Prime Mathematics 8

Prime Mathematics Book - 8

Equality of ratios is called proportion.

Thus if a : b = c : d, the terms a, b, c and d are in proportion and is also written as

a : b :: c : d

The terms a, b, c, d are called proportional.

Here a is called first proportional, b second proportional, c third and d fourth proportional.

First proportional a and a : b :: c : d
Fourth proportional d are called extremes and second means
Proportional b and third proportional c are called means.

We get, extremes

Products of extremes = Proportional of means.

i.e. a × d = b × c

This means the fourth numbers a,b,c,d are in proportion when the product of ... ... is

same as the product of means.

Example 1: Check weather the ratios 18 : 45 and 26 ; 65 are in proportion or not.
Solution: Here, given ratios 18 : 45 and 26 : 65

Let 18 : 45 :: 26 : 65
Product of means = 45 × 26 = 1170
Product of extremes = 18 × 65 = 1170
Since product of means product of extreme, the ratios are in proportion.
Alternate method:
Given ratios 18 : 45 and 26 : 65

18 2 26 2
First ratio = 45 = 5 = 2 : 5 Second ratio = 65 = 5 =2:5

∴ 18 : 45 = 26 : 65 ∴ 18 : 45 :: 26 : 65

Example 2: If 18, 30, 24 and x are in proportion, find the value of x

Solution: Since 18, 30, 24 and x are in proportion,

18 = 24 85
or, 30 x 24 × 30

or, 18x = 24 × 30 or, x = 18 3 or, x = 8 × 5 = 40 .

A. Continued proportion:

Three quantity of same kind a, b, c, are such that a : b = b : c, the terms are said
to be in continued proportion. The term a is called first proportional, b is second
proportional and c is called third proportional. In the proportion 4 : 12 :: 12 : 36, 4
is the first, 12 is the second and 36 is the third proportional. The second proportional
in case of continued proportion is also called mean proportion.

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Prime Mathematics Book - 8

Example 3: If 9 and 12 are the first two terms in a continued proportion, find the third

proportional.

Solution: Let the third proportion be x then 9, 12 and x are in continued proportion

or, 9x = (12)2 or, x = 1494
or, 9 = 12 or, x = 16
12 x

∴ The third proportional is 16.

Example 4: If the first and third proportional of the terms in continued proportion are
12 and 108, find the mean proportional.

Solution: Here, first proportional = 12, third proportional = 108
Let the mean proportional = x, then 12, x and 108 are in continued proportion.

or, 12 = x or, x2 = 12 × 108 or, x2 = 1296
x 108

∴ x = 1296 = 36

Types of proportions

Observe the following:

If a pen costs Rs. 160, then 2 pens cost Rs.2 × 160 = Rs.320. Here, ratio of pens 1:2
1

Rs.160 1
Ratio of their costs = Rs.320 = 2 = 1 : 2

2
Similarly, 2 pens cost Rs. 320, then 5 pens cost 5 × 160 = Rs. 800
Here, ratio of pens 2 : 5

Rs.320 2
Ratio of their costs = Rs.800 = 5 = 2 : 5

We get the ratio of the number of pens = ratio of the cost of the pens in each case.
Thus as the number of pens increase or decrease, the cost of pens also increase or
decrease proportionally.

If a quantity increases, the other quantity also increases proportionally and if a quantity
decreases, the other quantity also decreases prportionally the proportion is called direct
proportion and the variation (change) is called direct variation. Again, if 1 man takes 10
days to do a work then 2 men take 5 days to do the work.

Here, ratio of men = 1 : 2 Ratio of days taken = 10 : 5 = 2 : 1
Ratios are equal but inversely

Ratio, Proportion and Percentage 197

Prime Mathematics Book - 8

1
i.e. ratio of number of men = ratio of number of days taken
If a quantity increase or decrease, other quantity decrease or increase in opposite

proportion, the proportion is called indirect or inverse proportion and the variation
is called indirect variation. No. of objects and their costs, No. of objects and their
weights, No. of workers and quantity of work e.t.c. vary directly. No. of men and days,
working hours per day and time taken e.t.c vary inversely. Idea of direct and indirect
proportions are used for solving problems of unitary method.

Example 5: If the cost of 6 pens is Rs. 1080, find the cost of 11 pens.

Solution: No. of Pens Cost (Rs.)
6 1080
11 X

Let the cost of 11 pens is x, then as the number of pens and their costs vary

directly. Ratio of the number of pens = ratio of costs.

6 1080 or, 6x = 1080 × 11 or, x = 1080 × 11 or, x = 1980
or, 11 = x 6

∴ 11 pens cost Rs.1980

Process:
(i) Tabulate number of books and their costs.
(ii) Suppose unknown value as X.
(iii) Identify the variation (proportion) whether direct or indirect.

(iv) Equate the ratio and find the unknown term.

Example 6: If 18 men can accomplish a piece of work in 30 days. How many days do 12
men take to accomplish the same work?

Solution:

No. of men No. of days

18 30
12 X

Let 12 men can accomplish the work in x days.

Since number of men and number of day taken vary indirectly.

Ratio of number of men = ratio of 1 of days taken
number

95
18 × 30
or, 18 = x or, x = 12 2 or, x = 45
12 30

∴ 12 men can accomplish the work in 45 days.

198 7 Ratio, Proportion and Percentage

Prime Mathematics Book - 8

Example 7: A garrision of 480 men has provision for 24 days. After 4 days 80 men left the

garrision. For how long will the remaining provision last for the remaining men ?

Solution: For 480 men, the provision lasts for 24 days
After 4 days, for 480 men it will last for 20 days
Remaining number of men = 480-80 = 400
let the remaining provision will last for x day

After 4 days No. of days
No. of men 20
480 X
400

Since, number of men and days taken are indirect proportion.

x = 480 or, x = 480 × 20 or, x = 24
or, 20 400 400

∴ x = 24

∴ The remaining provision will last for another 24 days for the remaining men.

Exercise 1.2

1. Check whether the following ratios are in proportion or not.
(a) 15 : 9 and 35 : 21 (b) 6 : 8 and 54 : 90
(c) 49 : 63 and 96 : 108 (d) 120 : 135 and 88 : 110

2. If the following numbers are in proportion, find the value of x.
(a) 9, 12, x, 28 (b) x, 64, 36, 72 (c) 6k, 11k, 126, x (d) 45, x, 50, 80
3. Find the mean proportion of
a) 4,9 b) 2,32 c) 3,27 d) 9, 16

4. (a) If the first and mean proportional of a continued proportion are 4 and 6, find the
third proportional.
(b) If the mean proportional and third proportional of three terms in continued proportion
are 28 and 56, find the first proportional.
(c) First and third proportional of the terms in continued proportion are 72 and 128. Find
the mean proportional.

