Is 253 exactly divisible by 3? ++=

d) Is 414 exactly divisible by 3? ++=

e) Is 153 exactly divisible by 9? ++=

3.16 Factors and multiples

Let's investigate the ideas of factors and multiples from the following

examples.

In how many ways can you In how many ways can you
make 12 by multiplication? make 18 by multiplication?

1 × 12 2×6 1 × 18 2×9

12 18

3×4 3×6

So, 1, 2, 3, 4, 6 and 12 are the So, 1, 2, 3, 6, 9 and 18 are the
factors of 12. factors of 12.
12 is the multiple of 1, 2, 3, 4, 18 is the multiple of 1, 2, 3, 6,
6 and 12. 9 and 18.
63Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
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The factors of a number can exactly divide the number.

12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2 and 12 ÷ 12 = 1

18 ÷ 1 = 18, 18 ÷ 2 = 9, 18 ÷ 3 = 6, 18 ÷ 6 = 3, 18 ÷ 9 = 2 and 18 ÷ 18 = 1

3.17 Prime Factors
All possible factors of 15 are 1, 3, 5 and 15. Among these factors 3 and 5 are

the prime factors because 3 and 5 are prime numbers.
Similarly, all possible factors of 20 are 1, 2, 4, 5, 10 and 20. Among these

factors, 2 and 5 are the prime factors.

Classwork - Exercise

1. Let's find the multiples. Say and write all possible factors of the
multiples. Then list the prime factors.

a) 1 × 4 = , 2 × 2 =

All possible factors of 4 are , and

The prime factors of 4 is

b) 1 × 6 = , 2 × 3 =

All possible factors of 6 are , , and

The prime factors of 6 are and

c) 1 × 10 = , 2 × 5 =

All possible factors of 10 are ,, and
The prime factors of 10 are and

2. Let's say and write all possible factors of these numbers. Then, circle
the prime factors.

a) All possible factors of 8 are , , and

b) All possible factors of 9 are , and

c) All possible factors of 15 are , , and

d) All possible factors of 20 are , , , , and

3.18 Process of finding prime factors

Let's study these examples and investigate the idea of finding prime factors

of the given numbers.

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Let's find the prime factors of 12.

Here, 12 ÷ 2 = 6 and 6 ÷ 2 = 3 Factor tree

2 12 → 12 ÷ 2 = 6 12

2 2 → 6 ÷ 2 = 3 2×6
2× 2×3
3
So, 12 = 2 × 2 × 3

So, 12 = 2 × 2 × 3

Thus, to find the prime factors of a given number, we should start to divide

the number by the lowest prime number. We should continue division till

the quotient becomes a prime number. Factor tree
Again, let's find the prime factors of 24. 24

2 24 → 24 ÷ 2 = 12 2 × 12
2 12 → 12 ÷ 2 = 6

2 6 → 6 ÷ 2 = 3 2 × 2 ×6

3 2 × 2× 2 × 3

So, 24 = 2 × 2 × 2 × 3 So, 24 = 2 × 2 × 2 × 3

Classwork - Exercise

1. Let's divide the given numbers by the prime numbers till the quotient

becomes a prime numbers.

a) 2 18 b) 3 27 c) 2 30

3 3

18 = × × 27 = × × 30 = × ×

2. Let's say and write the correct numbers in the empty circles. Then, write
the number as the product of its prime factors from the factor tree.

a) 18 b) 28 c) 36

×9 2× × 18

× ×3 × ×7 ××

2× × ×3

18 = × × 28= × × 36 = × × ×
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FuFnudnadmamenetnatlaOl Opepreartaitoinosns- I- II

3.19 Process of finding multiples of a given number
Let's study these examples and investigate the process of finding multiples

of a given number.

2 × 1 = 2 2 × 2 = 4 2 × 3 = 6 2 × 4 = 8 2 × 5 =10
So, 2, 4, 6, 8, 10, ... are the first five multiples of 2.

3 × 1 = 3 3 × 2 = 6 3 × 3 = 9 3 × 4 = 12 3 × 5 =15
So, 3, 6, 9, 12, 15, ... are the first five multiples of 3.

Exercise - 3.3
Section A - Classwork
1. Let's say and write the correct answer as quickly as possible.

a) Numbers between 59 and 71 which are exactly divisible by 2 are

, , , , ,

b) Numbers between 89 and 100 which are exactly divisible by 3.

, , ,

c) Any 5 three-digit numbers which are exactly divisible by 4.

, , , ,

d) Any 5 three-digit numbers between 100 and 150 which are exactly
divisible by 5.

, , , ,

e) Any 5 numbers between 10 and 50 which are exactly divisible by 6.

, , , ,

f) Any 5 three-digit numbers which are exactly divisible by 9.

, , , ,

g) Any 5 three-digit numbers less than 300 and exactly divisible by 10.

, , , ,

vedanta Excel in Mathematics - Book 4 66 Approved by Curriculum Development CentreSanothimi, Bhaktapur

FFuunnddaammeennttaallOOppeerraattioionnss--II

2. Let's say and write these numbers as the product of their prime factors.

a) 4 = × b) 6 = × c) 8 = × ×

d) 9 = × e) 10 = × f) 12 = × ×

g) 14 = × h) 15 = × i) 20 = × ×

Section B

3. Let's use the rules of divisibility test. Then, identify which of the
following numbers are exactly divisible by 3, or 3 and 9 both.

a) 153 b) 276 c) 387 d) 489 e) 5967

4. Rewrite and complete these Factor Trees. Then, write the numbers as
the product of their prime factors.

a) 16 b) 24 c) 36
2×
2× × 12 ×

×× × 2× × ×
×
2× × ×3 2×

16 = 2 × 2 × 2 × 2 24 = 36 =

d) 30 e) 40 f) 54

2× 2× 2 × 27

×× ×× ××

×× ×5 ×× ×

40 = 54 =
30 =

5. Let's use the process of finding prime factors of these numbers. Then,
write the numbers as the product of their prime factors.

a) 2 16 b) 12 c) 15 d) 18 e) 20 f) 24

2 8 g) 27 h) 28 i) 30 j) 32 k) 35

2 4 l) 36 m) 40 n) 42 o) 45 p) 48

2 ? vedanta Excel in Mathematics - Book 4
So, 16 = 2 × 2 × 2 × 2
67
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Fundamental Operations - I

Unit Order of Operations

4

4.1 Simplification - A single solution of a mixed operation

Classwork - Exercise

1. Let's listen to your teacher! Say and write the answer as quickly as
possible.

a) Add 6 and 8. Then subtract 5. 6 + 8 – 5 = 14 – 5 = 9

b) Add 4 and 7. Then subtract 3. +–=–=

c) Subtract 7 from 15. Then add 9. –+=+=

d) Subtract 6 from 16. Then again subtract 4.

