Prime Mathematics 5

Percentage

Out of 100 squares, 40 are shaded. We say

40 percentage of the squares is shaded.

In symbol 40 percentage is written as

40%. The shaded part is also expressed
40 25.
in fraction as 100 = Percentage is a

fraction with denominator 100. 40 2
100 5

Conversion of fraction into percentage:
• Divide 100 by denominator.

• Multiply numerator and denominator of the fraction by the quotient so
obtained.

• This changes the given fraction into the fraction having denominator
100. Numarator is taken as percentage.

To convert 2 into percentage.
5
2 2 × 20 40
5 = 5 × 20 = 100 [ ∴ 100 ÷5 = 20]

∴ 2 = 40%
5

Alternative process.

2 means,
5

Out of 5 is 2

∴ out of 1 is 2 2 × 100
∴ out of 100 5 5
is

i.e. out of 100 is 40 which is 40% [ 2 × 100 = 40]
5
Thus, fraction × 100 = percentage.

× 100

Fraction/ Percentage
decimal

÷ 100

146 Prime Mathematics Book - 5

Example 1: Express 4 as percentage.
5
20
Solution: 4 = 4 × Alternative process
5 5 100%

= 80% 4 = 4 × 20 = 80 = 80%
5 5 × 20 100

∴ 4 = 80 % Note: All the numbers in denominator can
5 not be converted into 100 easily.

Example 2: Express 45% into fraction.

Solution: 45%

= 45 = 9 45 9
100 20 100 20
9
∴ 45% in fraction is 20 .

Example 3: Convert 45% into decimal.

Solution: 45%

= 45 = 0.45
100

Exercise: 7.8

1. Express each of the following fraction into percentage.

a. 27 b. 1 c. 1 d. 1
100 4 5 2

e. 1 f. 1 g. 2 h. 125
8 100 1 100

2. Convert the following percentage into fraction in the simple form.

a. 15 % b. 41 % c. 45 % d. 90%

e. 9 1 % f. 6 1 % g. 18 3 % h. 1 %
2 4 4 2

3. Express the following decimal into percentage.

a. 0.35 b. 0.06 c. 0.5 d. 0.75

e. 15.5 f. 1.25 g. 0.005 h. 1.05

4. Convert the following percentage into decimal.

a. 25% b. 67% c. 100% d. 300%

e. 1 1 % f. 4 1 % g. 1 % h. 5 %
2 4 2 4

Prime Mathematics Book - 5 147

Applications of percentage I have some chocolates.
If she has 100. I get 50. I’m giving your 50%.

If she has 10, I get 5 only.

If she has 2, I get 1 only.

I cannot
say how many I will get !

50 % simply has no meaning. It cannot tell exact quantity. 50% of 10
50
chocolates = 100 × 10

= 5 chocolates.

To express quantities, percentage is very useful in daily life.

Example 1: What is 25% of 80?
Solution: 25% of 80

= 25 × 80
100

= 20

∴ 25 % of 80 is 20.

Example 2: What percentage of 300 is 45? Alternative process

Solution: 45 out of 300 = 45 Let x % of 300 is 45
300
then x% of 300 = 45
45
Percentage = 300 × 100% or, x × 300 = 45
100
45
=15% or, x = 3
∴ 45 is 15 % of 300.
or, x = 15

∴ 45 is 15% of 300

Example 3: Reshma got 71 % marks in an examination of 8 subjects each
of 100 marks. What marks did she obtain in total?

Solution: Total full marks = 8 × 100 =800 marks.
Now 71% of 800

= 71 × 800
100

= 71 × 8 = 568
∴ The total mark she obtained is 568.

148 Prime Mathematics Book - 5

Exercise: 7.9

1. Evaluate the following: b. 10 % of 17 c. 8 % of 100
a. 32% of 25

d. 40% of 300 e. 150% of 8000 f. 90 % of 8848

2. Sg.olve3t3h13e%foofll9o0w0ing: h. 0.01 % of 10000000

a. What percentage of 80 is 32?
b. What percentage of 35 is 35?
c. What percentage of 16 is 80?
d. What percentage of 600 is 100?
e. What percentage of an hour is 40 minutes?
f. What percentage of a month is 6 days?
g. What percentage of a year is 3 months?

3. Find the following:
a. 4 % of what sum is Rs 4000?
b. 30% of what number is 150?
c. 25% of how many students is 300 students?
d. 50% of how much land is 1235 m2?

4. a. A fruit seller brought 200 oranges. If 10% of them are rotten, how
many of them are good?

b. There are 40 students in a class. If 20% of them are boys, how
many boys are there ?

c. 60% of the students of a class of 35 students are boys. Find the
number of girls in the class.

d. While buying a shirt costing Rs.700, a discount of 35%is given. Find
the actual cost to be paid for the shirt?

e. In an examination, Subash obtained 776 marks. If there were 8
subjects each of 100 full marks, what is his percentage?

f. Sanjay earns Rs. 8000 in a month. He spends 20% of it for education,
30% on food and 10% for rent. How much does he save?

g. In a unit test Tenzing got 80% while Sonam got 164 marks out of
200 full marks. Who got more marks?

h. Now, a litre of petrol costs, Rs. 120. What will be the price if it
hikes by 5% ?

Prime Mathematics Book - 5 149

Unit Revision Test

1. Convert the following into like fractions:
1 5 3 5 1
a. 4 and 6 b. 4 , 8 and 6 .

2. Simplify: 1 1
1 1 4 4 1
a. 8 2 - 4 4 b. 1 +3 -2 4

3. Find the value of:
4 4
a. 5 of Rs 30 b. 20 ÷ 5

4. Compare: b. 51.52 and 51.6
a. 246.8215 and 246.822

5. Carry out the multiplication:

a. 5.12 × 8 b. 0.8 × 0.6

6. Round off 245.3289 to b. nearest hundreds
a. nearest tens

7. Express:
1
a. 8 into percentage b. 25 percent into decimal
b. What percentage of 40 is 8?
8. Evaluate:
a. 40% of 300

9. Solve the following;

a. There are 40 students in a class. If 45% of them are boys, how
many boys are there?
141 . 2
b. The product of two fractions is If one fraction is 1 3 , find the
other fraction.

150 Prime Mathematics Book - 5

Answers:

Exercise: 7.1 (f) unlike
1. (a) unlike (b) like (c) like (d) unlike (e) like

2. (a) 8 and 3 (b) 2 and 5 (c) 9 and 2 (d) 5 and 6 (e) 1326,3260 and 27
12 12 6 6 12 12 8 8 36
1248, 15 16 12 15 18 7
(f) 24 and 24 (g) 5 and 5 (h) 14 and 14

3. (a) 3 (b) 12 (c) 24 (d) 5 (e) 6 (f) 20 (g) 1 (h) 1
4 13 25 8 7 27 2
5 1 7 2 3 1
4. (a) 12 (b) 3 (c) 15 (d) - 7 (e) 11 (f) - 7

5. (a) 13 (b) 1172 (c) 1274 (d) 13 (e) 1221 (f) 11121 (g) 118 (h) 1158
20 18
1 7 11 5 11 1 (g) 231 (h) 421
6. (a) 2 (b) 12 (c) 48 (d) 12 (e) 30 (f) 14

7. (a) 134 (b) 132 (c) 531 (d) 575 (e) 321 (f) 132 (g) 112 (h) 414 (i) 71307

8. (a) 1 (b) 2 (c) 7 (d) 7 (e) 7 (f) 1 (g) 2 (h) -1118
9 10 15 12 12 5
9. (a) 141 (b) 121 (c) 2 (d) 241 (f) 434 (h) 1318
(e) 1 (g) 1

10. (a) 3kg (b) 1052 m (c) 2110 (d) 15kg (e) 5kg (f) 2125
3
(g) Rajat purchased more by 20 kg

1. (a) 4 (b) 12 (c) 2 (d) 7 Exerc2i.se:(a7).232 (b) 15 (c) 123 (d) 6
3 9 15 16
3 3 1 1 (i) 3230 (j) 131 (k) 271 (l) 223
(e) 16 (f) 20 (g) 8 (h) 21

3. (a) 6 (b) 35 (c) 14kg (d) 14 (e) 8 (f) 1 (g) Rs. 20 (h) 36 students
2
1 1
4. (a) 33 (b) 2 work (c) 1 foot (d) 4

1. (a) 1 (b) 1 (c) 6 Exercise: 7.3 1 (b) 3 (c) 11 (d) 5
5 12 (d) 20 2. (a) 5 7 9

3. (a) 712 (b) 3423 (c) 25 (d) 28 (e) 3 (f) 5 (g) 5 (h) 1
4 52 6 5
(i) 119 10 (k) 131 (l) 271 4. (a) 112 2 (c) 141m (d) Rs. 64
(j) 13 (b) 5

Exercise: 7.4
1. Show it to your teacher 2. Show it to your teacher 3. Show it to your teacher
4. (a) 34.52 < 34.6 (b) 246.9 < 248.999 (c) 23.675 < 23.68
(d) 246.8215 < 246.822 (e) 220.005 < 220.01 (f) 5791.555 < 7581.559
(g) 7.531 > 6.987 (h) 207.3057 > 207.30509

5. (a) 5 (b) 24 (c) 225 (d) 425 (e) 1675 (f) 2005 (g) 500435 (h) 13025
10 10 1000 100 100 100 10000 1000
1 (b) 421 (c) 851 (d) 535 1 (f) 114 (g) 4290 (h) 121256
6. (a) 2 (e) 4