(d) Find the mean proportional to 1 , 9 .
16 36

5. (a) What number must be added to each of the number 7, 17 and 47 so that they will be
in continued proportion ?

(b) What number must be subtracted from each of the numbers 18, 78, 19 and 83 so that
the remainders will be in proportion ?

(c) To make the numbers 4, 6, 24 and 29 proportional,what should be added to 29 ?

6. (a) If 25 copies cost Rs. 1250, find the cost of 40 copies.
(b) A motorbike travels 161 km for 3.5 litres of petrol, what distance will it travel for 12

litres of petrol ?

Ratio, Proportion and Percentage 199

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(c) An agent received a commission of Rs. 1920 on sales totalling Rs. 24000. What should
be the total sales to get commission of Rs. 5000 ?

(d) Indian currency Rs. 56000 is exchanged for Nepali currency Rs. 89600, how much
Nepali currency can be exchanged for Indian currency Rs. 100?

7. (a) A car takes 8 hours to complete a journey with average speed of 60 km/hr. What
should be the average speed to complete the same journey in 6 hours ?

(b) 24 men can do a piece of work in 16 days. How many men can complete the same
work in 12 days?

(c) A garrison on 200 men has provision for 48 days. For how many men would the
provision last for 30 days ?

(d) Operating 8 hours a day a battery can be used for 96 days. For how long can the
battery be used operating 6 hours a day ?

8. a) If a : b = 2:5 and b:c = 3:7, find i) a:c ii) a:b:c
b) If x:y = 1:2 and y:z = 2:3, find i) x:z ii) x:y:z
c) If c:d = 3:7 and d:e = 4:5, find i) c:e ii) c:d:e

1.3 Percentage

A. Introduction:

The word "Percent" is derived from the latin word "Percentum" which means per
hundred or out of hundred. It is denoted by %. Thus 35% = 35 out of 100. Percentage
is a fraction with denominator 100. The numerator of the fraction is called rate
percent or simply percent.

If Shailaja secured 95 marks out of 100 marks in mathematics paper in an
examination, her percentage is 95 which is written as 95%.

100

Remember the following: 1

→ To convert percent to fraction or decimal, replace % with 100 .
15 3

Example: 15% = 100 = 20 = 0.15



→ To convert given fraction or decimal into percent, multiply it by 100

33
Example: 5 = 5 × 100% = 60%
x
X as percentage of y = y × 100%


Percentage increase/decrease = increase/decrease in quantity × 100%
original quantity

200 7 Ratio, Proportion and Percentage

Prime Mathematics Book - 8

Example 7: Express 45% as fraction.

Solution: 45% = 9 × 19
45 100 20 = 20

Example 8: Express 80% as decimal
80

Solution: 80% = 100 = 0.8

Example 9: Convert 7 into percentage.
32

7 = 7 25 175
Solution: 32 32 × 100% = 8 % = 21.875%

8

Example 10: Express 0.295 as percentage.
Solution: 0.235 = 0.235 × 100% = 23.5%

Example 11: Find 18% of Rs.3400

Solution: 18% of Rs. 3400 = 18 × Rs.3400 = Rs.18 × 34 = Rs. 612
100

∴ 18% of Rs. 3400 is Rs. 612

Example 12: Express Rs. 114000 as a percentage of Rs. 190000 114 × 10%
19
Solution: Rs. 114000 as percentage of Rs. 190000 = 114000 × 100% = = 60%
190000

Example 13: If 42% of a number is 1176, find the number.

Solution: Let the number be x, then

42% of x is 1176 or, 42 × x = 1176 or, x = 1176 × 100 ∴ x = 2800
100 42

Example 14: Price of an article reduced from Rs. 80 to Rs. 60. Find the percentage
decrease in the price.

Solution: As the price is reduced from Rs. 80 to Rs. 60,
Decrease in price = Rs. 80 - Rs. 60 = Rs. 20

Now, Pe rcenta ge dec rease i n price = deoc rriegainsael ipnrpicreice × 100% = Rs.20 × 25
100%
Rs.80
4
= 25%
∴ The Percentage decrease in price = 25%

Ratio, Proportion and Percentage 201

Prime Mathematics Book - 8

Example 15: If 70% of the students of a school are boys and the number of girls is 276,
find the number of boys in the school.

Solution: 70% are boys
∴ 100% -70% = 30% are girls
Let total number of students in the school be x.
As number of girls = 276
or, 30% of x = 276
30
or, 100 × x = 276

92
or , x = 276 × 100

30

or, x = 92×10 or, x = 920

∴ The number boys in the school = total number of students - number of girls

= 920 - 276 = 644

Example 16: Marked price of a pair of shoes is Rs. 1650 and the shopkeeper allows a
discount of 20%. How much should a customer pay for it?

Solution: Marked price = Rs. 1650 Discount = 20%

∴ Discount amount = 20% of Rs.1650

= 20 × Rs.1650
100
5

Rs.1650
= 5 = Rs.330

∴ Price after discount = Marked price - Discount amount = Rs.1650 - Rs.330 = Rs.1320
∴ The customer should pay Rs. 1320

- The price fixed is marked price also called
Rs.1650 labelled or quoted or tagged price.
- Marked price is fixed more than its actual cost price de-
pending upon tax, transportation (shipping) e.t.c. such that
after giving discount (reduction) the seller can make profit.

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Example 17: After discount of 25% a mobile set was bought for Rs. 28800. What was the

price of the mobile before discount?

Solution: Here, discount = 20% Price after discount = Rs. 28800

let the price before discount = x then,

∴ Price before discount - discount = Price after discount

or, x - 25% of x = Rs. 28800 or, x × 3 = Rs. 28800
4
25
or, x - 100 × x = Rs. 28800 9600 4
or, x = Rs.28800 ×3
( (or, x 25 1
1004
1- = Rs. 28800 or, x = Rs. 9600 × 4

( (4-1 or, x = Rs. 38400

or, x 4 = Rs. 28800 ∴ x = Rs. 38400

∴ The price of the mobile set before discount = Rs. 38400

Example 18: Bijondra wants to buy a laptop costing Rs.80,000. But he has to pay 15%
tax also. How much does he pay for it ?

Solution: Price of laptop = Rs. 80,000

Tax rate = 15%

Now, Price with tax = Price + Tax

= Rs.80,000 +15% of Rs.80,000
( (= Rs.80,000
1 + 15 3
100
20
4000 23
= Rs.80000 × 20

= Rs.4000 × 23

= Rs.92000

∴ He has to pay Rs.92000 including tax

Example 19: An andriod mobile set cost Rs. 40680 including 13% VAT. What is the cost of

the mobile set without VAT ?