– – = – =

e) Multiply 5 and 4, then add 7. ×+=+=

f) Divide 27 by 3 and subtract 6. ÷–=–=

The problems given above are the mixed operations, because these problems
have more than one operations. To workout these mixed operations, we
should perform division, multiplication, addition, and subtraction in order
to get a single answer. The order of performing such mixed operation to get
a single and simple answer is known as simplification.

2. First divide, then multiply, add, and subtract.

a) 8 ÷ 2 × 3 + 4 – 6 b) 10 + 7 – 15 ÷ 5 × 4

= × + – =+–×

= + – = + –

= – = –

= =

vedanta Excel in Mathematics - Book 4 68 Approved by Curriculum Development CentreSanothimi, Bhaktapur

FundamOenrdtaelrOopf eorpaetriaotnison- sI

4.2 Order of operations

Order of addition and subtraction

Example 1: Simplify a) 12 + 6 – 8 b) 12 – 8 + 6 c) 15 – 6 – 3

Solution

a) 12 + 6 – 8 = 18 – 8 b) 12 – 8 + 6 = 4 + 6 c) 15 – 6 – 3 = 9 – 3

= 10 = 10 =6

Order of multiplication, addition, and subtraction

Example 2: Simplify a) 4 × 7 + 2 b) 2 + 4 × 7 c) 20 – 5 × 3

Solution 4 × 7 + 2 = 4 × 9 = 36

a) 4 × 7 + 2 = 28 + 2 Which is the wrong order!

= 30 2 + 4 × 7 = 6 × 7 = 42
Which is the wrong order!
b) 2 + 4 × 7 = 2 + 28
20 – 5 × 3 = 15 × 3 = 45
= 30 Which is the wrong order!

c) 20 – 5 × 3 = 20 – 15

=5

Order of division and multiplication

Example 3: Simplify a) 24 ÷ 4 × 2 b) 2 × 24 ÷ 4

Solution 24 ÷ 4 × 2 = 24 ÷ 8 = 3

a) 24 ÷ 4 × 2 = 6 × 2 Which is the wrong order!

= 12 Another process
2 × 24 ÷ 4 = 48 ÷ 4 = 12
b) 2 × 24 ÷ 4 = 2 × 6

= 12

Example 4: Simplify 3 × 10 ÷ 5 + 12 – 4

Solution

3 × 10 ÷ 5 + 12 – 4 Another process Short process

= 3 × 2 + 12 – 4 3 × 10 ÷ 5 + 12 – 4 3 × 10 ÷ 5 + 12 – 4

= 6 + 12 – 4 = 30 ÷ 5 + 12 – 4 = 3 × 2 + 8

= 18 – 4 = 6 + 12 – 4 =6+8

= 14 = 18 – 4 = 14 = 14

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OFrudnedraomf oepnetaral tOiopnesrations - I

4.3 Use of brackets in simplification

Let's read these illustrations carefully and learn to use brackets in
simplifications.

Example 5: Multiply the sum of 5 and 4 by 7.
Solution

Here, the mathematical expression is (5 + 4) × 7 but not 5 + 4 × 7

(5 + 4) × 7 = 9 × 7 But 5 + 4 × 7 = 5 + 28 = 33 and it is the
wrong answer for the given problem
= 63

In this problem, at first, we need to find the sum of 5 and 4. Then, the sum
is multiplied by 7. Therefore, to find the sum at first, we enclose 5 + 4 in
the brackets ( ).

Example 6: Find 5 times the difference of 16 and 12.
Solution

Here, the difference of 16 and 12 = 16 – 12

So, 5 × (16 – 12) = 5 × 4

= 20

Example 7: Simplify a) 40 ÷ 5 × 2 + 12 – 7 b) 40 ÷ (5 × 2) + 12 – 7
Solution

a) 40 ÷ 5 × 2 + 12 – 7 b) 40 ÷ (5 × 2) + 12 – 7

= 8 × 2 + 12 – 7 = 40 ÷ 10 + 12 – 7

= 16 + 5 = 4 + 5

= 21 = 9

Exercise - 4.1

Section A - Classwork

1. Let's read the instructions and make mathematical expressions. Then
simplify.

a) Add 5 and 7, then, subtract 8. +–=–=

b) Subtract 6 from 18, then, add 4. – + = + =

c) Subtract 9 from 20, then, subtract 3. – – = – =

d) Multiply 6 and 7, then, add 10. × + = + =

e) Divide 72 by 9, then, multiply by 5. ÷ ×=×=
70
vedanta Excel in Mathematics - Book 4 Approved by Curriculum Development CentreSanothimi, Bhaktapur

FundaOmrdenertaolfOoppeerraattiioonnss- I

2. Let's simplify. Say and write the answer as quickly as possible.

a) 4 + 7 – 2 = – = b) 10 – 3 + 8 = + =

c) 16 – 5 – 6 = – = d) 4 × 6 + 3 = + =

e) 5 × 7 – 10 = – = f) 9 + 7 × 2 = + =

g) 30 – 4 × 3 = – = h) 27 ÷ 9 × 6 = × =

i) 4 × 10 ÷ 5 = × = j) 4 × 10 ÷ 5 = ÷ =

3. Let's insert the appropriate sign (+, – , ×, or ÷) in the blank spaces to
get the given answer.

a) 4 5 2 = 7 b) 4 5 2 = 18

c) 4 5 2 = 22 d) 4 5 2 = 14

e) 3 4 2 = 6 f) 3 4 2 = 5

g) 3 4 2 = 10 h) 3 4 2 = 24

i) 12 6 3 = 3 j) 12 6 3 = 5

k) 12 6 3 = 10 l) 12 6 3 = 9

Section B b) 5 × 7 – 20 + 10
Let's simplify these mixed operations.
d) 60 ÷ 6 + 3 × 4
4. a) 6 × 4 + 8 – 9 f) 10 × 4 ÷ 2 – 5 + 6
h) 54 ÷ 6 – 5 × 3 + 7
c) 30 ÷ 5 × 2 + 7 j) 4 × 18 ÷ 9 – 3 × 5 + 12
e) 9 × 8 ÷ 4 – 3
g) 56 ÷ 7 + 4 × 5 – 8
i) 30 + 2 × 10 ÷ 5 × 3 – 15