7. (a) 1.2 (b) 0.7 (c) 0.25 (d) 0.04 (e) 0.008 (f) 1.246 (g) 0.246 (h) 0.5
(i) 0.25 (j) 0.4 (k) 0.15 (l) 2.56 (m) 0.125 (n) 12.375 (o) 15.0625 (p) 0.016

Prime Mathematics Book - 5 151

1. (a) 13.56 Exercise: 7.5
(g) 4.934 (b) 133.536 (c) 206.88 (d) 84.135 (e) 122.272 (f) 29.48
(h) 238.835
2. (a) 3.3 (b) 3.8 (c) 4.14 (d) 433.25 (e) 5.2 (f) 10.74
(g) 71.75 (h) 91.13
3. (a) 98 (b) 5.916 (C) 1.142 (d) 185.78 (e) 946.592 (f) 469.665
(g) 2485.59 (h) 2166.09kg
4. (a) 6.436 (b) 29.541 (c) 4.36 (d) 69.523 (e) 962.037 (f) 99.999
(g) 123.579 (h) 0.725
5. (a) 27.724 (b) 577.144 (c) 1.79 (d) 0.62 (e) 11.492 (f) 1.793
(g) 18.769 (h) 47.687
6. (a) 10m (b) Rs. 38.95 (c) 29.40cm (d) Rs. 254.50 (e) 14.565l (f) 26.64
(g) 10cm (h) Rs. 299.90

1. (a) 2.47 (b) 0.85 Exercise: 7.6 (d) 120.05 (e) 43.8
(c) 24.52 (j) 347
(f) 0.4 (g) 573.1 (h) 3405.06 (i) 2.5 (e) 89.268
(k) 7654.3 (l) 632210.8 (j) 1508.268
2. (a) 5.6 (b) 5.76 (c) 4.815 (d) 12.8 (e) 2.8782
(f) 218.946 (g) 0.06 (h) 450.688 (i) 71.162 (j) 486.75
(k) 1989.92 (l) 6159.78 (e) 26.66cm2
3. (a) 0.7 (b) 0.63 (c) 5.76 (d) 24.0744
(f) 2.38136 (g) 0.0099 (h) 28.382 (i) 21.2625
(k) 0.09 (l) 442.68
4. (a) Rs. 603.75 (b) 403.4cm (c) 10750gm (d) Rs. 282
(f) 321.3cm3 (g) 47.728cm (h) Rs. 688.275

1. (a) 57 Exercise: 7.7(d) 520
(b) 162 (c) 238
2. (a) 750 (b) 1230 (c) 490 (d) 10
3. (a) 100 (b) 500 (c) 4600 (d) 12400
4. (a) 1.3 (b) 25.9 (c) 0.3 (d) 245.1
5. (a) 42.36 (b) 72.58 (c) 105.32 (d) 0.15
6. (a) 0.123 (b) 2.781 (c) 59.987 (d) 215.001
7. (a) 8.2 (b) 21.43 (c) 12.346 (d) 125.2935
8. (a) 146 (b) 150 (c) 100 (d) 146.3 (e) 146.35
(f) 146.349 (g) 146.349

1. (a) 27% (b) 25% (c) 20% (d) E51x900e%rcis((eee:))71.281209210% (f) 1%
2. 3 9 (d) 1
(b) 41 9 (h) 1
(a) 20 100 (c) 20 (f) 16 (g) 16 200

3. (a) 35% (b) 6% (c) 50% (d) 75% (e) 1550% (f) 125% (g) 0.5% (h) 105%
4. (a) 0.25 (b) 0.67 (c) 1 (d) 3 (e) 0.015 (f) 0.0425 (g) 0.05 (h) 0.0125

Exercise: 7.9
1. (a) 8 (b) 1.7 (c) 8 (d) 120 (e) 12000 (f) 7963.2 (g) 300 (h) 1000
(b) 100% (c) 500% (d) 16 32% (e) 6632%
2. (a) 40% (f) 20% (g) 25%

3. (a) 100000 (b) 500 (c) 1200 (d) 2470m2
4. (a) 180 (b) 8 (c) 14 (d) Rs. 455 (e) 97% (f) Rs. 3200
(g) Sonam (he got 82%) (h) Rs. 126

152 Prime Mathematics Book - 5

Un8it Unitary Method, Simple
Interest, Profit and Loss

Estimated periods − 9

Objectives

At the end of this unit, the students will be able to:
• find the value of one object when the total value of objects is known
• find the value of given number of objects when the value of one obect is known
• solve simple problem on simple interest
• solve simple problem on profit and loss

Teaching Materials
Price list, price tags

Activities
It is better to:
• explain the process of finding the value of single objects when value of a set of some
objects is given and finding the value of a set of objects when the value of a single object
is given.
• explain the concept of simple interest and apply unitary method to find interest .
• explain concept of selling and buying and profit and loss.

Prime Mathematics Book - 5 153

Learn the following Unitary method

1 pencil costs Rs. 6. Rs. 6
How much do 5 pencils cost?

Rs. 6 + Rs. 6 + Rs. 6 + Rs. 6 + Rs. 6 = Rs. 30
Or, 5 × Rs. 6 = Rs. 30
If the cost or value of 1 article is known, we can find the cost or value of
more articles by multiplying the price by the number of articles.
Proceed like this –
1 pencil costs Rs. 6
∴ 5 pencils cost 5 × Rs. 6 = Rs. 30
Thus, the cost of 5 pencils is Rs. 30.

6 apples cost Rs. 24. What
is the price of an apple?

Rs. 24
Re. 1 Re. 1 Re. 1 Re. 1 Re. 1 Re. 1 -Rs. 6

= Rs. 18
Re. 1 Re. 1 Re. 1 Re. 1 Re. 1 Re. 1 -Rs. 6
= Rs. 12
Re. 1 Re. 1 Re. 1 Re. 1 Re. 1 Re. 1 -Rs. 6
= Rs. 6
Re. 1 Re. 1 Re. 1 Re. 1 Re. 1 Re. 1 -Rs. 6
=×
∴ Rs. 24 ÷ 6 = Rs. 4
If the cost of certain number of articles is known, we can find the cost or
value of 1 article by dividing the total cost by the number of articles.
We proceed like this
The cost of 6 apples is Rs. 24
∴ The cost of 1 apple is Rs. 24 ÷ 6 = Rs. 4
Therefore, the cost of an apple is Rs. 4

154 Prime Mathematics Book - 5

Unitary method is method of finding the value of 1 (unit). If value of 1
article is known, the value of any numbers of the articles can be found.

To solve the unitary method problem:
Let's see an example.

If 5 pens cost Rs. 400, how many pens can be brought with Rs. 960?

Known part unknown part

The question consists two parts, known
part and unknown or asked part.

What do we have to find number of the
pens or the cost? Write the known part
as it is or slightly derived such that the

term to be found is to the right.

Find the value of unit and then find
the value of any number. If more we

multiply, If less we divide.

Solution: For Rs. 400, we can buy 5 pens. 5 12
400
∴ For Rs. 1, we can buy 5 pens. × 960
400
5 8
400
∴ For Rs. 960 we can buy × 960 pens.

=12 pens.
Therefore, 12 pens can be bought for Rs. 960

Example 1: If the cost of 6 notebooks is Rs. 168, find the cost of 1 note-
book.

Solution: The cost of 6 notebooks is Rs. 168
∴ The cost of 1 notebook is Rs.168 ÷ 6
= Rs. 28
Therefore, the cost of 1 notebook is Rs. 28.

Example 2: If the cost of 1 chocolate is Rs. 3.50. Find the cost of 7 such
chocolates.

Solution: The cost of 1 chocolate is Rs. 3.50
∴ The cost of 7 chocolates is 7 × Rs. 3.50
= Rs. 24.50
∴ The cost of 7 chocolates is Rs. 24.50.

Prime Mathematics Book - 5 155

Example 3: The cost of 4 board markers is Rs. 112. What is the cost of 7
such markers?
Solution: The cost of 4 board markers is Rs. 112
112
∴ The cost of 1 board marker is Rs. 4

∴ The cost of 7 board markers is Rs. 112 × 7
4

=Rs. 28 × 7

=Rs. 196

∴ The cost of 7 board markers is Rs. 196

Exercise 8.1
1 (a) The cost of 1 book is Rs. 155. Find the cost of 12 such books.
(b) Raju earns Rs. 12000 in a month. How much does he earn in a

year?
(c) A bike runs 40 kilometers with 1 liter of petrol. How far does it run

with 9 liters of petrol ?
(d) A man walks 4 kilometers in an hour. How far can he walk in 8

hours ?

2. (a) The cost of 6 chocolates is Rs.48. What is the cost of 1 chocolate?
(b) 5 cups of coffee cost Rs.60. Find the cost of a cup of coffee.
(c) 144 plants are planted equally in 12 rows. How many plants are

there in a row?
(d) 9 buses can carry 540 passengers. How many passengers can a bus

carry?

3. (a) The cost of 7 pencils is Rs. 56. What is the cost of 12 such
pencils?

(b) A factory produces 1610 sets of goods in a week. How many sets of
goods can be produced in 10 days?

(c) 5 sets of uniform cost Rs.3625. Find the cost of 2 sets of such
uniform?