Solution: VAT rate = 13% Cost with VAT = Rs. 40680

Let the cost of the mobile set without VAT = X then,

Cost + VAT = Rs. 40680

or, X + 13% of x = Rs. 40680

Ratio, Proportion and Percentage 203

Prime Mathematics Book - 8

( (or, X1+ 13 = Rs. 40680
100

113
or, X × 100 = Rs. 40680
100

or, X = Rs. 40680 × 113 = Rs. 36000
∴ The cost of the mobile set without VAT is Rs. 36000

Example 20: The cost of bicycle is Rs. 15000. If 15% VAT charged,what will be the cost
of the bicycle including VAT ?

Solution: Here, cost of the bicycle without VAT = Rs. 15000 VAT rate = 15%
= Rs. 150 × 15 = Rs. 2250
∴ VAT amount = 15% of Rs. 15000 = 15 × Rs.15000
100

∴ Cost including VAT = Rs. 15000 + Rs. 2250 = Rs. 17250

Example 21: Price of a watch is Rs. 9000. Including VAT, it costs Rs. 10170. Find the VAT rate.

Solution: Here, cost of watch excluding VAT = Rs.9000
Cost of the watch including VAT = Rs.10170
∴ VAT amount = Rs.10170 - Rs.9000 = Rs.1170

VAT Amount
Now, VAT rate = Cost excluding VAT × 100%

= Rs.1170 × 100% 117
Rs.9000 = 9 % = 13%

∴ The VAT rate is 13%

Exercise 1.3

1. Express the following as fraction.

(a) 22.5% (b) 72% (c) 120% (d) 12.5%

2. Express the following as decimal.

(a) 90% (b) 32.5% (c) 150% (d) 40%

3. Convert the following into percentage.

(a) 5 (b) 3 (c) 0.475 (d) 2.05
84

204 7 Ratio, Proportion and Percentage

Prime Mathematics Book - 8

4. (a) Find 55% of Rs.7500
(b) If 72% of a number is 10440, find the number.
(c) Express 224 as the percentage of 1792.
(d) What percentage of 492 is 123 ?

5. (a) The price of petrol reduced from Rs. 135 to Rs. 127.68. Find the percentage reduction
in the price.

(b) Price of gold at present is Rs. 49000. If price of gold hiked by 6%, find the new price
of the gold.

(c) A's salary is 25% above the salary of B's. Find by how much percent is B's salary less
than A's salry.

(d) A man spends 65% of his monthly income and saves Rs.10,500. What is his monthly
income ?

6. (a) Last year, Radhika's monthly salary was Rs. 12000. This year her monthly salary is
Rs. 15000. What is the percentage increment in her salary?

(b) The monthly income of a person is Rs. 24000 from which he spends 20% on food,15%
on education, 25% on miscellaneous expenditure and the rest he save. Find his
monthly expenditure on each topics.

(c) The monthly income of a family is Rs. 60,000 in which 40% is from business, 30% is
from service and the rest house rent. Find the incomes of each source.

(d) In a village these are 1000 old peoples, 3600 adults and 3400 children. Find the
percentage of each type of people.

7. (a) A man gave 40% of his money to his son, 30% to his daughter and remaining
Rs.25200 to his wife. Find how much did his son and daughter got ?

(b) In an examination Santosh got 27 marks in mathematics and filed by 5 marks. If
pass marks is 40%, find the full marks.

(c) In S.L.C. Examination, an examinee needs 80% to secure first division with
distinction and Malala got 632 marks and could not secure distinction by 8 marks.
Find the full marks of the examination. Also find her percentage marks.

(d) There were two candidates in an election of a municipality, the winner candidate
got 52% votes of total vote casted and on the election with the majority of 1092
votes. Find the total number of votes casted and the number of votes received by
the looser candidate.

8. (a) A costumer bought a radio set for Rs.4600 with 20% discount. What was the
price of the radio quoted with ?

(b) Kepil got 18% discount while buying a basket ball costing Rs.2700. How much did
he pay for it?

Ratio, Proportion and Percentage 205

Prime Mathematics Book - 8

(c) Ashutosh bought a android mobile set for Rs.30400 which was marked Rs.38000.
Find the discount percentage.

(d) Dealer A allows discount of Rs.3000 and the dealer B allows 6% discount on
a laptop costing Rs.60,000. Where will you buy it?

9. (a) A cardigan costs Rs.5000 but 13% VAT is levied. What amount should a customer
pay for it including VAT?

(b) A steel cupboard costs Rs.13560 including 13% VAT. Find the marked price of the
cupboard.

(c) A wooden idol of Buddha costs Rs.9600 and Tashi paid Rs.11040 with VAT. Find the
VAT percentage?

(d) A car costs Rs.3,00,000 in India. How much will it cost in Nepal including 200%
custom duty and 10% shipping charge?

206 7 Ratio, Proportion and Percentage

Prime Mathematics Book - 8

Unit -Ar1e7a: RReevvisisioionnTeTsets(tRatio, Proportion and Percentage)

1. (a) Write the ratio 750ml to 4l in simple form.
(b) What should be added to the terms of the ratio 1 : 3 to make it 3 : 1 ?
(c) Check whether the ratios 18 : 12 and 39 : 26 in proportion or not.
(d) If 6, 11, 126 and x are in proportion, find the value of x.

2. (a) Two number are in the ratio 7:5. When 5 is added to the each number, the ratio
becomes 6:5. Find the numbers

(b) First and third proportions of the terms in continued proportion are 72 and 128. Find
the mean proportional.

3. (a) In a family income from service and rent are in the ratio 2:5. If income from service
is Rs. 24000, find the income from rent.

(b) IC Rs. 56000 can be exchanged with NC Rs. 89600. How much NC can be exchanged
with IC Rs.100 ?

(c) 24 men can accomplish a piece of work in 16 days. How many men will accomplish
the work in 12 days ?