5. a) 9 + (8 – 3) b) 15 – (7 + 5)

c) 4 × (20 – 12) d) 36 ÷ (8 + 4)
e) (50 – 20) ÷ 6 f) (6 + 4) × 8 ÷ 2
g) (5 + 3) × (7 + 2) h) (12 + 16) ÷ (18 – 11)
i) 72 ÷ (15 – 6) × (12 – 8) j) 80 ÷ (14 – 9) × 2 + 7
k) (8 + 4 × 8) ÷ (6 × 3 – 10) l) (60 – 3 × 4) ÷ (7 × 5 – 29)

71Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

FuOnrdaemr oefnotapleOrapteiorantsions - I

6. Let's enclose the operation to perform it first by using brackets. Then,

simplify to get the given answer.

a) (3 + 2) × 5 = 25 b) 4 + 3 × 2 = 14 c) 4 × 10 – 8 = 8

d) 24 ÷ 4 × 2 = 3 e) 20 ÷ 7 + 3 = 2 f) 49 – 7 ÷ 7 = 6

g) 45 + 18 ÷ 9 = 7 h) 3 × 6 + 4 ÷ 2 = 15 i) 24 ÷ 6 – 2 × 5 = 30

7. Let's make mathematical expressions and simplify.

a) You have Rs 10 and mother gives you Rs 5 more. If you spend Rs 8, how
much money do you have now?

Solution

Rs10+Rs5–Rs8=

So, I have Rs now.

b) Bishwant had 9 sweets. He ate 4 sweets and again he bought 7 sweets.
How many sweets did he have now?

c) There are 25 students in a school bus. 6 students get down at one place
and 8 students get down at another place. How many students are left
in the bus?

d) There are 7 rows of 5 chairs in each row and 4 more chairs in a room.
How many chairs are there in the room?

Solution

7 × 5 + 4 =

So, there are chairs in the room.

e) Bina Magar had 7 color pencils. She bought a few more colour pencils
for Rs 40 at the rate of Rs 8 each. How many colour pencils does she
have now?

f) On Friday, there were 27 students in class four. 16 of them were girls
and the rest were boys. If only 3 boys were absent on that day, find the
number of boys in class four.

g) Ram, Hari and Laxmi were in a race. When Ram finished the race, Hari
was 5 metres behind Ram and Laxmi was 9 metres behind Hari. How
far away from the finish line was Laxmi?

h) After buying 5 chocolates at Rs 10 each, Kalpana Rai had Rs 25 left.
How much money did she have at first?

Let's make mathematical expressions using brackets. Then simplify.

8. a) The sum of 6 and 10 is subtracted from 25.
Solution

vedanta Excel in Mathematics - Book 4 72 Approved by Curriculum Development CentreSanothimi, Bhaktapur

FundamOenrdtaelrOopf eorpaetrioantiso-nIs

25 – (6 + 10) =
b) The sum of 4 and 5 is multiplied by 6.
c) The difference of 20 and 8 is divided by 3.
d) The product of 6 and 4 is divided by 8.
e) 56 is divided by the sum of 3 and 4.

9. a) A sick person takes 20 ml of medicine twice a day. How much medicine
does she take in a week?

Solution
7 × (20 ml + 20 ml) =

So, she takes ml of medicine in a week.

b) The distance between your house and your school is 4 km. How many
kilometres do you travel in 6 days?

c) The highway distance between Birendra Bazar and Lahan is 55 km.
Mr. Mahato travelled 25 km by a taxi, 27 km by a bus and the remaining
distance he walked. How many kilometres did he walk?

d) You buy a pen for Rs 30 and a box for Rs 85. If you give Rs 150 to the
shopkeeper, what changes does the shopkeeper return you?

e) There are 4 girls and 3 boys in each group. How many students are
there in 5 groups?

f) Mrs. Shrestha earns Rs 3500 in a week. She spends Rs 200 everyday to
run her house. How much money does she save in a week?

g) Teacher divided 50 sweets equally between 12 boys and 13 girls of
class four. How many sweets does each student get?

It's your time - Project work!
10. a) Let's rewrite these simplifications and find the mistakes. Then,

complete the simplification in the correct way. You can display your
work on wall-magazine of your school.

20 – 10 – 5 5 + 4 × 3 14 – 5 × 2 30 ÷ 3 × 2
= 20 – 5 = 9 × 3 = 9 × 2 = 30 ÷ 6
= 15 = 27 = 18 =5

b) Let's make any four your own mixed expressions by using all 4 signs
(+, –, ×, ÷) in each expressions. Simplify them and get the correct answer.

? vedanta Excel in Mathematics - Book 4

73Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Fraction Fraction

Unit

5

5.1 Fraction - Looking back!

Classwork - Exercise

1. Let's say and write the answer in the blank spaces as quickly as possible.

a) A pizza is divided into equal parts.

41 b) Fraction of 1 part is

c) Fraction of 2 parts is

d) Fraction of 3 parts is e) Fraction of 4 parts is

f) How many quarters make a whole? 1 Numerator
2. Let's write the stories of these fractions. 2 Denominator

a) 1 2 Numerator
2 3 Denominator

b) 2 2 1 An object is divided into equal parts and part is taken.
3
An object is divided into equal parts and parts are taken.
c) 3
5 An object is divided into equal parts and parts are taken.

3. Let's read the stories and write the fractions.

a) Mother cuts an apple into 4 equal pieces and you eat 3 pieces.

What fraction of the whole apple do you eat?

b) Father cuts a bread into 6 equal slices and sister eats 4 slices.

What fraction of the whole bread does she eat?

c) Teacher divides a rectangle into 8 equal parts and she shades 5 parts.