(d) The cost of distempering a wall of area 25m2 is Rs. 375. Find the
cost of distempering 30m2 of wall with same quality distemper.

4. Find the total cost from the following:

Price No. of items Total cost
(a) Rs. 12.50 10 ?
(b) Rs. 125 20 ?
(c) Rs. 44 16 ?
(d) Rs. 224 7 ?

156 Prime Mathematics Book - 5

5. Find the cost of an item from the following:

No. of items Total cost cost per item
(a) 7 Rs.105 ?
(b) 15 Rs. 360 ?
(c) 27 Rs. 2754 ?
(d) 40 Rs. 2496 ?

6. (a) There are 1584 pages in 8 books. How many pages are there in a
book ?

(b) A mobile set costs Rs. 6540. How many such mobile sets can be
bought with Rs. 71940?

(c) A car consumes 1 liter of petrol to run 12 km. If the car moved 288
km, how much petrol was consumed ?

(d) A porter can carry 80 kg of rice. How many porters are needed to
carry 1280 kg of rice ?

Simple interest
Learn the following:

Sahu ji,
I need some money. Will you
please lendi me Rs. 10,000 for 2

years?

Ok. You are my trusted. I
can lend you the sum but I

do take 20 % interest.

In need, one customer might borrow money from other or lend money to
other. If we have saved money we deposit it in the bank. And the bank gives
back the sum deposited along with an extra money after certain period. If we
need some money we borrow it from bank or a finance company or a person
and give back the sum along with some extra money.
The money borrowed or lent is called principal (P)
The additional amount paid along with the sum borrowed or lent is called the
interest (I)
The duration for which the money is lent or borrowed is called time (T)
The borrower and the lender make an agreement with a condition, the extra
money to pay for Rs. 100 for 1 year is called rate (R). We can solve problems
related to simple interest using unitary method.

Prime Mathematics Book - 5 157

Example 1: The interest of Rs. 100 for 1 year is Rs 12. What is the interest
of the sum for 3 years?

Solution: Interest of Rs. 100 for 1 year is Rs. 12.

∴ Interest of Rs. 100 for 3 years is Rs. 12 × 3
=Rs. 36
Therefore, the interest of Rs. 100 for 3 years is Rs. 36

Example 2: If the interest of Rs. 100 for 1 year is 8, find the interest of
Rs. 5000 for 1 year.

Solution: Interest of Rs.100 for 1 year is Rs. 8
8
∴ Interest of Rs.1 for 1 year is Rs. 100

∴ Interest of Rs. 5000 for 1 year is Rs. 8 × 5000
100

= Rs. 400
Therefore, the interest of Rs. 5000 for 1 year is Rs. 400

Example 3 : If the interest of Rs. 100 for 1 year is Rs. 18. Find the interest
of Rs. 10,000 for 2 years.

Solution: The interest of Rs. 100 for 1 year is Rs. 18

∴ The interest of Rs. 1 for 1 year is 18
100
18
∴ The interest of Rs. 10,000 for 1 year is Rs 100 × 10,000

∴ The interest of Rs. 10,000 for 2 years is Rs. 18 × 10,000 × 2
100

= Rs. 3600.
Therefore, the interest of Rs. 10,000 for 2 years is Rs. 3600

Example 4: Find the interest of Rs. 20,000 for 3 years at the rate of 15%.

Solution: The rate 15% means-

The interest of Rs. 100 for 1 year is Rs. 15
15
∴ The interest of Rs. 1 for 1 year is Rs. 100
∴ The interest of Rs. 20,000
for 1 year is Rs. 15 ×20,000
100
15
∴ The interest of Rs. 20,000 for 3 years is Rs. 100 ×20,000×3

=Rs. 9000

Therefore, the interest of Rs. 20,000 for 3 years at the rate of

15% is Rs. 9000

158 Prime Mathematics Book - 5

Exercise 8.2

1. If the interest of Rs. 100 for 1 year is Rs. 10, find the interest of Rs.
100 for 5 years.

2. If the interest of Rs. 100 for 1 year is Rs. 12, find the interest of Rs.
100 for 7 years

3. If the interest of Rs. 100 for 1 year is Rs. 20, find the interest of Rs.
800 for 1 year.

4. If the interest of Rs. 100 for 1 year is Rs. 15, find the interest of Rs.
12000 for the same period.

5. If the interest of Rs. 100 for a year is Rs. 8, find the interest of Rs.
14000 for 6 years.

6. If the simple interest of Rs. 100 per year is Rs. 6, find the interest of
Rs. 25000 borrowed for 5 year.

7. Find the simple interest of:

a. Rs. 12500 for 5 years at 10% per year.

b. Rs. 2500 for 8 years at 8.5% per annum.

c. Rs. 1600 for 4 years at 8% p.a

d. Rs. 7500 for 6 years at 9% p.a.

Profit and loss

Buy goods for low price Ram
and sell for high price and Lakhanwa is
get profit. It is the basic from Terai. He
principle of business. is honest and
Buying price is also called laborious. He
cost price (C.P.). If selling carries fruits
price (S.P.) is greater than on his bicycle
C.P., there will be profit. and sells in
Thus, we take streets.
profit = S.P. – C.P.
Some times it is necessary He buys fruits in Kalimati and sells than in the
to sell the goods for price streets for a little higher price. It is the business.
less than cost price (C.P.).
In such case, there is loss

∴ Loss price = C.P. – S.P.

Prime Mathematics Book - 5 159

Profit and loss are expressed as percentage of cost price (C.P.)

∴ Profit percentage = Actual profit × 100%
Cost price

= Profit ×100%
C.P.

and loss percentage = Loss × 100%
C.P.

Example 1: Ram Lakhanwa bought some fruits for Rs. 1260 and sold for
Rs. 1540. Find his profit or loss .

Solution : Here C.P. of the fruit = Rs. 1260
S.P. of the fruit = Rs. 1540

Since S.P. > C.P., there is profit
∴ Profit = S.P. – C.P.
= Rs. 1540 – Rs. 1260
= Rs. 280

Therefore, his profit is Rs. 280

Example 2: Find profit or loss when C.P. = Rs. 2480 and S.P. = Rs. 2105.
Solution Here,

C.P. = Rs. 2480
S.P. = Rs. 2105
C.P > S.P. , so
Loss = C.P. – S.P.

= Rs. 2480 – Rs. 2105
= Rs. 375

Example 3: Sneha bought a mobile set for Rs. 7500 and made a profit of
Rs.1125. Find her profit in percentage .

Solution Here,
C.P = Rs. 7500

Profit = Rs. 1125
∴ profit percentage = PCro.Pf.it × 100%

160 Prime Mathematics Book - 5

= Rs.1125 × 100%
Rs7500

= 15%
∴ Here profit percent is 15 %

Example 4: If C.P. = Rs. 3440 and profit = 10% find S.P.

Solution Here,

C.P. = Rs. 3440

Profit = 10 %

∴ Actual Profit = 10 % of C.P.

= 10 % × Rs. 3440

= 10 × Rs. 344 0
100
= Rs. 344

Now , S.P. = C.P.+ Profit

= Rs. 3440 + Rs.344

= Rs. 3784

Exercise 8.3
1. Find the profit or loss in each of the following :

a. C.P. = Rs. 1800, S.P. = Rs. 2150
b. C.P. = Rs. 540, S.P. = Rs. 700
c. C.P. = Rs. 3225, S.P = Rs. 3100
d. C.P. = Rs. 4350, S.P. = Rs. 4150
2. Find profit or loss percentage when
a. An article is bought for Rs. 200 and sold for Rs. 240
b. A jacket is bought for Rs. 3500 and sold for Rs. 3185
c. A dictionary is bought for Rs. 1150 and sold for Rs.1288
d. A blanket bought for Rs. 4000 is sold for Rs. 3750
3. Find selling price when
a. C.P. = Rs. 560 and profit = 25%
b. C.P. = Rs. 12540 and profit = 20%
c. C.p. = Rs. 1680 and loss = 10%
d. C.P. = Rs. 2700 and loss = 15%

Prime Mathematics Book - 5 161

4. Solve the following problems:
a. A man purchased an article for Rs. 1120 and sold it by taking 20%
profit. Find his profit amount and selling price of the article.
b. A fruit seller bought some fruits for Rs. 1840 and sold for Rs. 2300.
Find his profit or profit percentage.
c. Rajesh bought a motorbike for Rs. 1,50,000 but he had to sell it soon
for Rs. 132000. What was his loss? Also find his loss percentage
d. For what price a mobile set bought for Rs. 7800 should be sold for
making 10% profit?

Unit Revision Test

1. If the cost of 5 copies is Rs. 125, find the cost of 1 copy.
2. Rita earns Rs. 9500 in a month. How much does she earn in a year?
3. The cost of 8 pencils is Rs.64. What is the cost of 3 pencils ?
4. If the interest of Rs. 100 for 1 year is Rs. 10, find the interest of Rs. 100

for 4 years .
5. If the simple interest of Rs. 100 is Rs. 12, find the interest of Rs 5000 for

1 year.
6. Find the interest of Rs. 8000 for 2 years at the rate of 12%.
7. Find the profit or loss when an article bought for Rs. 400 is sold for

Rs. 512.
8. Find the profit or loss percent when a jacket bought for Rs.1600 is sold

for Rs. 1880.