4. Express the following as fractions:

a) 12.5% b) 36% c) 70%

5. Convert the following into percentage.

a) 7 b) 3 c) 0.375
12 8

6. A man spends 60% of his income and saves Rs. 12,500. What is his income?

7. A customer bought a cycle for Rs. 7,500 which 15% discount. What was the price of the
cycle before discount.

8. A table costs Rs. 2500 including 13% VAT. Find the marked price of the table.

Ratio, Proportion and Percentage 207

Prime Mathematics Book - 8

Answers

Exercise 1.1 (Ratios)

1. (a) 2:7 (b) 40:3 (c) 3:16 (d) 1:24 (e) 20:1 (f) 3:32 (g) 7:4 (h) 13:60

2. (a) −3 (b) 7 (c) 18 and 26 (d) 21 and 15 3. (a) 32 (b) 1.6km (c) Rs.4200 (d) Rs.25000

4. (a) Rs.30, Rs.48 (b) Rs.350000 and Rs.250000 (c) 30°, 60°, 90° (d) Rs.160000, Rs.200000, Rs.240000

Exercise 1.2 (Proportion)

1. (a) Yes (b) No (c) No (d) No 2. (a) 21 (b) 32 (c) 231 (d) 72

3. (a) 6 (b) 8 (c) 9 (d) 12 1

4. (a) 9 (b) 14 (c) 96 (d) 8 5. (a) −2 (b) 3 (c) 7

6. (a) Rs.2000 (b) 552km (c) Rs.62500 (d) Rs.160

7. (a) 80km/hr (b) 32 (c) 320 (d) 128 days

8. (a) (i) 6:35 (ii) 6:15:35 (b) (i) 1:3 (ii) 1:2:3 (c) (i) 12:35 (ii) 12:28:35

9 18 Exercise 1.3 (Percentage)
1. (a) 40 (b) 25
2. (a) 0.9 (b) 0.325 (c) 1 1 (d) 1
(c) 1.55 8
(d) 0.4

3. (a) 62.5% (b) 75% (c) 47.5% (d) 205%

4. (a) Rs.4125 (b) 14500 (c) 12.5% (d) 25%

5. (a) 5.42% (b) Rs.51940 (c) 20% (d) Rs.30000

6. (a) 25% (b) food Rs.4800, education Rs.3600, miscellaneous Rs.6000, saving Rs. 9600

(c) biz Rs.24000, service Rs.18000, rent Rs,18000 (d) old 12.5%, adults 45%, children 42.5%

7. (a) son Rs.33600, daughter Rs.25200 (b) 80 (c) 800, 79% (d) 27,300 and 13164

8. (a) Rs.5750 (b) Rs.2214 (c) 20% (d) B

9. (a) Rs.5650 (b) Rs.12000 (c) 15% (d) Rs.930000

208 Answers

At the end of this unit the student will be able to Estimated periods - 8
Objectives:
● find the profit or loss when selling price and cost price are given.
● find the profit and loss in percentage.
● find the selling price or cost price when profit or loss in percentage is given.
● find the discount when discount percent and marked price are given.
● find the VAT when VAT percent is given.
● Solve the problems of profit and loss including discount and VAT percent.

Teaching Materials:

chart of market where the people are buying the things, price tag, chart of formula for profit and loss,
chart of VAT percent etc.

Activities:

It is better to
● demonstrate the activities of buying and selling the things in the classroom to give the concept of cost

price and selling price.
● display the chart of the market where the people are buying and selling the things to give the concept of

the cost price, selling price, profit and loss including percentage, discount and VAT.
● derive the formula to find the profit and loss, selling price and cost price, discount amount and VAT

amount.
● derive the formula to find the profit and loss in percentage.
● say the students to drill the problems by using the formula.

Prime Mathematics Book - 8 Profit and loss Estimated periods: 8

Unit - 1

1.1 Review

We have been discussing about the profit and loss from grade 4. But in this grade,
we are discussing more advance than previous classes about this chapter. In this grade
we discuss about discount and VAT with profit and loss. Generally the words profit and
loss are related with business. Generally a businessman wants to make a profit on his
business. But sometimes he may bear a loss in his business.

The businessman invests the money to purchase goods which is called cost price. It is
denoted by c.p. The money which he gets after selling the goods is called selling price.
It is denoted by s.p.

If the selling price is more than the cost price, there is a profit (gain).

∴ Profit = selling price − cost price = S.P. − C.P.

If the selling price is less than the cost price, there is a loss.

∴ Loss = cost price − selling price = C.P. − S.P.

Profit and loss can also be expressed in percentage.

∴ Profit Percent = Actual Profit (S.P. − C.P.) × 10%
C.P.

∴ Loss Percent = Actual loss (C.P. − S.P.) × 100%
C.P.

Let us discuss the application of above formulae in the following examples.

Example 1: A man bought an ox for Rs.13000 and sold it after some days for Rs.14300.
How much profit or loss percent did he make ?

Solution: Here,

Cost price(C.P.) of an Ox = Rs.13000

Selling price(S.P.) of the Ox = Rs.14300

Since S.P. > C.P. So, he made profit.

Profit = S.P. − C.P. = Rs.14300 − Rs.13000 = Rs.1300

Profit percent = Profit × 100% = Rs.1300 × 100%
C.P. Rs.13000

= 10%

210 8 Profit and Loss

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Example 2: A Shopkeeper purchased 12 crate eggs at the rate of Rs.270 per crate. 15
eggs were broken in transaction. The number of eggs in each crate are 30.
If he sold the remaining eggs at Rs.11 each, find his gain or loss percent.

Solution: Here,
Total number of eggs which he purchased = 30 × 12 = 360
C.P. of 360 eggs = Rs.270 × 12 = Rs. 3240
No. of broken eggs = 15
∴ No. of remaining eggs = 360 − 15 = 345
S.P. of 345 eggs = Rs.11 × 345 = Rs. 3795
Since S.P. > C.P. So, he made profit.

∴ Profit(gain) = S.P. − C.P. = Rs.3795 − Rs.3240 = Rs.555

Profit percent = Profit × 100% = Rs.555 × 100% = 2775 % = 17.13%
C.P. Rs.3240 162

Exercise 1.1

1. Find the profit or loss for each of the following:

Cost price(C.P.) Selling price(S.P.)

(a) Rs.500 Rs.700

(b) Rs.5000 Rs.4500

(c) Rs.3500 Rs.2880

(d) Rs.7000 Rs.7700

Also find profit or loss percent.

2. Ram bought a radio for Rs.1500 and sold it for Rs.1200. What was his profit or loss
percent ? Find it.

3. A farmer bought an ox for Rs.12000 and sold it for Rs.10500 after a few months.
What was his profit or loss percent ? Find it.

4. A stationer bought 5 dozen ball-pens at the rate of Rs.108 per dozen. He found 5 of
them unfit for sale. He sells the remaining at Rs.12 each. Find his gain or loss in percent.

5. Ram bought a second hand cycle for Rs.1150 and spent Rs.200 on its repair. when he
sold it after few days, he got Rs.1250. What was his gain or loss percent ? Find it.

6. A sari purchased for Rs.4300 is sold at a profit of Rs.670. What is its selling price. Find it.
7. If a radio bought for Rs.1925 was sold at a loss of Rs.215, what is its selling price ?

Find it.
8. A man makes a profit of Rs.750 while selling an article for Rs.2175. What was the

cost price of the article ? Find it.
9. Meena purchased 100 eggs for Rs.900. Out of them, 8 were rotten. If she sold the

Profit and Loss 211

Prime Mathematics Book - 8

remaining eggs at Rs.10.50 each, what is the percentage gain or loss ? Find it.
10. A fruit seller buys 50 apples at Rs.12 per piece and sells them at Rs.10.50 per piece.