What fraction of the whole rectangle does she shade?

vedanta Excel in Mathematics - Book 4 74 Approved by Curriculum Development CentreSanothimi, Bhaktapur

Fraction

4. Let's say and write the fractions of the shaded and the non-shaded parts.

Shaded Shaded Shaded Shaded

Non-shaded Non-shaded Non-shaded Non-shaded

5. What do numerator and denominator tell? Let's write the correct
answers in the blank spaces.

a) In 1 , numerator tells part is taken, denominator tells total parts.
3 parts are taken, denominator tells total parts.

b) In 3 , numerator tells
4

6. Let's write the fractions of these fraction names.

a) Two-third = b) One-quarter = c) Three-fifth =

d) Five-sixth = e) Four-seventh = f) Nine-tenth =

7. Let's estimate which one is exactly half, one-third, or one quarter. Write
1, 1, or 1 below the exact diagram.
23 4



8. Let's join the equal parts and make the whole shape. Then, say and
write the answer as quickly as possible.

a) How many halves make a whole? halves

b) How many thirds make a whole? thirds

c) How many quarters make a whole? quarters

75Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Fraction

Let's say and write the answers in the blank spaces. parts

9. Sunayana eats 3 parts of a whole bread.
8

a) In how many parts is the bread divided?

b) How many parts does Sunayana eat? parts.

c) What is the fraction of the parts not eaten?

5.2 Reducing fractions to their lowest terms

Let's study the following examples and learn the process of reducing a
fraction to its lowest terms.

Example 1: Reduce the fractions a) 2 b) 6 c) 8 to their lowest
4 9 10
terms.

Solution 2
4
a) 24 = 21 = 21 2 ÷ 2 = 1 1
42 4 ÷ 2 2 2

b) 6 = 6 2 = 2 6 ÷ 3 = 2 6
9 9 3 3 9 ÷ 3 3 9
2
3

c) 8 = 84 = 4 8÷2 = 4 8
10 10 5 5 10 ÷ 2 5 10

Example 2: Reduce the fractions a) 20 4
60 5
terms.
b) 800 to their lowest
1200

Solution

a) 20 = 20 = 2 1 = 1 Innu26m00e,rcaatnocrealnthdedeeqnuoaml ninuamtobre. rTohfezne,r2osafnrdom6
60 60 6 3 3
2÷2 1
are divided by 2. 6÷2 = 3

b) 800 = 800 = 82 = 2 Two zeros from 800 and 1200 are cancelled.
1200 1200 12 3 3
Then 8 and 12 are divided by by 4.

8÷4 = 32.
12 ÷ 4

vedanta Excel in Mathematics - Book 4 76 Approved by Curriculum Development CentreSanothimi, Bhaktapur

Fraction

Classwork - Exercise

1. Let's reduce these fractions to their lowest terms.

a) b) c)

2 = 3 = 2 =
4 6 6

d) e) f)
4 6 6
8 = 9 = 8 =

2. Let's divide the numerator and denominator by the same common factor
and reduce the fractions to the lowest terms.

a) 4 = 4 ÷ = b) 6 = 6 ÷ = c) 4 = 4 ÷ =
6 6 ÷ 8 8 ÷ 8 8 ÷

d) 3 = 3 ÷ = e) 8 = 8÷ = f) 10 = 10 ÷ =
9 9 ÷ 12 12 ÷ 15 15 ÷

3. Let's cancel the equal number of zeros from numerator and
denominator. Then, reduce the fraction to the lowest terms.

a) 10 = b) 20 = c) 30 = d) 10 =
30 30 40 50

e) 20 = = f) 20 = = g) 40 = = h) 60 = =
60 80 60 90

Exercise - 5.1

Section A - Classwork

1. Let's say and write the fractions of the shaded parts in each pair of
diagrams.

a) 84 b)
1
2

77Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Fraction

2. Let's shade the parts of each pair of diagram to show the given fractions.

a) 2 b) 4
5
4 6
10 8

12

3. Let's say and write the lowest terms of these fractions.

a) 2 = b) 2 = c) 3 = d) 3 =
4 8 6 9
10 30 70 800
e) 50 = f) 70 = g) 100 = h) 900 =

Section B
4. Let's reduce these fractions to their lowest terms.

a) 24 b) 26 c) 63 d) 48 e) 4 f) 5
12 10

g) 93 h) 155 i) 120 j) 4 k) 360 l) 7
16 35

5. Let's cancel the equal number of zeros from numerator and
denominator of each fraction to reduce them to their lowest terms.

a) 10 b) 10 c) 20 d) 5200 e) 30 f) 30
20 40 30 40 50

g) 5400 h) 7400 i) 6500 j) 670000 k) 490000 l) 700
800

6. Let's divide the numerator and denominator of each fraction by their
common factor and reduce them to the lowest terms.

a) 46 b) 68 c) 96 d) 140 e) 6 f) 8
10 10

g) 182 h) 192 i) 195 j) 10 k) 1126 l) 12
15 18

m) 14 n) 1202 o) 1250 p) 1206 q) 18 r) 21
21 24 28

7. Let's cancel the equal number of zeros from numerator and
denominator and reduce the fractions to their lowest terms.

a) 40 b) 60 c) 60 d) 14000 e) 16000 f) 80
60 80 90 100

vedanta Excel in Mathematics - Book 4 78 Approved by Curriculum Development CentreSanothimi, Bhaktapur

Fraction

g) 18200 h) 112000 i) 115000 j) 120500 k) 1920000 l) 1200
1500

It's your time - Project work!

8. Let's take a rectangular sheet of paper and fold it into halves. Shade
one-half part. Again, fold it into halves two times. Then, answer the following
questions.

First folding Second folding Third folding

a) What is the fraction of the shaded part in the first folding?

b) What is the fraction of the shaded parts in the second folding?

c) What is the fraction of the shaded parts in the third folding?

d) Are the shaded parts of the three folding equal?

5.3 Like and unlike fractions
Classwork - Exercise

Let's say and write the answer as quickly as possible. A

1. a) In how many equal parts is the rectangle A divided? B
b) In how many equal parts is the rectangle B divided?

c) Are both the rectangles divided into the same number of parts?

d) What is the fraction of the shaded parts in A?

e) What is the fraction of the shaded parts in B?