Answers:

1. (a) Rs.1860 (b) Rs. 144000 Exercise: 8.1 (d) 32 km
2. (a) Rs. 8 (c) 360 km (d) 60
3. (a) Rs. 96 (b) Rs. 12 (c) 12 (d) Rs. 450
4. (a) Rs. 125 (b) 2300 (c) Rs. 1450 (d) Rs. 1568
5. (a) Rs 15 (b) Rs. 2500 (c) Rs. 704 (d) Rs. 62.4
6. (a) 198 (b) Rs. 24 (c) Rs. 102 (d) 16
(b) 11 (c) 24 l

1. Rs. 50 2. Rs. 84 Exercise: 8.2 4.Rs. 1800
5. Rs. 6720 6. Rs. 7500 3. Rs. 160 (b) Rs. 1700
(d) Rs 4050 7. (a) Rs. 6250
(c) Rs. 512

Exercise: 8.3 (d) Loss Rs. 200
1. (a) Profit Rs. 350 (b) Profit Rs. 160 (c) Loss Rs. 125 (d) Loss 6.25%
2. (a) Profit 20% (b) Loss 9% (c) Profit 12% (d) Rs. 3105
3. (a) Rs. 700 (b) Rs. 1548 (c) Rs. 1848 (c) Rs. 18000, 12%
4. (a) Rs. 224,Rs. 1344 (b) Rs. 460 , 25%
(d) Rs.8580

162 Prime Mathematics Book - 5

Un9it Charts, Bills, Bar-graph
and Co-ordinates

Estimated periods − 15

Objectives

At the end of this unit, the students will be able to:
• prepare the charts, bills and bar graphs.
• receive information from the given chart, bill, and bar graph.
• read the coordinates of the points and to locate the points in the coordinate plane.

Teaching Materials
Models of chart, bill and bar graph etc .

Activities
It is better to:
• show the bar graph, bill and chart to the students and ask them to get informations.
• ask the students to prepare the bar graph , chart and bill with supplied information.
• ask the students to locate the points in the coordinate plane.
• ask the students the coordinates of the given points in the coordinate plane.

Prime Mathematics Book - 5 163

Charts

Observe the given menu (a price chart) kept in a restaurant.

S. N. Items Rate
01. Tea Rs. 15.00
02. Coffee Rs. 25.00
03. Vegetable momo Rs. 60.00
04. Chicken momo Rs. 90.00
05. Vegetable chowmein Rs. 85.00
06. Chicken chowmein Rs. 95.00
07. Pizza Rs. 110.00
08. Burger Rs. 100.00
09. Chicken tandoori Rs. 130.00

The menu given above is an example of a chart. By observing it, we can
decide which item should be ordered.

Let's observe another chart

Price list of fruits pasted in a fruit shop:

S. N. Item Rate
01. Apple Rs. 80.00/kg.
02. Orange Rs. 110.00/kg
03. Grapes Rs. 140.00/kg
04. Mango Rs. 75.00/kg
05 Banana Rs. 60.00/dozen
06. Guava Rs. 55.00/kg

07. Pineapple Rs. 170.00/kg

Answer the following questions:
1. Which fruit is the cheapest?
2. Which fruit is the most expensive?
3. What is the cost of three dozens of bananas?

Lets learn to make a chart with the help of following information. Aman has
scored marks in the terminal examination as follows:

Nepali 60 Mathematics 100

English 85 Social studies 55

Science 95

The full marks of each subject is 100 and the pass marks is 40

164 Prime Mathematics Book - 5

Student's name : Aman

S. N. Subject F.M. P.M. Marks obtained
100 40 60
01. Nepali 100 40 85
100 40 95
02. English 100 40 100
100 40 55
03. Science

04. Mathematics

05. Social studies

Exercise: 9.1
1. The price list of vegetables of a vegetable shop is given below:

S. N. Item Rate
Rs.45.00/kg
1. Potato RS.60.00/kg
Rs.55.00/kg
2. Tomato Rs.95.00/kg
Rs.115.00/kg
3. Radish Rs.35.00/kg
Rs.65.00/kg
4. Cauliflower

5. Peas

6. Pumpkin

7. Cabbage

Answer the following questions:
(a) Which vegetable is the cheapest one and what is its rate?
(b) What is the cost of 3 kg of cauliflower ?
(c) What is the difference in the rates of peas and cabbage ?
(d) Which vegetable is the most expensive?

2. The number of students enrolled in different grades in a certain
year is as follows:
Grade I - 30 Grade II - 28 Grade III - 40
Grade IV - 38 Grade V - 60 Grade VI - 50
Represent the above information in a chart.

3. In a certain day, traffic police has taken action against the vehicles
for violating the traffic rules is as follows:

S. N. Types of vehicles Number of vehicles
1. Motor cycles 25
2. Taxi 30
3. Gas tempo 28
4. Bus 10
5. Micro bus 22
6. Private car 5

Prime Mathematics Book - 5 165

Answer the following questions:
(a) How many taxis were caught for violating the rules ?
(b) What is the number of motor cycles caught for violating the rules?
(c) Which vehicle is caught least for violating the rules?
(d) How many vehicles were caught for violating the rules ?

Bill
When you buy goods from a shop, you will get a piece of paper from the
shopkeeper for the payment as shown below.

Karki Stationery

Kathmandu - 14

Bill No. : 169 Date: 16/05/2065

Name of customer: Shim Thapa
Address: Asan, Kathmandu

S.N. Description Quantity Rate (Rs.) Amount (Rs.)
1. Pen 5 75.00 375.00
2. Geometry box 7
3. Pencil 15 150.00 1050.00
4. Chart paper 20 8.00 120.00
4.00 80.00

Total 1625.00

In words: One thousand six hundred and twenty five rupees only.

Customer's signature Seller's Signature

Paper with details of sale and purchase is called a bill. The bill gives the
following informations.
1. Name of the shop and its address. 2. Bill number.
3. Date of purchase. 4. Name of the customer and address.
5. List of purchased articles. 6. Quantity of purchased articles.
7. The rate of each article. 8. The total price of each article.
9. The total amount to be paid. 10. Signature of customer.

Some examples of other bills are bill sent by a school for the fee, water bill,
electricity bill, telephone bill, etc. Let's learn to prepare a bill.

Isha bought two shirts for Rs. 1050.00 each, three pants for Rs. 1450.00, one
jacket for Rs. 2000.00, three T-shirtsfor Rs. 450.00 from U.F.O. Let's prepare
a bill for Isha.

166 Prime Mathematics Book - 5

U.F.O.

Kathmandu

Bill No. : 196 Date: 16/05/2068

Name of customer: Isha Quantity Rate (Rs.) Amount (Rs.)
Address: Baneshwor 2 1050.00 2100.00
2 1450.00 2900.00
S.N. Description 1 2000.00 2000.00
1. Shirt 3 1350.00
2. Pants 450.00
3. Jacket
4. T - shirt

Total 8350.00

In words: Eight thousand three hundred and fifty rupees only.

Customer's signature Seller's Signature

Exercise 9.2
1. Anshu bought 10 exercise books for Rs. 7 each, 9 pencils for Rs. 8 each,

12 pens for Rs. 70 each, 5 chart papers for Rs. 6 each from Sharma
Stationery on 16-05-2068. Prepare a bill for Anshu.
2. From a shop, Riya bought 3 cans of apple juice for Rs. 40 each, 5 bottles
of cold drink for Rs. 75 each, 6 red bull juice for Rs. 55 each and 8 ice-
creams for Rs. 60 each on 14-04-2068. Prepare a bill for Riya.
3. Raj bought 4 shirts for Rs. 900 each, 2 pants for Rs. 1400 each, 7 T-shirts
for Rs. 850 each, 1 suit for Rs. 6500 and 3 ties for Rs. 90 each from Karki
Store. Prepare a bill for her.
4. Salman bought 3 kg of apples for Rs. 80 per kg, 5 kg of grapes for Rs. 90
per kg, 9 kg of oranges for Rs. 110 per kg and 2 kg of guavas for Rs. 40
per kg. Prepare a bill for him.
5. John of Sitapaila purchased 5 kg of sugar for Rs. 55 per kg, 10 kg of
rice for Rs. 65 per kg, 3 kg of vegetables for Rs. 85 per kg, 8 kg of flour
for Rs. 90 per kg and 20 l of cooking oil for Rs. 130 per litre from Hira
grocers of Swayambhu. Prepare a bill.

Prime Mathematics Book - 5 167

Bar graphs

50 Number of students enrolled in
45 different classes of ABC school.

No. of students 40

35

30

25

20

15

10

5

0 II III IV V
Classes
I

Observe the diagram given above. It shows the number of students enrolled
in different classes of ABC school. The number of students enrolled in class I
is 15, in class II is 25, in class III is 35, in class IV is 20 and in class V is 50.

The diagram shown above is called bar diagram or bar graph. It is a method
to show the numerical information of different events.
Note:
i) Width of the bars should be same.
ii) Gap between the bars should be same.
iii) Event is shown along the horizontal line.
iv) Numerical information is shown along the vertical line.

Let’s learn how to construct a bar diagram of the data given below.

No. of students 20 25 30 35 40 45
graduated in S.L.C

Year 2060 2061 2062 2063 2064 2065

Here, the year is an event so it should be taken along the horizontal line.
The number of students graduated in SLC is the numerical information so it
should be taken along the vertical line with suitable scale.