What is his gain or loss percent ? Find it.
11. 50 litres of petrol was purchased at Rs.116 per litre. Out of them 10 litres are lost

by leakage and the remaining is sold at Rs.114 per litre. Find the gain or by loss
percent.
12. A fruit seller buys 300 kg of mangoes from Janakpur at Rs.80 per kg and pays a
transport charge Rs.3 per kg. If he/she sells them at Rs.85.50 per kg, find his/her
gain or loss percent ?

13. A shopkeeper gains cost price of 2 cups by selling 10 cups. Find his/her gain percent.

1.2 To calculate S.P. when C.P. and gain or loss percent are given:

From the given C.P. and gain or loss percent, we can find the actual gain or loss at first.
Then we use the following formula for the calculation of S.P.

S.P. = C.P. + actual gain or S.P. = C.P. − actual loss

Let us discuss the application of above formula in the following examples.

Example 3: At what price a cycle of cost Rs.15000 is sold to gain 10% ?

Solution: Here, Cost price(C.P.) = Rs.15000

Gain = 10%

Actual gain = 10% of C.P. = 10 × Rs.15000 = Rs.1500
100
Now,

S.P. of the cycle = C.P. + Actual gain = Rs.15000 + Rs.1500 = Rs.16500

Example 4: Mohan bought a watch for Rs.350 and sold it at a loss of 12%. Find the selling
price of the watch.

Solution: Here, Cost price(C.P.) = Rs.350 Loss = 12% Selling price(S.P.) = ?

Actual loss = 12% of C.P.

= 12 × Rs.350 = Rs.42
Now, 100

S.P. of the watch = C.P. − Actual loss = Rs.350−Rs.42 = Rs.308
Example 5: Ashim Miya bought 120 electric bulbs at the rate of Rs.240 per dozen. 15

bulbs were damaged in transit and he sold the remaining bulbs so that his
profit was 12% on his outlay. Find the selling price of each bulb.
Solution: Here, No. of bulbs which Ashim Miya bought = 120

C.P. of 1 dozen bulbs = Rs.240

C.P. of 120 bulbs = Rs. 240 × 120 [1dozen = 12 piece]
12

= Rs.2400

212 8 Profit and Loss

Prime Mathematics Book - 8

No. of damaged bulbs = 15

∴ No. of good (remaining) bulbs = 120−15 = 105

Profit percent = 12%

Actual profit = 12% of C.P.

= 12 × Rs.2400 = Rs.288
Now, 100

S.P. of the 105 bulbs = C.P. + Actual profit = Rs.2400 + Rs.288 = Rs.2688

∴ S.P. of 1 bulb = Rs. 2688 = Rs.25.60
105

Exercise 1.2

1. Find the selling price(S.P.) if,

(a) C.P. = Rs.480 and profit percent = 20%.

(b) C.P. = Rs.320 and loss percent = 5%.

(c) C.P. = Rs.2500 and profit percent = 18%.

2. Dolma Sherpa bought an electric fan for Rs.1250 and sold it at a profit of 15%. At what

price did she sell it ?

3. At What price an umbrella should be sold at a loss of 20% when it was bought for

Rs.275 ? 2

4. At what price saeccaolncudlahtaonrdshmouolbdilbeefsoorldRsa.t9a5l0o0ssaonfd6sp3e%ndwshResn.1it5w00astoboruegphatirfoirt.RHs.e75s0el?ls
5. A boy buys a

it after a few days and makes a profit of 15%. At what price does he sell it ?

6. Anup bought 1500 oranges at the rate of Rs.7 each. He found 60 of them rotten and sold

the good oranges to make the gain of 16%. At what rate did he sell the good oranges ?

7. Rupadevi bought 20 pieces saries at the rate of Rs.700 each. She sold 12 pieces series at

the rate of Rs.850. At what rate should she sell the rest as to gain 20% on her outlay ?

1.3 To find C.P. when S.P. and gain or loss percent are given:

From the given S.P. and gain or loss percent, we can find C.P. But we know

that the gain or loss percent is always calculated on the cost price(C.P.). So, we

suppose the cost price by using any variable (letter) and then we calculate actual gain
or loss from the given gain or loss percent in terms of the variable. After that we use
the formula to calculate C.P.

CP = SP x 100 Or CP = SP x 100
100 + P% 100 - L%

So, actual gain or loss = gain or loss percentage of C.P.
Then C.P. = S.P. − gain or S.P. + loss
The above situation is illustrated in the following examples

Profit and Loss 213

Prime Mathematics Book - 8

Example 6: A shopkeeper gained 20% by selling a radio for Rs.1200. At what price did he
purchase it ? Also find the amount of gain.

Solution: Here,
S.P. of the radio = Rs.1200, gain percent = 20%
C.P. of the radio = ?, gain amount = ?
Let C.P. be Rs.X
Then,

actual gain = 20% of x = 20 ×x = x
100 5

Now,

C.P. = S.P. − gain or, x = 1200− x or, x+ x = 1200 or, 6x = 1200
5 5 5

or, x = 1200 × 5
6

∴ x = 1000

Therefore the cost price of the radio is Rs.1000

The amount of gain = Rs.1000 = Rs.200
5

Example 7: Mohan bought 500 chicken and 75 of them died due to cold. He sold the

remaining chicken at Rs.120 each, so that he made a profit of 2.5%. At what

price did he purchase each ? Find it.

Solution: Here, No. of chicken he purchased = 500
No. of chicken which died = 75
∴ The remaining chicken = 500−75 = 425
S.P. of 1 chicken = Rs.120
∴ S.P. of 425 chicken = Rs.120 × 425 = Rs.51000
profit percent = 2%

Let total C.P. be Rs.x

Then, 2 x
100 50
Actual gain = 2% of x = × x =

By using formula,

C.P. = S.P. − Profit
oorr,, xx =+ 55x1000=05−1050x00

or, 51x = 51000
50

or, x = 51000 × 50
51

214 8 Profit and Loss

Prime Mathematics Book - 8

∴ x = 50000

Now,

C.P. of 500 chicken = Rs.50000

∴ C.P. of 1 chicken = Rs. 50000 = Rs.100
500

Hence, he purchased 500 chicken at Rs.100 each.

Example 8: A man sold an electric Iron for Rs.4000 losing there by 20%. At what price did

he purchase it ?