Here, both the rectangles are divided into the same (like) number of

parts. So, 1 and 2 are like fractions. Like fractions always have the same
4 4

denominators. 51, 25, and 3 are also the like fractions.
5

79Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Fraction P

2. a) Are the rectangles P and Q divided into the same number
of parts?

b) What is the fraction of the shaded parts in P? Q

c) What is the fraction of the shaded parts in Q?

Here, the rectangles P and Q are divided into the different (unlike) number

of parts. So, 2 and 2 are unlike fractions. Unlike fractions have the different
3 4

denominators. 1 , 3 , and 2 are also the unlike fractions.
2 4 5

Classwork - Exercise

1. Let's say and write whether the fractions of the shaded parts in these
pairs of diagrams are like or unlike fractions.

a) and are fractions.

b) and are fractions.

2. Let's list the like and unlike fractions separately.

a) 2 , 3 , 1 , 2 Like fractions Unlike fractions
3 4 3 5 a)

b) 3 , 5 , 3 , 5 b)
8 6 10 8

5.4 Comparison of like fractions

Classwork - Exercise
1. Let's compare the shaded parts of each pair of like fractions. Then

compare the fractions using the symbols '<' or '>'.

a) b)

43 5 6
55 8 8

Could you investigate the rule of comparing the like fractions?

Like fractions have the same denominators. So, the rule to compare like
fractions is 'just compare the numerators' and compare the fractions.

vedanta Excel in Mathematics - Book 4 80 Approved by Curriculum Development CentreSanothimi, Bhaktapur

Fraction

2. Let's insert the appropriate symbols '<' or '>' between these like
fractions and compare them.

a) 1 2 b) 3 2 c) 5 4 d) 3 7
3 34 47 7 10 10

Vedanta ICT Corner
Please! Scan this QR code or
browse the link given below:

https://www.geogebra.org/m/e8uq9amu

5.5 Proper and improper fractions

Let's investigate the ideas about proper and improper fractions from the
given illustrations.

Bishnu and Laxmi shared 1 bread Shiva and Parbati shared 2 breads

equally between them. equally between them.
1 Each got 2 ÷ 2 = 1 bread.
Each got 1 ÷ 2 = 2 of the bread.

Sita and Ram shared 3 breads equally

between them. 3 1
2 2
Each got 3 ÷ 2 = = 1 and bread.

Here, 1 of the bread has only a half part of the whole bread. So, 1 is a proper
2 2
3 1 3
fraction. But 2 of breads have 1 whole bread and 2 of a bread. So, 2 is an

improper fraction.

Now, let's compare the numerators and denominators of these proper

and improper fractions. Then, investigate the rule to identify proper and

improper fractions.

Proper fractions Improper fractions

2 → Numerator is smaller than 3 → Numerator is greater than
3 denominator. 2 denominator.

3 → Numerator is smaller than 5 → Numerator is greater than
5 denominator. 3 denominator.

Thus, in a proper fraction, numerator is less than denominator.

In an improper fraction, numerator is greater than denominator.

81Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Fraction

5.6 Mixed numbers

Let's read the stories of a few improper fractions.

Story of 3 : A rectangle is divided into 2 equal parts and 3 parts are shaded!
2 Is it possible?

Story of 5 : A rectangle is divided into 2 equal parts and 5 parts are shaded!
2 Is it possible?

Story of 7 : A rectangle is divided into 3 equal parts and 7 parts are shaded!
3 Is it possible?

1 whole half

3 1 + 1 = 121 . So, 121 is a mixed number.
2 2

2 whole half

5 2 + 1 = 212 . So, 221 is a mixed number.
2 2

2 whole one-third

7 2 + 1 = 231 . So, 231 is a mixed number.
3 3

Thus, a fraction made up of a whole number and a proper fraction is called
mixed number (or mixed fraction).

aWnedrseoaodn1. 21 as 'one whole and half' , 231 as 'two whole and one-third',
Classwork - Exercise

1. Let's select the correct answer and fill in the blanks.

a) 9 is a/an fraction. (proper/improper)
4 fraction. (proper/improper)

b) 4 is a/an
9

c) In a proper fraction denominator is than numerator.

(less/greater)

d) In an improper fraction denominator is than numerator.
(less/greater)

e) We get a mixed number only from a/an fraction.
(proper/improper)

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Fraction

2. Let's circle the proper fractions and tick the mixed numbers.

43, 43, 113, 27, 225, 72, 95, 312, 9
5

5.7 Process of changing improper fraction to mixed number

Let's study these illustrations and investigate the rule of changing an

improper fraction to a mixed number. I got it!

3 = 3 ÷ 2 = 1Q 1R = 112 Mixed number is
2
112 Quotient Remainder
Divisor

8 = 8 ÷ 3 = 2Q 2R = 223 223
3

Could you investigate the rule? Discuss with your friends.

5.8 Process of changing mixed number to improper fraction

Let's study the stories of these mixed numbers. Then investigate the rule of
changing a mixed number to an improper fraction.

121 1 time 2 parts 1 × 2 + 1 = 2 + 1 = 3
and 1 more part 2 2 2

232 2 times 3 parts 2 × 3 + 2 = 6 + 2 = 8
and 2 more parts 3 3 3

Could you investigate the rule? Discuss with your friends.

Exercise - 5.2

Section A - Classwork

1. Which of the following pairs of diagrams represent 'like' or 'unlike'
fractions? Let's write 'like' or 'unlike' below the diagrams.

a) b) c)

fractions fractions fractions

83Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Fraction

2. Let's list the like and unlike fractions separately.

a) 45 , 4 , 3 , 5 Like fraction Unlike fraction
7 5 6 a)

b) 23 , 14 , 1 , 1 b)
8 3

3. Let's write the fractions of the shaded parts and compare the fractions
by using symbols '<' or '>'.

a) b)

c) d)

4. Let's compare these like fractions by using the symbol '<' or '>'.

a) 2 1 b) 3 4 c) 6 5 d) 4 7
3 35 57 7 10 10

5. Let's write the fractions of the shaded parts in these diagrams. Then,
say and write whether they are proper or improper fractions.

a) b)

is fractions is fractions
c)
d)

is fractions is fractions

6. Let's say and write the improper fractions and the mixed numbers.

a) b) c)

= = =

7. Let's list the proper, improper, and mixed fractions separately.

4 , 152 , 3 , 234 , Proper fractions Improper Fractions Mixed fractions
3 4

7 , 2 , 313 , 29, 5
5 9 7

vedanta Excel in Mathematics - Book 4 84 Approved by Curriculum Development CentreSanothimi, Bhaktapur