Y 40 45
2064
No. of students 50 20 30 35
45 2060 25 2063
40
35 2061 2062 2065 X
30
25 Years
20
15
10

5

X' 0
Y'

168 Prime Mathematics Book - 5

Exercise 9.3

1. The marks scored by Alisha in first term in different subjects are
given below.

Subjects Nepali English Maths Science Social std.
85 55
Marks scored 65 80 95

Show the above information in a bar graph.

2. The number of students of class V present in different days of a week
are given as follows.

Day Sunday Monday Tuesday Wednesday Thursday Friday

No. of students 25 40 45 42 38 28
present

Present it in a bar graph.

3. Shreya bought some articles from a stationery shop and is given below.

Items Pencils Copies Pens Erasers Books Geometry boxes
12 18
No. of 16 22 8 32
items

Represent the given data in a bar graph.

4. Study the given bar graph and answer the following questions.

50 Y Bar graph showing the average temperatures of different
45 months of Kathmandu valley.
Temperature oC
40

35

30

25

20

15

10

5

X' 0 Baisakh Jestha Asadh Shrawan Bhadra X

Y' Months

i) Which month has maximum temperature? What is it?
ii) Which month has minimum temperature and what is the

temperature?
iii) What is the average temperature of Bhadra?
iv) What is the average temperature of Shrawan?

Prime Mathematics Book - 5 169

5. Bar graph showing the favorite subjects of the students of class 5.

Number of Students Y

55 Maths English Nepali Science Social X
50 Subjects Studies
45
40
35
30
25
20
15
10

5

X' 0
Y'

Study the bar graph given above and answer the following questions:
i) Which subject is the most favorite subject of the students? What

is the number of students who liked this subject?

ii) Which subject is liked by least number of students and what is the
number of students who liked this subject?

iii) Which is the next favorite subject of the students? Can you mention
the number of students who liked this subject?

iv) What is the number of students who liked Science? What is the
number of students who liked Social Studies? Can you say the total
number of students who liked Science and Social studies?

Ordered Pairs and Co-ordinates
Ordered pair
(Socks ,shoes); (Vest, shirt); (Shirt, coat) are some examples of pair of
objects. These pairs have fixed order. For example, first we put on socks
and then shoes. Similarly, first we put on vest and then shirt. Such pairs of
objects are called ordered pair.

Co-ordinates:
Lets see the seat arrangement of students of a class.

170 Prime Mathematics Book - 5

Column 1 (C1) Column 2 (C2) Column 3 (C3)
Ayusha Abin
Row 1 (R1) Subina
Shreya
Abinash

Row 2 (R2)

Shristy

Row 3 (R3)

Amod Aman Amar

Abinash is in the 1st row 1st column i.e. (R1, C1)
So, his position is (1, 1)

Similarly, Ayusha's position is (R1, C2) i.e. (1, 2).
Abin's position is (R1, C3) i.e. (1, 3).
Shristy's position is (R2, C1) i.e. (2, 1).
Subina's position is (R2, C2) i.e. (2, 2).
Shreya's position is (R2, C3) i.e. (2, 3).

In this way the position can be shown by a pair of numbers.

In the graph given alongside; Y
OX is called x-axis and OY is
called y-axis. Some points
are shown in the graph.
Let's locate their positions. C

First, let's find the position
of A. To reach A, first we
go 1 unit along x-axis from B

O. Then, we go 2 units up A E
along vertical line above D
x-axis. So, it’s position is (1,
2). Here, 1 is called x co- X' O X

ordinate and 2 is called y Y'
co-ordinate of A. (1, 2) is called co-ordinates of A. To locate the position of
B, first we go 3 units along x-axis from O. Then, we go up i.e. 4 units above
x-axis. Therefore, the position of B is (3, 4) so, (3, 4) is the co-ordinates of B.

Similarly, the co-ordinates of C is (5, 6); the co-ordinates of D are (7, 1) the
co-ordinates of E is (9, 3).

Prime Mathematics Book - 5 171

Exercise: 9.4

1. Plot the following points on a square paper.
A (1, 2), B (3, 5), C (1, 5), D (2, 7), E (3, 4),
F (2, 4), G (3, 6), H (4, 5), I (5, 7), J (6, 2),

2. Write the co-ordinates of following points.

Y

FJ

C E G
B D I
A
H
X' O Y'
X

Unit Revision Test

1. Students taking different club activities of a school are shown
below:
Judo – 50 Classical dance – 60
Modern dance – 90 Sitar – 70
Taekwondo – 75 Vocal – 40
Represent the above information in a chart.

2. Naina bought 4kg of cauliflower at the rate of Rs. 60/kg, 5kg of
potatoes at the rate of Rs. 75/kg, 3kg of raddish at the rate of Rs.
55/kg, 2kg of tomatoes at the rate of Rs. 95/kg, 6kg of cabbage at the
rate of Rs. 80/kg from a vegetable shop. Prepare a bill for Naina.

3. Students graduated from a school in the S.L.C. exam in different
years is given below. Represent the information in bar diagram.

Year 2050 2051 2052 2053 2054 2055

No of students 60 70 50 80 75 90

4. Plot the following points on a square paper.
A (3,5) B (4,2) C (6,3) D (1,4)
E (5,4) F(2,4) G (4,1) H (3,6)

172 Prime Mathematics Book - 5

U1n0it Sets

Estimated periods − 9

Objectives
At the end of this unit, the students will be able to:
• distinguish well defined collection and not well defined collection.
• use of the symbol ∈ and ∉
• write the sets in descriptive method, listing method and set builder form.
• know the types of sets depending upon the number of elements contained in it
• know the relationship between disjoint sets, overlapping sets, equal sets and equivalent
sets.

Teaching Materials
Collection of objects of same kind.

Activities

It is better to:
• explain with many proper examples for well defined collection.
• explain how to use ∈ and ∉.
• explain the description method, listing method and set builder form
• explain different types of sets
• explain the relationship between disjoint sets and overlapping sets, equal sets and

equivalent sets.
• design a bame using Table Tenise ball on additional subrtraction and equation. divide

the class in groups and let them play

Prime Mathematics Book - 5 173

Set

(i) (ii) (iii)
First figure is the collection of flowers. So, it is a set of flowers and each
flower is the member of this set.
Second figure is the collection of fruits. So, it is a set of fruits and each fruit
is a member of this set.
Third figure is the collection of vegetables. So, it is a set of vegetables and
each vegetable is a member of this set.

Rojina Reshma Rokina

If we consider above figure as a set of slim girls, it may lead to a confusion.
For example, Rokina is slim than Reshma and Rojina is slim than Rokina. So,
collection of slim girls is not well defined. So, it is not a set.

Thus, a collection of well defined objects is called a set.

Set should be denoted by using capital letters A, B, C, P, Q, ........ etc.

Elements of set should be denoted by using small letters a, b, c, p, q, r, .....
etc.

Members of a set P= a, e,
Given collection is a set of vowels of English i, o, u
alphabets. The members or elements of this
set are a, e, i, o, u.

174 Prime Mathematics Book - 5

∴ a ∈ P i.e. a belongs to the set P.
e ∈ P i.e. e belongs to the set P.
i ∈ P i.e. i belongs to the set P.
o ∈ P i.e. o belongs to the set P.
u ∈ P i.e. u belongs to the set P.
But,
b ∉ P i.e. b doesn’t belong to the set P.

Method of writing set:
There are various ways of expressing the sets.
i. Description method.
ii. Listing method
iii. Set-builder or rule method.

Descriptive method:
• The set of natural numbers less than 10.
• The set of districts of Narayani Zone.
• The set of first five prime numbers.
Above examples are some of sets expressed in description method. In this
method common property of the members of the set are described.

Listing method:
A = { 1, 2, 3, ……………, 9}.
B = { Parsa, Bara, Rautahat, Chitwan, Makawanpur}
C = { 2, 3, 5, 7, 11}

In the above examples, the members of the set are listed inside the curly
bracket { }. The members are separated by commas and the sets are named by
using capital letters. So, this method of writing set is called listing method.

Set- builder or rule method:
A = { x: x is a natural number less than 10}.
B = { y: y is a district of Narayani Zone}.
C = { z: z is a first five prime numbers}.

In this method, the members of the set are represented by variable x, y, z etc
and the common property of the elements are described.
The members 2, 3, 5, 7, 11 of a set C are represented by the variable z. The
common property of the members (first five prime numbers) is described by z.
C = { z: z is a first five prime number} is read as "C is a set of all values of z such
that z is a first five prime numbers." The symbol (:) is read as "such that".

Prime Mathematics Book - 5 175

Exercise: 10.1

1. Which of the following collections are well defined?
(a) A collection of first six even numbers.
(b) A collection of tall girls of class X.
(c) A collection of good football players.
(d) A collection of wild animals.
(e) A collection of days of a week.

2. Use the symbols ∈ or ∉:
(a) i ........... {a, e, i, o, u}
(b) 2 ........... {1, 3, 5, 7}
(c) Cat ........... {set of domestic animals}
(d) Tiger ........... {set of wild animals}
(e) Apple ........... {set of fruits}

3. List the members of the following sets and name them:
(a) The set of first four natural numbers.
(b) The set of first five prime numbers.
(c) The set of vowels of English alphabets.
(d) The set of four domestic animals.
(e) The set of four green vegetables.

4. List the elements of the following sets:
(a) A = {x: x is a prime number less than 10}.
(b) B = {y: y is an even number less than 11}.
(c) C = {z: z is a composite number less than 15}.
(d) D = {p: p is an odd number less than 9}.
(e) E = {q: q is a natural number between 5 and 10}.