Solution: Here,

S.P. of Iron = Rs.4000, Loss percent = 20%

Let C.P. of the Iron be Rs.x

Then, actual loss = 20% of x = 20 × x = x
100 5

Now,

C.P. = S.P. + Actual loss

or, x = 4000 + x
5

or, x − x = 4000
5

or, 4x = 4000
5

or, x = 4000 × 5
4
∴ x = 5000

Hence, he purchased the electric iron for Rs.5000

Exercise 1.3

1. Find the cost price(C.P.) if:

(a) S.P. = Rs.1350 and profit = 12 1 % (b) S.P. = Rs.405 and loss = 15%
2
(c) S.P. = Rs.168 and loss = 20%

2. By selling an article for Rs.60, a man makes a profit of 12%. Find the cost price of

the article.

3. A man sold a black & white TV for Rs.6000 losing there by 20%. At what price did he

purchase it ? Find it.

4. By selling a computer set for Rs.40000, a shopkeeper makes a profit of 25%. At what

price did the shopkeeper purchase it ? Find it.

5. A man sold an article for Rs.560 losing there by 20%. Find the cost price of the

article and the amount of loss.

Profit and Loss 215

Prime Mathematics Book - 8

6. Mohan gained 20% by selling a radio for Rs.1260. At what price did he
purchase it ? If he had to gain 30%, at what price should have he sold it ? Find it.

7. Kushal sold two bicycles for Rs.3500 each gaining 25% on one and losing 20% on the
other. Find his gain or loss percent on the whole.

8. A shopkeeper bought 200 apples. But 40 of them were rotten and he sold the rest
at Rs.9 each making a loss of 4%. What was the cost price of each apples ? Find it.

1.4 Discount Date: 2071/10/25

Study the given Bill and try to find the Shyam Electronic Shop
answer of the following questions.
Mahendra chowk, Biratnagar
(a) What is the price of the phone set
before discount? Name: Anima Mathema

(b) What is the price of the phone set S.No. Name of Items Rate(in Rs.) Quantity Cost(in Rs.)
before discount?
1 Phone Set 850 1 850
(c) How much discount amount does
Anima get? Total 850

(d) How much did Anima pay for the Discount amount with 14% 119
phone set?
Total amount after discount 731

In words: Seven hundred thirty one only.

Buyer Seller

Find the answer of the above questions by discussing in your group and write your conclusion.

In order to sell the goods, the shopkeeper fixed the price of the goods. The price
marked (fixed) on the goods is called the marked price or labelled price. The shopkeeper
sold the goods deducting certain amount in the marked price of the goods. That deducted
amount is called the discount. Discount is always given in percentage. It is always in the
marked price of the goods.

The price at which the goods are sold after allowing the discount is called the net
(actual) selling price.

So, Discount amount = Discount percent of marked price (M.P.) = Discount percent x M.P.

Actual selling price = Marked price - discount amount.

Example 9: The marked price of an article is Rs.2500. If it is sold at a discount of 12%,

what is the selling price of the article?

Solution: Here, marked price (M.P.) = Rs.2500 Discount= 12%

So, discount amount =12% of M.P. = 12 × Rs.2500 = Rs.300
100
Now, Selling price = M.P. - discount amount = Rs. 2500 - Rs.300 = Rs.2200

Example 10: After allowing a discount of 8%, a book was sold for Rs.184. What was its

marked price?

Solution: let the marked price (M.P.) of the book be Rs. x.

Then, discount amount = 8% of x = 8 × x = 2x
100 25

216 8 Profit and Loss

Prime Mathematics Book - 8

Now, selling price = M.P. – discount

or, 184 = x – 2x or, 184 = 23x or, x = 184×25 x = Rs.200
25 25 23

Therefore, the marked price of the book is Rs. 200.

Example 11: A shopkeeper marked the price of an article at Rs. 1200. He allowed a

discount of 12% and still earned a profit of 10% . How much did the seller

pay for it?

Solution: Here, marked price (M.P.) = Rs.1200
12
Discount amount = 12% of M.P. = 100 × 1200 = Rs.144

Selling price = M.P. - discount = Rs.1200 - Rs.144 = Rs.1056

Profit = 10%

Now, cost price = 100 S.P × 100 = Rs.1056 × 100 = Rs.105600
+ profit percent 100 + 10 110
= Rs.960
Therefore, the cost price of the shopkeeper is Rs. 960.

Tax:

A country has to spend money for military for security of country, police force for
social harmony, education, health, transportation and other welfare services for the
benefit of the country. To meet this expenses, government levies money (revenue) from
the people under different subjects which is called Tax.

Value Added Tax (VAT):
Value Added Tax (VAT) is a modified and scientific form of sales tax. This tax is levied

on import and sales of all the goods (except those exempted by the law), which passes
through many hands.

To understand the concept of VAT, observe the following example:

Suppose a manufacturer sells a mobile set for Rs.10,000 to a wholesaler. The
wholesaler sells it to a dealer for Rs.12,000. The dealer sells it to a retailer for Rs.
15,000 and the retailer in turn sells it the customer for Rs.16,000. If the VAT rate is 10%,
amount of VAT to be paid in different stages are shown as:

Supply Selling Value Tax on Tax on tax to be paid

Stages Price Added S.P. C.P. to government

Manufacturer Rs.10,000 - Rs.1000 - Rs.1000

Wholesaler Rs.12,000 Rs.2000 Rs.1200 Rs.1000 Rs.1200-Rs.1000 = Rs.200

Dealer Rs.15,000 Rs.3000 Rs.1500 Rs.1200 Rs.1500-Rs.1200 = Rs.300

Retailer Rs.16,000 Rs.1000 Rs.1600 Rs.1500 Rs.1600-Rs.1500 = Rs.100

Profit and Loss 217

Prime Mathematics Book - 8

VAT (Value Added Tax) Date: 2071/09/15

Study the given bill and discuss with your Raju Fashion Shop
friends. Here, the marked price of the Jeans Khichapokhari, Kathmandu
pant is Rs.2500. Discount percent is 8%, So
Name: Mohan Yadav

S.No. Name of Items Rate(in Rs.) Quantity Cost(in Rs.)

1. Jeans Pant 2500 1 2500

the discount amount is Rs.8% of Rs.2500 =

Rs.200. So, the selling price of the paints after Total 2500
discount is Rs.2300 VAT Percent is 13% which is
fixed by the government. Discount amount with 8% 200

So, the VAT amount is 13% of Rs.2300 = Selling Price 2300
Rs.299.
VAT amount with 13% 299
Now, the final selling price (cost price
Total amount after discount 2599

In words: Two Thousand five hundred ninety nine only.

Buyer Seller

by the customer) of that pant is Rs. 2599 with VAT amount. Value Added Tax (VAT) is

modified form of sales tax. This tax is levied on import and sales of all the goods, which

passes through many hands. VAT is fixed by the government in percent and it goes to the

government revenue. It is always added to the actual selling price. The customer always

pays the money for the goods after adding VAT.