Fraction

Section B

8. Let's draw diagrams and show each pair of like fractions by shading
the parts.

a) 1 and 32 b) 1 and 43 c) 2 and 54 d) 3 and 5
3 4 5 6 6

9. Let's arrange these fractions in ascending and descending orders.

a) 73 , 2 , 5 , 47 b) 5 , 1 , 7 , 69 c) 4 , 7 , 3 , 9
7 7 9 9 9 10 10 10 10

10. Let's convert these improper fractions to mixed numbers.

a) 3 b) 5 c) 72 d) 9 e) 4 f) 5
2 2 2 3 3

g) 130 h) 45 i) 94 j) 56 k) 152 l) 15
4

Let's convert these mixed numbers to improper fractions.

a) 112 b) 113 c) 123 d) 221 e) 223

f) 143 g) 314 h) 135 i) 235 j) 315

It's your time - Project work!

12. a) Let's write three proper fractions of your own. Show the fractions by
shading parts in rectangles.

b) Let's write three improper fractions of your own. Show them by shading
parts in rectangles. Then, write improper fractions in the form of mixed
numbers.

c) Let's write three mixed numbers of your own. Show them by shading
parts in rectangles. Then, write the mixed numbers in the forms of
improper fractions.

5.9 Addition and subtraction of like fractions

Classwork - Exercise

1. Let's read carefully and answer the given questions.

Teacher folded a sheet of paper into 8 equal

parts. She cut 2 parts and gave to Pratik.

Again, she cut 3 parts and gave to Pinky.

85Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Fraction

a) How many parts did the teacher give them altogether? + =

b) What is the fraction of 5 parts of the whole sheet of paper?

c) What is the fraction of the parts did Pratik get?

d) What is the fraction of the parts did Pinky get?

So, 82 + 3 We got it!
8
While adding like fractions, we simply
= 2 + 3 add the numerators and we write the
8 common denominator as it is.

= 5 → Total of numerators
8 The same denominator

2. Let's write the fractions of the green and pink shaded parts. Then, find

the total of the shaded parts.

a) 3 b) 5
4 6

+ = 2 + 1 = += =
4

Could you investigate the rule of addition of like fractions?

Of course, the sum of like fractions = Total of numerators
The same denominator

3. Let's write the fractions of the shaded parts. Then, subtract the fraction
of the crossed parts.

a) b)
×××
××× ××× ×××
××× ××× ×××
××× ×××
××× ×××

– = 3 – 1 = –= =
4

Could you investigate the rule of subtraction of like fractions?

The difference of like fractions = difference of numerators
The same denominator

vedanta Excel in Mathematics - Book 4 86 Approved by Curriculum Development CentreSanothimi, Bhaktapur

Fraction

Exercise - 5.3
Section A - Classwork
1. Let's say and write the sum or difference as quickly as possible.

a) 13 + 1 = b) 1 + 2 = c) 3 + 1 = d) 2 + 3 =
3 4 4 5 5 6 6

e) 32 – 31 = f) 4 – 2 = g) 6 – 2 = h) 5 – 3 =
5 5 9 9 7 7

2. Let's say and write the missing fraction in the blank space.
a) 52 + 3 3 5 5 8
= 5 b) 8 + = 8 c) 9 + = 9

d) 54 – = 2 e) 5 – = 1 f) 7 – = 4
5 7 7 10 10

Section B
3. Let's write the fractions of the green and pink shaded parts. Then, find

the sum of the fractions.

a) b) c)

4. Let's write the fractions of the shaded parts and subtract the fractions
of the crossed parts.

a) b)×××××××× ××× ××× ××× c) ××× ×××
××××× ××× ××× ××× ××× ×××
××××× ××× ××× ××× ××× ×××
××××××× ××× ××× ××× ××× ×××

5. Let's add these like fractions.

a) 1 + 25 b) 2 + 36 c) 3 + 71 d) 5 + 28 e) 3 + 5
5 6 7 8 9 9

6. Let's subtract these like fractions.

a) 3 – 26 b) 6 – 47 c) 7 – 82 d) 8 – 4 e) 9 – 6
6 7 8 9 9 10 10

7. Let's simplify these like fractions.

a) 1 + 2 + 73 b) 3 + 4 + 120 c) 6 – 1 – 28 d) 7 – 3 – 1
7 7 10 10 8 8 9 9 9

Vedanta ICT Corner
Please! Scan this QR code or
browse the link given below:

https://www.geogebra.org/m/kusjechj

87Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Fraction

8. Let's convert the mixed numbers into improper fractions. Then, add

or subtract.

Example: Add 131 + 231 = 3 × 1 + 1 + 3 × 2 + 1
3 3

= 3 + 1 + 6 + 1 = 4 + 7 = 4 + 7 = 11 = 332
3 3 3 3 3 3

a) 131 + 113 b) 114 + 114 c) 115 + 125 d) 213 + 313 e) 225 + 115

f) 231 – 113 g) 341 – 141 h) 421 – 312 i) 352 – 215 j) 434 – 314

Let's read these problems carefully and solve them.

9. a) You cut your birthday cake into 8 equal pieces. You gave 2 pieces to
your brother and 3 pieces to your friends. What fraction of the whole
cake did you give them altogether?

28 + 3 = = So, I gave of the whole cake.
8

b) Mother cut a bread into 6 equal slices. You ate 2 slices and sister

ate 2 slices. What fraction of the whole bread did you and sister eat

altogether?

c) Teacher asked you to fold a rectangular sheet of paper into 4 equal
parts. Then, she asked you to colour 1 part with green and 2 parts with
blue. What fraction of the whole sheet of paper did you colour?

10. a) Father cut a bread into 7 equal slices and he gave you 5 slices. If you ate
3 slices, what fraction of the whole bread is left with you?

57 – 3 = = So, of the whole bread it left with me.
7

b) Sunayana cuts a pizza into 8 equal pieces. She gives 3 pieces to Bishwant.
What fraction of the whole pizza is left with her?

88 – 3 = = So, of the whole pizza is left.
8

c) Teacher divided a sheet of paper into 9 equal parts. She cut 4 parts and

gave to a student. What fraction of the whole sheet of paper was left

with her?