5. Write the given sets in set builder form:
(a) A = {1, 3, 5, 7, 9}. (b) B = {2, 3, 5, 7, 11, 13}.
(c) C = {1, 2, 3, 4}. (d) D = {5, 10, 15, 20, 25}.
(e) E = {2,4,6,8}.

6. Write the following sets in description method:
(a) A = {a, e, i, o, u}.
(b) M = {bus , jeep , car, van}.
(c) N = {grape, apple, orange, guava}.
(d) B = {2, 4, 6, 8, 10}.
(e) D = {cat , cow, dog, buffalo}.

176 Prime Mathematics Book - 5

Types of sets:
i. Empty or null set:

A = {x: x is a boy student of Padma Kanya College}.
M = {y: y is a man with three eyes}.
N = {z: z is an even number between 10 and 12}.
These are examples of empty sets. These sets contain no elements.
Empty sets are denoted by { } or φ. It is also known as void set.

Thus, a set containing no element is called an empty set.

ii. Unit or singleton set:
B = {largest ocean of the world} = {Pacific ocean}.
C = {smallest district of Nepal} = {Bhaktapur}.
D = {highest mountain of the world} = {Mount Everest}
Above sets contain only one element. Such sets are called unit sets.

Thus, a set containing only one element is called an unit set.

iii. Finite set:
A = {1, 2, 3, 4}
B = { 2, 4, 6, 8, 10 }
C = {2, 3, 5, 7, 11, 13}
The set A is containing four elements, set B, five elements and C, six
elements. These sets are containing finite elements. Such sets are
called finite sets.
Thus, the set containing finite number of elements is called finite set.

iv. Infinite set:
M = {1, 2, 3, ………………..}
N = {2, 4, 6, …………………}
Q = {2, 3, 5, ………………..}
We can’t count the number of elements of the sets M, N and Q. Such
sets are called infinite sets.
Thus, set containing infinite number of elements is called infinite set.

Prime Mathematics Book - 5 177

Relationship of sets

i. Equal sets:
Consider, A = {1,3,5,7} and B = {3,7,1,5}.
The sets A and B contain same elements and equal number of elements.
Such sets are called equal sets.

ii. Equivalent sets:
Consider, A = {2, 4, 6, 8, 10} and B = {a, e, i, o, u}
The set A contains 5 elements and also the set B contains 5 elements.
Thus, the sets A and B contains equal number of elements. Such sets are
called equivalent sets.

iii. Overlapping sets:
Consider, A= { 2, 4, 6, 8, 10} and B = {4, 8, 12, 16, 20}.
AB

2 4 12
6 8 16
10 20

The elements 4 and 8 are present in both sets A and B. So, A and B are called
overlapping sets because they contain common elements.
iv. Disjoint sets:

Consider, A = {2, 4, 6, 8} and B = {1, 3, 5, 7}.
AB

24 13
68 57

The sets A and B do not contain any common elements. So, they are called
disjoint sets.

178 Prime Mathematics Book - 5

Exercise: 10.2
1. Observe the following sets and mention the type of the sets depending

upon the number of elements contained by them:
(a) P = { x : x is a day of a week whose first letter is M }
(b) Q = { y : y is the largest zone of Nepal}
(c) R = { a man with three hands }
(d) A = { an even number between 8 and 10}
(e) B = {1, 2, 3, ……………}
(f) C = {5, 10, 15, ………….}
(g) M ={ 3, 6, 9, 12 }
(h) N = {2, 3, 5, 7, 11}

2. State whether the following sets are equal or equivalent:
(a) A = {3, 6, 9, 12} and B = {9, 6, 3, 12}.
(b) C= {a, e, i, o, u} and D = {b, c, d, f, g}.
(c) E = {mango, orange, banana} and F = {banana, mango, orange}.
(d) M = {1, 4, 9, 16} and N = {1, 8, 27, 64}
(e) P = {prime factors of 6} and S = {prime factors of 12}.

3. Find the over-lapping and disjoint sets in the following:
(a) A = {1, 3, 5, 7} and B = {3, 7, 11, 15}.
(b) C = {a, e, i, o, u} and D = {a, b, f, u, h}.
(c) E = {1, 3, 5, 7} and F = {2, 4, 6, 8}.
(d) P = {a, b, c, d } and Q = {e, f, g, h}.

Unit Revision Test

1. Use the symbols ∈ or ∉ E in the following :

a. 4 …….. { Set of even numbers}
b. Dog ………{ Set of wild animals}
c. 3 ………….{ 2,4,6,8}

2. List the members of the following sets :
a. A = { The set of four wild animals}
b. B = { The set of first five odd numbers.}
c. C = { The set of five fruits}

Prime Mathematics Book - 5 179

3. Write the following sets in set builder form:

a. A = { 3,6,9,12,15} b. B = { 0,1,2,3,4,5}

c. C = { a,e,i,o,u}

4. Write the following sets in description method:
a. P = { 1,3,4,7,9}
b. Q = { potato, onion, pea, bean}

5. Define the following sets with an example:

a. Null set b. Singleton set

c. Infinite set d. Equivalent set

e. Disjoint set

6. Write the types of the following sets:
a. A = { a man with three eyes}
b. B = { an odd number between 10 and 12. }
c. C = { x:x is a day of a week whose first letter is F. }
d. D = { 2,4,6,8 }
e. E = { 5,10,15,20,….. }

7. State whether the following sets are equal or equivalent:
a. A = { 5,10,15,20 } and B = { 5,15,20,10}
b. M = { letters of word Nepal } and N = { letters of word panel}
c. D = { letters of word pen } and E = { letters of word nip}

8. Find the overlapping and disjoint sets in the following:
a. A = { 2,4,6,8} and B = { 3,6,9,12}
b. A = { letters of word put} and B = { letters of word but}
c. A = { 2,4,6,8 } and B = { 1,3,5,7 }.

Answers:
Exercise 10.1 and Exercise 10.2 [show your teacher]

180 Prime Mathematics Book - 5

U1n1it Algebra

Estimated periods − 18

Objectives
At the end of this unit, the students will be able to:
• identify the variables and constants
• classify the algebraic expressions according to the number of terms.
• find the numerial values of algebraic expressions by substituting the values of variables.
• perform the four fundamental mathematical operations on algebraic expressions.
• write the simple word problems in algebraic expressions.
• solve the simple algebraic equation of single variable.
• express the simple mathematical problems in the form of algebraic equations and solve
them.

Teaching Materials
Chart of algebraic expression, tiles, beam balance, etc.

Activities
It is better to:
• revise the concept of variable, constant, coefficient, terms and algebraic expressions
before starting the lesson.
• discuss the way of writing the algebraic expression of simple word problems.
• discuss the activities of four fundamental mathematical operations on algebraic
expressions by using tiles.
• illustrate the concept of equation on the basis of beam balance.
• drill the equation problems of a single variable on the basis of balance.

Prime Mathematics Book - 5 181

Algebra

Historical fact
Diophantus (believed to live somewhere between 150 B.C. to 250 AD in
Alexandria, Greek) is regarded as the father of Algebra. We know that in
algebra we use symbols for unknowns. He was the first to use symbols for
unknowns in his book "Arithmetica". But the word 'algebra' came from 'Al-
Jabr Wal Muqabalah' a book by al-khowarizmi of Arabia (9th century A.D.).

My dear students, let's solve
the following problems for revision

of class IV and preparation for
class V.

1. Identify whether the following are variables or constants:
a) x represents the number of students of class V in your school.
b) a represents the number of teachers in Kathmandu valley.
c) y represents the number of districts of Bagmati Zone.
d) z represents odd numbers between 7 and 14.

2. Identify whether the following are terms or expressions:
a) 4x b) 7x2y c) 2x + 3
d) a - b + c e) 5axy f) x - 2y + 3

3. Identify whether the following are like or unlike terms:
a) 2x and 2a b) 4ab and -3ab c) a2xy and a2xz
d) 4x2y and 3x2y e) -2x and -2 f) 3y2 and 3y

4. Add: b) 2x + 3x + 7x
a) a + 2a + 3a d) a2b + 2a2b + 7a2b
c) 3b + 2b + 4b f) x + y and 2x + 3y
e) a + b and 2a + b b) 3ab from 7ab
g) 5a + 2b + c and 2a + 4b + 4c d) 3a2 from 9a2
f) 4xy from 15xy
5. Subtract:
a) 2a from 5a
c) 4y from 7y
e) 4z from 12z
g) 3a2b from 11a2b

182 Prime Mathematics Book - 5

6. If a = 2 and b = 3, find the value of the following expressions:
a) 3ab b) 2a-b c) 3a2 - b
d) a + b e) 7(a2 - 2b) f) b2 - 2a
g) 2(a + 2b) h) 5(b - a) i) 3(7a - b2)

7. Write the following in algebraic expressions:
a) x is added to 7
b) 3 is subtracted from x
c) x is multiplied by 2y
d) 2 is added to the product of x and 3y
e) The sum of 2x and 3y is divided by 5y

Constants:
Let's take a letter x which stands for the longest river of the world. Then x
represents only one name of the river. It means the letter x does not represent
more than one value. So, the letter which always stands for only one value is
called constant. For examples, 3 represents only three number of quantities
and it never represents any other number of quantities. So, 3 is a constant.