VAT amount = VAT percent of S.P., VAT amount = Final S.P. – actual S.P.

VAT percent = VAT amount x 100%
S.p.

Example 12: The cost of a radio is Rs.3000. A shopkeeper allows a discount of 10% for

selling it and adds 13% VAT on it. Find the price of the radio to be paid by

the customer.

Solution: Here, marked price = Rs.300, discount=10%

Discount amount= 10% of Rs.3000 = 10 × Rs.3000 = Rs.300
100

S.P. = M.p. – discount = Rs.3000 - Rs.300 = Rs.2700

VAT amount = 13% of S.P. = 13 × Rs.2700 = Rs.351
100

S.P. with VAT = Rs. 2700 + Rs.351 = RS.3051

Hence, the customer pays Rs. 3051 for the radio.

Exercise 18.4

1. The marked price of a good is Rs. 1500. If a discount of 12% is allowed, what is the
selling price of it?

2. The marked price of a book is Rs.400. If a shopkeeper sold it by giving 14% discount,
how much did the customer pay for it?

218 8 Profit and Loss

Prime Mathematics Book - 8

3. Find the price to be paid after discount: Item Marked price Discount
a) Cycle Rs. 7000 10%
b) Watch Rs. 3500 12%
c) Calculator Rs. 1500 7%

4. The price of a good after allowing the discount of 8 % is Rs. 1840. What is marked
price of the good?

5. The marked price of a radio is Rs. 1500 and it is sold for Rs. 1200 after allowing the
certain discount. Final the discount percent.

Item Discount Price after
discount
6. Find th e marked price of the a) Electric fan 8% Rs. 2760
goods which are given in the adjoining table: b) Mobile 12% Rs. 5280
c) Heater 6% Rs. 1128

7. Find the discount percent on the items Items Marked Price Price after discount
which are given in the adjoining table a) Book Rs. 1200 Rs. 1104
b) Pen Rs. 120 Rs. 110
c) Geometry box Rs. 400 Rs. 380

8. A shopkeeper marked the price of a good at Rs. 850. He allowed a discount of 12%
and still earned a profit of 12 %. Find the cost price of the good.

9. A dealer allows his customer a discount of 20% and still he gains 25%. Find the
marked price of an article which costs for the dealer Rs.820.

10. A shopkeeper leavied 13% VAT after allowing 3% discount on an article of marked
price Rs.1600. How much did the customer pay for it?

11. Find the final selling price of the items which Items Marked Price Discount VAT
13%
are given in the adjoing table. a) Calculator Rs. 1600 4%

b) Watch Rs. 2400 12% 13%

c) Radio Rs. 5000 15% 13%

12. A cycle marked as Rs.15000 is sold for Rs. 14916. If the customer paid 13% as VAT,
what was the rate of discount?

13. The marked price of an article at the dealer is Rs.4000. The dealer allows a discount
of 20% to the retailer. At what price should the retailer purchases the article? What
would be the final price if 13% VAT is imposed?

14. The marked price of a good is Rs.250. If a shopkeeper sold it for Rs.192 allowing the
discount of 36% and leavied the certain rate of VAT, find the rate of VAT.

Profit and Loss 219

Prime Mathematics Book - 8

Area Revision Test (Profit and Loss)

1. (a) Mohan bought a radio for Rs. 3000 and sold it for Rs. 3500. What was his profit or
loss percent? Find it.

(b) An article purchases for Rs. 4300 is sold at the profit Rs. 670. What was its selling
price find it.

(c) A shopkeeper purchased 360 tea cups at the rate of Rs. 25 per each. 15 cups were
broken in transaction. If he sold the remaining cups at Rs. 35 per each, find his
gain or loss percent.

2. Hari bought a cycle for Rs. 5000 and sold it at a loss of 15%. Find the selling price of
the cycle.

3. Chandra Miya bought 120 electric bulbs at the rate of Rs.240 per dozen. 15 bulbs were
damaged in transit and he sold the remaining bulbs so that his profit was 12% on his
outlay. Find the selling price of each bulb.

4. Sohan bought 600 chicken and 35 of them died due to cold. He sold the remaining
chicken at Rs.130 each, so that he made a profit of 3%. At what price did he purchase
each ? Find it.

5. A man sold an article for Rs. 700 losing there by 20%. Find the cost price of the article
and amount of loss.

6. The marked price of an article is Rs.3000. If it is sold at a discount of 12%, what is the
selling price of the article?

7. A shopkeeper marked the price of an article at Rs. 1500. He allowed a discount of 12%
and still earned a profit of 10% . How much did the seller pay for it?

8. A shopkeeper charged 13% VAT after allowing 5% discount on an article of marked price
Rs. 2000. How much did the customer pay for it.

9. The marked price of an article at the dealer is Rs.5000. The dealer allows a discount
of 18% to the retailer. At what price should the retailer purchases the article? What
would be the final price if 13% VAT is imposed?

Answers

Exercise 1.1 (Profit and Loss)
5
1. a) Rs. 200, 40% b) Rs. 500, 10% c) Rs. 620, 17 75 % d) Rs. 700, 10%

2. 20% 3. Loss = 12.5% 2 711% 7. Rs. 1710
4. Gain = 22 9 % 5. 5 6. Rs. 4970
1 1
8. Rs. 1425 9. 7 3 % 10. 12.5% 11. 21 29 % 12. 3 83 %, 13. 20%

b) Rs. 304 Exercise 1.2
1. a) Rs. 576 c) Rs. 2950
2. Rs. 1437.5 3. Rs. 220 4. Rs. 700 5. Rs. 12650 6. Rs. 8.45
7. Rs. 825
Exercise 1.3
1. a) Rs. 1200 b) Rs. 476.47 c) Rs. 210
2. Rs. 53.57 3. Rs. 7500 4. Rs. 32000 5. Rs. 700, Rs. 140
6. Rs. 1050, Rs. 1365 7. Loss = 2.5% 8. Rs. 7.5

1. Rs. 1320 Exercise 1.4 (Discount)
2. Rs. 344 3. a) Rs. 6300 b) Rs. 3080 c) Rs. 1395
4. Rs. 2000 5. 20% 6. a) Rs. 3000 b) Rs. 6000 c) Rs. 1200
1
7. a) 8% 17b3)58.683 % c) 5% 8. Rs. 667.85 9. Rs. 1281.25 10. Rs. 1753.76
11. a) Rs. b) Rs. 2386.56 c) Rs. 4802.5 12. 12% 13. Rs. 3200, Rs. 3616
14. 20%

220 8 Profit and Loss

UNITARY METHOD

At the end of this unit the student will be able to Estimated periods - 6
Objectives:

● find the value of certain number of an object when the value or cost of one object is known.
● find the value of cost of one object when the value or cost of certain number of objects is known.