11. a) A painter mixed 1 litre of red paint and 2 litre of yellow paint to get an
5 5
orange paint. Find the amount of orange paint.

vedanta Excel in Mathematics - Book 4 88 Approved by Curriculum Development CentreSanothimi, Bhaktapur

Fraction

b) Harka Bahadur completed three-tenth of his maths homework and
four-tenth of his science homework. How much homework did he
complete altogether?

c) A pole 514 metre high is standing on the ground. The length of the pole
under the ground is 3 metre. Find the length of the pole above the
ground. 4

d) Maya Limbu ate four-seventh parts of a pizza. What fraction of the
whole pizza was left?

12. It's your time - Project work!

a) Let's take three rectangular sheets of paper. Fold them separately into
quarters, sixths and eighths. Then, colour the parts to represents the
following sums.

(i) 1 + 42 (ii) 2 + 63 (iii) 3 + 4
4 6 8 8

b) Let;s write two like fractions to find the given sums or difference.

(i) + = 4 (ii) + = 5 (iii) – = 2
5 8 7

c) Let's write three pairs of proper like fractions of your own choice. Then,
find their sums and differences.



5.10 Repeated addition of like fractions

Classwork - Exercise

1. Let's say and write how many times are the like fractions added. Then,
find the products.

a) 1 + 21 = 2 times 1 = 2 × 1 = 2×1 = 2 = 1 (whole)
2 2 2 2 2

b) 1 + 1 + 21 = times 1 = × = ==
2 2 2

c) 1 + 1 = times 1 = × = ==
3 3 3

89Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Fraction

5.11 To find the value of fraction of a number in a collection

Classwork - Exercise

1. Let's study the given illustrations. Then, complete the sums and find

the correct answer. I got it!

a) Half of 4 apples Half of 4 = 1 × 4 and
2
= 121× 42= 2 apples
1 × 4 = 4÷ 2 = 2!!
2

b) Half of 6 apples c) One-third of 6 pencils
=×= apples = × = pencils

d) Two-third of 9 cherries e) Three-quarter of 12 dots

= × = cherries = × = dots

Could you investigate the rule of finding the value of fraction of a number in
a collection?

The value of the given fraction of a number = fraction × number

Exercise - 5.4

Section A - Classwork

1. Let's say and write how many times are the like fractions added. Then,

find the products.

a) 1 + 1 + 1 = × 1 = b) 1 + 1 = × =
2 2 2 2 3 3

c) 1 + 1 + 1 + 1 = × = d) 1 + 1 + 1 = × =
3 3 3 3 4 4 4

e) 1 + 1 = × = f) 1 + 1 + 1 + 1 = × =
5 5 6 6 6 6

2. Let's draw dotted lines to divide the beads into the given fractions of
the number of beads. Then, find the values of the fractions.

a) 1 of 6 beads = beads b) 1 of 6 beads = beads
2 3

vedanta Excel in Mathematics - Book 4 90 Approved by Curriculum Development CentreSanothimi, Bhaktapur

c) 1 of 8 beads = beads d) 1 of 8 beads= Fraction
2 4
beads

e) 1 of 12 beads = beads f) 1 of 12 beads= beads
3 4

3. Let's say and write the values as quickly as possible.

a) 21 of 8 children = children b) 1 of Rs 10 = Rs
2

c) 1 of 12 eggs = eggs d) 1 of 15 kg = kg
3 3 girls

e) 1 of 16 l = l f) 1 of 30 girls =
4 5

Section B
4. How many times is each shaded part? Then, find the product.

a) 2 times 14 b) c) d)

= 2 × 1 = 21 = 1
4 42 2

e) f) g) h) i)

5. The given marbles are equally divided by the dotted lines into the
fractions of the number of marbles. Let's find the values of fractions of
the number of marbles.

a) 1 of 6 marbles b) c) d)
3

= 1 × 62 = 2 marbles
3
1

e) f) g) h) i)

91Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Fraction

6. Let's find the values of the fractions of the numbers.

a) 12 of Rs 12 b) 1 of 9 kg c) 2 of Rs 15 d) 1 of 20 boys
3 3 4

e) 3 of 28 students f) 1 of 30 l g) 4 of 25 km h) 3 of Rs 500
4 5 5 10

Let's read these problems carefully and solve them.

7. a) You save 1 of your pocket money everyday. What fraction of your pocket
4

money do you save in 3 days?

b) Workers construct 1 part of a road everyday. What fraction of the road
10

do they construct in a week?

c) What is the fraction of a half part of the half of a whole bread?

d) What is the fraction of one-third part of the half of a whole pizza?

8. a) 2 of 30 students in a class are girls. How many girls are there in the
5

class?

b) A vegetable shopkeeper sells 3 of 40 kg of vegetables in each day. How
4

much vegetables does she/he sell everyday?

c) Mother and father earn Rs 800 in a day. They ssppeenndde83veorfytdhaeyir? earning
to run their family. How many rupees do they

d) The distance from Dhangadi to Dadeldhura is 140 km. Mr. Joshi travelled
4
7 part of the distance by a taxi and the remaining distance by a bus.

(i) How many kilometres does he travel by taxi?

(ii) How many kilometres does he travel by bus?

e) There are 450 students in a school and 3 of them are girls.
5

(i) Find the number of girls.

(ii) Find the number of boys.

vedanta Excel in Mathematics - Book 4 ?

92 Approved by Curriculum Development CentreSanothimi, Bhaktapur

Fraction

Unit Decimal and Percent

6

6.1 Tenths and hundredths - Looking back
Classwork - Exercise

1. Let's say and write the answer as quickly as possible.
a) How many cubes are there in the block?
b) What is the fraction of blue coloured cube?

c) Write this fraction in decimal.
d) What is the fraction of red coloured cubes?
e) Write this fraction in decimal.

f) What is the fraction of green coloured cubes?
g) Write this fraction in decimal.

It is one-tenth = 1 of the block of 10 cubes.
10
1
We write 10 = 0.1 and read it 'zero point one' or 'decimal one'.

It is two-tenths = 2 = 0.2 ('zero point two' or 'decimal two')
10

It is three-tenths = 3 = 0.3 ('zero point three' or 'decimal three')
10

It is a block of 100 cubes. So, each cube is

one-hundredth = 1 of the block. Here, blue
100

cube is one-hundredth of the block. Red cubes are

three-hundredths of the block. Green cubes are
twelve-hundredths of the block.

93Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Decimal and Percent

2. Let's learn from the given exaDmepclesi.mThaeln, say and write the fractions,

decimals, and decimal names.

a) One-hundredths = 1 = 0.01 Zero point zero one.
100

b) Three-hundredths = 3 = 0.03 Zero point zero three.
100

c) Two-hundredths = =

d) Five-hundredths = =

e) Twelve-hundredths = =

f) Fifteen-hundredths = =

6.2 Mixed number and decimal
Classwork - Exercise

1. Let's learn from the given illustrations. Then, say and write the mixed
numbers in decimals.

a) b) c)

1120 = 1.2 2150 = 3170 =
d) 4130 = e) 5140 = f) 10190 =
c) 814020 =
2. a) 11060 = 1.06 b) 61800 =

6.3 Place and place value of decimal numbers

Let's learn about the places and places values of decimal numbers from the
following examples.

3.43 tenths = 140 = 0.4

whole number
hundredths = 1300 = 0.03

Similarly, let's learn the places and place values of a few more decimal

numbers from the table given below:

vedanta Excel in Mathematics - Book 4 94 Approved by Curriculum Development CentreSanothimi, Bhaktapur

Decimal and Percent

Decimal Place and place value
numbers
Tenths Hundredths
0.56
5 = 0.5 6 = 0.06 Vedanta ICT Corner
0.19 10 100 Please! Scan this QR code or
browse the link given below:
0.24 1 = 0.1 9 = 0.09
10 100 https://www.geogebra.org/m/frk9ndv6

2 = 0.2 4 = 0.04
10 100

6.4 Comparison of decimal numbers

The whole number parts of decimal numbers are compared as like that of
whole numbers. To compare the decimal parts, at first, we should compare
the tenths place. If the tenths place digits are equal, we compare the

hundredths place.

Let's learn the comparison of decimal number from the examples given
below:

a) 2 .51 3.51 b) 0.56 0.48 c) 0.54 0.57

< > =
<

So, 2.51 < 3.51 So, 0.56 > 0.48 So, 0.54 < 0.57

Exercise - 6.1

Section A - Classwork

1. Let's say and write the fractions and decimals of the shaded parts.

a) b) c)

3 = = =
10

d) e) f)



= = =

95Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Decimal and Percent

2. Let's say and write the decimals of these fractions.

a) 7 = b) 7 = c) 1090 =
10 100

d) 25 = e) 36 = f) 81 =
100 100 100

3. Let's say and write the decimals of these mixed numbers.

a) 1180 = 1.8 b) 215040 = 2.54 c) 3170 = d) 411080 =

4. Let's say and write the fractions and decimals.

a) Nine-tenths = = b) Six-tenths = =

c) Five-hundredths = = d) Forty-two-hundredths= =

5. Let's say and write the fraction of tenths and hundredths of these
decimals.

a) 0.4 = 4 b) 0.8 = c) 0.7 d) 0.03 = e) 0.56 =
10

6. Let's write '<' or '>' symbol in the blanks and compare the decimal
numbers.

a) 0.4 0.2 b) 0.7 0.9 c) 0.65 0.58

d) 0.05 0.5 e) 0.12 0.09 f) 0.07 0.1

7. Let's say and write the place and place value of the digit in the decimal
number.

a) In 0.3, the place of 3 is and place value is

b) In 0.72, the place of 7 is and place value is

c) In 0.05, the place of 5 is and place value is

d) In 0.63, the place of 3 is and place value is

vedanta Excel in Mathematics - Book 4 96 Approved by Curriculum Development CentreSanothimi, Bhaktapur

Decimal and Percent

Section B

8. Let's write the decimal numbers of these number names.

a) zero point seven b) decimal four

c) zero point zero five d) decimal two six

9. Let's write the decimal number names of these decimals numbers.

a) 0.2 b) 0.02 c) 0.57

10. Let's write the decimal numbers of these fractions.
a) 150 b) 1060 c) 13020

11. Let's write these improper fraction in mixed numbers and in decimals.

a) 27 = 2170 = 2.7 b) 316 = 311060 = 3.16 c) 4150
10 100

d) 61 e) 113009 f) 512040
10

12. Let's write the fraction of tenths or hundredths of these decimals.

a) 0.3 = 3 b) 0.4 c) 0.7 d) 0.07 = 1070
10

e) 0.05 f) 0.09 g) 0.36 h) 0.81

13. The given ruler shows the whole numbers from 0 to 15 and their
tenths in order. The first arrow points 0.3. Which numbers do the
other arrows point to?

0.3 a b cd ef g h

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.511 11.512 12.51313.514 14.515

14. Let's rewrite the decimal numbers. Then, write the places and place

values of each digit.

0.27 = 0.2 a) 0.15 b) 0.76 c) 0.28 d) 0.55
tenths

h undre dths = 0.07

97Approved by Curriculum Development Centre, Sanothimi, Bhaktapur vedanta Excel in Mathematics - Book 4

Decimal and Percent

15. Let's compare these decimal numbers using '<' or '>' symbol.

a) 0.3 and 0.5 b) 0.3 and 0.05 c) 0.7 and 0.4

d) 0.07 and 0.4 e) 0.26 and 0.28 f) 0.26 and 0.028

16. Let's arrange the decimal numbers in ascending order.

a) 0.01, 0.1, 0.02 b) 0.25, 0.08, 0.5

17. Let's arrange the decimal numbers in descending order.

a) 0.2, 0.07, 0.3 b) 0.09, 0.18, 0.1

18. It's your time - Project work!

a) Let's draw a decimal-tree of tenths in a chart paper as shown in the given
model.

0.1 0.1 0.1 0.1

0.2 0.1 0.1 0.1 0.1 0.2

0.3 0.1 0.1 0.1 0.1 0.1 0.1 0.3

0.4 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4

0.5 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.5

0.6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.6

0.7 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.7

0.8 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.8

0.9 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.9
1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1

10 cm 10 cm

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

0.5
0.1

0.50 98 Approved by Curriculum Development CentreSanothimi, Bhaktapur
0.10



vedanta Excel in Mathematics - Book 4