Variables:
Let's take a letter x which stands for the weight of the students of class V in
your school. But we know that the weight of the students are different. So,
x represents the different values. Therefore, the letter which represents the
different values is called variable. The variables are generally denoted by
the letters x, y, z, etc.
If x represents the day of a week, then x maybe Sunday, Monday, ......,
Saturday. It means x does not represent fixed day. Therefore, x is a variable.

Algebraic expressions:
The result which is obtained after connecting the constants and variables
with different power or variables only with different power also by using any
four mathematical fundamental operations is called algebraic expression.
For example, the sum of x and 7 is x + 7, where x and 7 are connected by +ve
sign
So, x + 7 is an algebraic expression.
Similarly, other examples of the algebraic expressions are 9x2, 2x - 7y, x, x3 +
yz, 3x3 + y + 2, etc.

Prime Mathematics Book - 5 183

Algebraic terms:
The parts of an algebraic expression are called the algebraic terms. For
example, 2x + 3y is an algebraic expression. In this expression, 2x and 3y are
its parts. So, 2x and 3y are the terms.
Similarly, in the expression 4x2 + 3x - 7, 4x2, 3x and 7 are the terms.
Types of algebraic expression
There are the following algebraic expressions.
Monomial, Binomial, Trinomial and Multinomial expressions.

Monomial expression: An algebraic expression which contains only one term
is called monomial expression. For example: x, 3x, 2x2, 4xy, 7 etc. These are
also called terms. So, a term itself is a monomial expression.
Binomial expression: An algebraic expression which contains two terms is
called binomial expression. For example: a + 2b, 3x2 - y3, 2x + 7, etc.
Trinomial expression: An algebraic expression having three terms is called a
trinomial expression. For example: a + b - c, 2x + 7y - 5, etc.
Multinomial expression: An algebraic expression which contains more than
three terms is called a multinomial expression. For example: a + 2b - 3c + d,
x2 + 2y2 + z3 - 7, etc.

Coefficient, base and power of an algebraic term:
Let's consider a variable x and add it three times itself. Then, x + x + x =
3x. Here, 3 represents the repetition of x three times. So, 3 is called the
coefficient of the variable x.
What is the coefficient of y in a term y ?
If we consider a term ax where x is the variable, then a is called the coefficient
of x.
In term 5x, 5 is called numeral coefficient of x.
In term ax, a is called literal coefficient of x.
What is coefficient of x in the terms -7x and 3bx?
Let's consider a variable x and multiply it three times itself. Then, x × x × x
= x3.
In x3, the numeral 3 is called the power of the variable x and the variable x
is called the base.

184 Prime Mathematics Book - 5

Ram what are 5, x and
3 in 5x3?
5 is called the coefficient
of x3. x is called the base
Very good! and 3 is called the power

of x.

Like and unlike terms:
Let's discuss the like and unlike terms from the following examples.
a) 2x, 5x, -7x, 8x, are like terms
b) x3, 3x3, -5x3, 17x3 are like terms
c) 3xy, 4xy, -7xy, are like terms
d) 5a2b, 7a2b, -5a2b are like terms
e) 2x, 2y, 2z, 2w are unlike terms
f) x3, 3x2, -7x5 are unlike terms
g) 2xy, 3x2y, 2xy2 are unlike terms
h) 5a2b, 5ab2, 5bc, 7ac are unlike terms

From the above examples, it is clear that the terms
are like terms if the terms have the same base and
equal power of the variable.
The terms are unlike terms if the terms have the
different variables or different power of the same
base.

Evaluation of terms or expressions:
When we replace the variable of a terms or expression with numbers, the
value of the term or expression is obtained after use of mathematical
fundamental operation. It is called the evaluation of term or expression.

Let's learn it from the following examples:

Example: If x = 3, y = 2 and z = 4, find the value of

(i) 2x2y (ii) 3xy - 4z (iii) 3x2 - 2yz

Prime Mathematics Book - 5 185

Solution:
(i) 2x2y = 2×(3)2 × 4 = 2 × 9 × 4 = 72
(ii) 2xy - 4z = 3 × 3 × 2 - 4 × 4 = 18 - 16 = 2
(iii) 3x2 - 2yz = 3 × (3)2 - 2 × 2 × 4 = 3 × 9 - 16 = 27 - 16 = 11

Exercise: 11.1

1. Write constant or variable in the boxes:
a) x represents the number of your family. x is a
b) x represents the price of cakes. x is a
c) y represents the even number between 6 and 15. y is a
d) p represents the age of students of grade V. p is a

2. Write the terms and the number of terms in the following expressions:

a) 3xy b) 2x + 3yz c) 2a2b - 4bc + 2b2d

d) 4xy2 - 6yz

3. Write the types of expressions in the following expressions:

a) 3x - 2z b) 2a + 4y - 7

c) 4xy d) 2a2b + 3bc

e) 3x2yz f) 2x - 3y2 + 7z - 9

g) -7m2n h) 7x - 4p + 5y2 - 2z + 11

4. Write the coefficient, base and power of the following terms:

a) 2x5 b) -3p c) 7ax3
d) -9by7 2
e) 3 x2

5. Write the term having:

a) Base = x, power = 2 and coefficient = 3

b) Base = y, power = 5 and coefficient = 7

c) Base = z, power = 3 and coefficient = 5a

d) Base = z, power = 7 and coefficient = -2b

6. Identify the like or unlike terms in the following:

a) 2x, - 3x and 5x b) 2ab, -4ab, 6ab
c) 3a, 2a2, 3a3 d) 2ab, 3a2b, 4ab2

e) 3x, 2x2, 5xy f) x2y, xy, 3x2y2

186 Prime Mathematics Book - 5

7. If x = 2, find the values of:

a) 3x - 7 b) x + 7 c) x2 + 9

d) 2x2 + 5x e) 4x2 + 7x - 9

8. If a = 3 and b = 5, find the value of:

a) a + b b) 2a + 4b c) b - a

d) 3(a + b) e) a+b
4

9. Find the value of the following expressions when x = 3, y = 2 and
z = -4.

a) xy - z b) (x - y)2 + z c) 3xy - 2z

d) 2yz + 5xy e) 5xy + 3z

10. If a = 2, b = 3, c = -2 and d = 5, find the value of:

a) ac + bd b) 2bd - ab c) 2a2b - cd

d) a-c e) ad - bc
2 4

11. Write the perimeter of the following figures in the algebraic
expression. Also find the numeral value of the perimeter if x = 3,
y = 5 and z = 2.

a) z b) c) z
x y z 2x y

x+y x+z

Addition and subtraction of algebraic terms:

In the addition or subtraction of algebraic terms, we should add or subtract
the coefficients of like terms only. We can't add or subtract the coefficients
of unlike terms.

For example: y + y = 2y, 3x + 8x + 5x = 16x, 4x2 + 5x2 + x3 = 9x2 + x3

5m - 3m = 2m, 7a2 - 4a2 = 3a2, 8x4 - 2x3 - 3x4 = 5x4 - 2x3

Prime Mathematics Book - 5 187

Exercise: 11.2

1. Add or subtract the following expressions:

a) x + x b) x + x + x

c) y + y + y + y d) 2y + y + 3y

e) 4x + 3x + 2x f) 2a2 + 3a2

g) a2 + 4a2 + 7a2 h) 3a5 + 2a5 + 4a5

i) 2ab + 3ab j) 4ab + 7ab + 11ab

k) 7yz + 3yz + 12yz l) abc + 3abc + 5abc

m) 4x - x n) 5y - 3y

o) 6z2 - 2z2 p) 9x3 - 5x3

q) 6ab - 4ab r) 8a2b - 3a2b

s) 7xy - 5xy t) 15m3m2 - 19m3n2

2. Simplify: b) 2a + 7a - 4a
a) 5x + 3x - 2x d) 3ab - 5ab + 7ab
c) 7x - 12x + 19x - 4x f) 15ab2 - 12ab2 + 17ab2 - 3ab2
e) 4x2y + 9x2y - 12x2y
g) 18x3 - 2x3 + 7x3 - 5x3

3. Find the outputs using the following mathematical machines:

a) 3x b) 2x

+ x If x = 2 + 3x x = 5

output output

c) 7x d) 17x
- 3x
x=3 - 8x x=2
output
output

188 Prime Mathematics Book - 5

4. Find the perimeter of the following figures if x = 2cm.

a) b) c) 4x d) x x
2x 3x 2x 3x 2x
5x 2x+1 2x 2x
x 4x
3x

5. a) What should be added to 4x to get 17x?
b) What should be added to 9x to get 15x?
c) What should be subtracted from 13a to get 5a?
d) What should be subtracted from 17xy to get 12xy?
e) By how much is 3ab less than 11ab?
f) By how much is 15 ab more than 7ab

Addition and subtraction of algebraic expression:
In the addition or subtraction of two or more algebraic expressions having
two or more than two terms, we should arrange the like terms in the same
column. Then, we add or subtract their coefficient. We can add or subtract the
term of the algebraic expressions in vertical and horizontal arrangement.
Study and learn the following examples:

Example 1. Add: 3a + 7b - 2c and 4a + 2b + 5c.

Solution: Vertical arrangement

3a + 7b - 2c • Arrange the like terms in
(+) 4a + 2b + 5c same column.

7a + 9b + 3c • Add or subtract the
coefficient of like terms.
3a + 4a = 7a, 7b + 2b = 9b,
5c - 2c = 3c

Horizontal arrangement Put the like terms
3a + 7b - 2c + 4a + 2b + 5c together and then
perform the addition
= 3a + 4a + 7b + 2b - 2c + 5c and subtraction.
= 7a + 9b + 3c

Prime Mathematics Book - 5 189

Example 2. Add: 3a + 5b - 2c, 4b + 3c + 2a and a - 2b + 5c
Solution: Here,

3a + 5b - 2c
2a + 4b + 3c
(+) a - 2b + 5c
6a + 7b + 6c

Example 3. Subtract: 2x + 3y - 2z from 5x + 7y + 4z

Solution: Vertical arrangement

5x + 7y + 4z Arrange the like terms in same
(-)2x( +-) 3y( -+ )2z column.
Change the sign of each terms
3x + 4y + 6z of the second expression.
Perform the addition or
subtraction.

Horizontal arrangement
(5x + 7y + 4z) - (2x + 3y - 2z)

= 5x + 7y + 4z - 2x - 3y + 2z
= 5x - 2x + 7y - 3y + 4z + 2z
= 3x + 4y + 6z

Example 3. Simplify: 2x2 + 7x - 5y + 5x2 - 3 - 2x + 9y
Solution: Here,

2x2 + 7x - 5y + 5x2 - 3 - 2x + 9y
= 2x2 + 5x2 + 7x - 2x - 5y + 9y - 3
= 7x2 + 5x + 4y - 3

190 Prime Mathematics Book - 5

Example 4. What should be added to 2a + 4b - c to get 5a + 9b + 4c?

Solution: In this problem, we
should subtract 2a + 4b - c

From 5a + 9b + 4c.

Let's think, what should be
added to 7 to get 10?
It is 3. That means,
10 - 7 = 3. We use this
idea in algebra too.

5a + 9b + 4c
(-)2a(-+) 4b(+-)c
3a + 5b + 5c

∴ The required expression to be added is 3a + 5b + 5c.

Example 5. What should be subtracted from 7x + 4y + 9 to get 3x + 2y + 3?
Solution: In this problem, we
should subtract the

result 3x + 2y + 3 from
7x + 4y + 9.

Let's think, what should
be subtracted from 7 to
get 3? It's definitely 4. It's
7 - 3 = 4. We use this idea in

algebra two.

7x + 4y + 9
(-)3x (+-)2y (+−)3

4x + 2y + 6
∴ The required expression to be subtracted is 4x + 2y + 6.

Prime Mathematics Book - 5 191

Exercise: 11.3

1. Add:

a) a + 3 and 2a + 4 b) 3x + 2 and 2x + 7

c) 4x - 5 and 7x + 2 d) 3a + 5b and 2a + 3b

e) 4m - 7n and 7m + 2n f) 2x2 + 3y and 5x2 + 2y

g) 2ab + 9bc and 5ab - 2bc h) 2x + 5y + z and 3x + 2z + 3y

i) 3a - 7b + 2c and 5a + 2b + 5c

j) 4a2 + 5b2 + 7 and 5a2 + 3b2 + 9

k) 9x2 + 2xy + 3y2 and 2x2 + 7xy + 8y2

l) 3a2 - 2ab + 5b2 and 6a2 + 9ab + 2b2

m) x3 + 2x2 + 7x - 6 and 4x3 + 7x2 - 2x + 4

n) 2x + 3y, 4x - 2y and 5x + 7y

o) 3a - 2b, 4a + 5b, a + 7b and 3a + b

p) 2x2 + x + 3, 3x2 + 4x + 2 and x2 - 2x + 7

q) 2ab + 3bc + 4ac, 5ab + 2bc + ac and 3ab + 7bc + 9ac

2. Subtract:

a) x + 2 from 5x + 7 b) 2x + 3 from 7x + 9

c) 2a + 3b from 5a + 9b d) 4m - 2n from 9m + 3n

e) 3x2 + 2y2 from 5x2 + 11y2 f) 2a2 + 3b2 from 5b2 + 7a2

g) 2x + y + 7 form 5x + 3y + 11 h) 5x - 2y + z from 7x + 3y + 4z

i) 5a + 9b - 2c from 3a - 2b + 4c

j) 2a2 + 3ab + 3b2 from 5a2 - 2ab + 7b2

k) ab + 3bc - 4ca from 5ab - 7bc + 2ca

l) 4x3 + 2x2 + x - 1 from 7x3 + 5x2 - 2x + 7

3. Simplify: b) 3x - 2y + 6x + 9y
a) a + 2b + 3a - b d) 5m + 2n + 4p - 2m + 7n - 2p
c) 2x2 + 4x - 1 + 3x2 - 2x + 9 f) 4pq - 2qr + 3rp - 9pq + 4qr - 2rp
e) 3a2 - 6 + 2a - 2a2 + 13

4. a) What should be added to 2x + 7 to get 7x + 19?
b) What should be added to 3a + 2b to get 5a + 9b?
c) What should be added to 2x + 3y - 2 to get 7x + 9y + 6?
d) What should be added to 3a + 4b - 2c to get 9a + 7b + 6c?

192 Prime Mathematics Book - 5

5. a) What should be subtracted from 7x + 6 to get 4x + 2?
b) What should be subtracted from 9a + 4b to get 3a + 2b?
c) What should be subtracted from 7a + 6b - 2c to get 3a + 2b - 4c?

6. a) If x = 2a + 3b and y = 4a + 2b, find x + y and y - x.
b) If p = 4x + 7 and q = 9x - 3, find q - p and q + p.
c) If a = 3x - 2y + 1 and b = 2x + 5y + 3, find 2a + 3b and 2b - a.

7. If x = 3cm, y = 2cm and z = 3cm, 2y A C
calculate the perimeter of the given B 4z
triangle ABC.
2x+2

8. If a = 2cm and b = 3cm, calculate AD
the perimeter of the given rectangle 4a-b
ABCD.
B 3a+2b C

Multiplication of Algebraic Expressions

Product of 3x and 2x is 3x ×
2x = 3 × 2 × x1 × x1

= 6x1+1 = 6x2
Product of 4y2 and 3y3 is 4y2 ×
3y3 = 4 × 3 × y2 × y3
= 12y2+3 = 12y5

The product of algebraic expressions
means multiply the coefficients of terms and

add the power of the same base.

In the case of multiplication of algebraic expressions, the coefficients of the
terms are multiplied and the power of the same base (variable) are added.

Prime Mathematics Book - 5 193

Multiplication of Monomial Algebraic Expressions
We know that the monomial algebraic expression contains only one term. So,
when we multiply two or more monomial expressions, then first we multiply
the coefficients of all given terms and add the power of the same bases
(variables) of the given terms.

Example 1: Multiply 2x, x and 3x

Solution: Here, In 2x, coefficient is 2
In x, coefficient is 1
2x × x × 3x In 3x, coefficient is 3
= 2 × 1 × 3 × x1+1+1 So, first multiply 2, 1 and 3.
= 6x3 In each term, the base is x of
which power is 1. So, add the
power of x of each term.

Example 2: Multiply 4m2 and 7m5 Multiply the
Solution: Here, coefficients of terms
and add the power of
4m2 × 7m5 variable.
= 4 × 7 × m2 × m5 I remember that (+) ×
= 4 × 7 × m2+5 (-) = (-)
= 28m7 So, 5 × -9 = -45

Example 3: Multiply 5x3 by -9x5
Solution: Here,

5x3 × (-9x5)
= 5 × (-9) × x3 × x5
= -45 × x3+5 = -45x8

Note:

× - has no meaning. Negative term after × should be kept with in brackets.

Exercise: 11.4

1. Perform the following multiplications:

a) 3 × 4a b) 7x × 2x c) 3x × 4x
f) 2b × 5b2
d) a2 × a3 e) x2 × x i) -3x × 4x2
l) 5z2 × (-12z6)
g) 6m3 × 3m2 h) 12p3 × 4p7

j) 4x2 × (-5x4) k) 7q × 5q3

194 Prime Mathematics Book - 5

2. Simplify: b) 2x × x × 3x
a) a × a × a d) Y × 2y2 × 4y3
c) 4y × (-y) × 3y f) x × 2x × 5x2
e) 2b × 3b2 × (-b3) h) (-p) × 3p2 × (-2p3)
g) 2y2 × (-7y3) × y j) 2z × (-3z2) × 4z2 × (-9)
i) a × 2a2 × 4a3 × 3
k) 6x × (-2x2) × -3x × 7

3. Area of a square is side × side [i.e. (side)2 = l2]. Find the area of the
following squares. Also calculate the actual area if x = 3.
a) b) c) d)

x cm
2x cm
4x cm
3x cm

x cm 2x cm 4x cm 3x cm

4. Area of a rectangle is length (l) × breadth (b). Find the area of the
following rectangles. Also calculate the actual area if y = 2cm.
a) b) c)

y 2y 3y

2y 3y 5y

5. Volume of a cubiod is length (l) × breadth (b) × height (h). Find the
volume of the following cuboid. Also calculate the actual volume if
a = 2cm.
a) b) c)

h=2a
h=2a
h=3a

b=2a l=2a b=3a l=3a b=2a l=4a

6. a) If a = 3x, b = 2x2 and x = 2, find a × b.
b) If x = 2a, y = 3a and a = 3, find x × 2y.

Prime Mathematics Book - 5 195