Rs. 10/- Rs. ?

? hours 9 hours

Teaching Materials:

menu, price list etc.

Activities:

It is better to
● discuss about use of direct proportion and indirect proportion for solving unitary methods problems.
● Let the students solve the related problems by group discussion.
● say the students to drill the problems related to unitary method.

Prime Mathematics Book - 8 Unitary Method Estimated periods : 6

Unit - 1

1.1 Review:

In the pervious class (Class - 7) we already discussed about the direct variation
and indirect variation. The process in which we first find the value of unit quantity by
using division or multiplication in the case of direct variation and indirect variation
respectively is called the unitary method. Then, we find the value of the given required
quantity with the help of the value of unit quantity. Let us study the following examples.

Example 1: The cost of 9 chairs is Rs. 2700. What will be the cost of 5 chairs ?

Solution: Here, the cost of 9 chairs is Rs. 2700

2700 300 = Rs. 300
∴ The cost of 1 chairs is Rs. 9


∴ The cost of 5 chairs is Rs. 300 x 5 = Rs. 1500

By using the variation method:

Since the number of quantity and its cost are directly proportional(as the number
increases, the cost also increases),we must have ratio of numbers = ratio of corresponding

costs.

So, 9 = 27x00 , where x is the cost of 5 chairs.
5
3
or, x = 2700 x 5 = Rs.300 x 5 = Rs.1500
or, 9x = 2700 x 5 9

Hence, the cost of 5 chairs will be Rs.1500.

Example 2: 12 men can do a piece of work in 25 days. How many men can do it in 20 days ?
Solution: Here,
In 25 days, a piece of work can be done by 12 men
∴ In 1 day, the work can be done by 12 x 25 men

63 5
∴ In 20 days, the work can be done by 12 x 25
20 10 5 men = 15 men

222 9 Unitary Method

Prime Mathematics Book - 8

By the variation method:
Since the number of men and the number of days are inversely (indirect) proportion(as

the number of men decreases the number of days increases), we must have ratio of the
number of men = inverse of the ratio of the number of days.

So, 12 = 20 , where x is the number of men.
x 25

635
12 x 25
or, 20x = 12 x 25 or, x = 20 10 = 3 x 5 = 15 men

Hence, 15 men will do the work in 20 days. 5

Example 3: A garrison of 400 men have food enough to last for 50 days. How many men

should be added so that the food may last only for 40 days ? Find it.

Solution: Here,

The food is enough for 50 days in garrison of 400 men

∴ The given food will last 50 days for 400 men

∴ The given food will last 1 day for 400 x 50 men

10
400 x 50
∴ The given food will last 40 days for 40 = 10 x 50 men = 500 men

Hence, the number of men added = 500 - 400 = 100 men.

Example 4: The total cost of 5 pens and 7 geometry boxes is Rs. 855. If the cost of a pen
is Rs. 45, what will be the cost of 3 geometry boxes ? Find it.

Solution: Here,

The cost of 5 pens and 7 geometry boxes = Rs. 855

The cost of 1 pen = Rs. 45

∴ The cost of 5 pens = Rs. 45 x 5 = Rs. 225

Now,

The cost of 7 geometry boxes = Rs. 855 - cost of 5 pens

= Rs. 855 - Rs. 225 = Rs. 630

630 90 = Rs.90
∴ Tthe cost of 1 geometry box = Rs. 7


∴ The cost of 3 geometry boxes = Rs. 90 x 3 = Rs. 270

Unitary Method 223

Prime Mathematics Book - 8

Example 5: 15 men reap 20 hectares of the field in 6 days. How long will it take for 12

men to reap 24 hectares ?

Solution: Here,

∴ 15 men reap 20 hectares of the field in 6 days.

∴ 1 man reaps 20 hectares of the field in 6 x 15 days.

90
12
∴ 12 men reap 20 hectares of the field in days.

90
12 x 20
∴ 12 men reap 1 hectares of the field in days.

12 days.
∴ 12 men reap 24 hectares of the field in 90 x 24
12 x 20

= 9 days

By the variation method:

Let 12 men reap 24 hectares of the field in x days. Then, the statement may be written
as follows:

The variation between men and days is Men Area in hectares Days
indirect variation. The variation between 15 20 6

area and days is direct variation. 12 24 x

6 12 20 33 21
So, x = 15 x 24 or, x = 6 x 15 x 24
∴ The required days = 9 =3x3=9
1 12 x 20 10
21

Exercise 1.1

1. The cost of 6 kg. of rice is Rs.480 . What will be the cost of 11 kg. of rice ?
2. If the cost of 7 pens is Rs.490, find the cost of 12 pens.
3. 5 boxes of Muna Tea is Rs.510. What will be cost of such 7 boxes of Muna Tea ?
4. 8 workers can complete a piece of work in 21 days. How many workers can do it in

12 days ?

224 9 Unitary Method

Prime Mathematics Book - 8

5. The rate of income tax is 5 paisa per rupee and the taxable income of a service
holder is Rs. 50000. What is the amount of his income tax ?

6. A garrison of 300 men have food enough to last for 60 days. How many men should
be added so that the food may last for 45 days ?

7. A group of 800 people had provisions for 40 days. Some of them left the group after
10 days and the remaining food lasted for 48 days. How many people had left the
group ?

8. A hostel of 1500 students has food enough for 48 days. How long will the remaining
food last if 600 students join them after 13 days ?

9. 6 tables or 24 chairs cost Rs.7200. What will 4 tables and 11 chairs cost ?
10. 7 pens and 3 ink-pots cost Rs.270.60. If the cost of an ink-pots be Rs.20.20, find the

cost of 3 pens.
11. 6 goats or 4 cows can eat the grass of a field in 17 days. In how many days will 8 goats

and 6 cows eat it ?
12. The cost of 12 dot pens and 8 copies is Rs.264. If the cost of a dot pen is Rs.12, find

the cost of 9 dot pens and 13 copies.
13. If 5 men earn Rs.6500 in 6 days, how much will 4 men earn in 9 days ?
14. 21 men can do a piece of work in 24 days working 8 hours a day. In how many days

24 men do the same work working 6 hours a day ?
15. How much rice will be required by 35 men for 30 days if 45 men require 300 kg of

rice for 28 days ?
16. If 280 kg of potatoes can be carried 120 km for Rs.75, What weight can be carried

140 km for Rs.45 ?
17. If 20 pumps can raise 125000 litres of water in 5 hours, in how many hours can raise

75000 litres of water by 30 pumps ?

Unitary Method 225

Prime Mathematics Book - 8

Area Revision Test (Unitary Method)

1. The cost of five pens is Rs. 200. What will be the cost of 17 pens?
2. 9 workers can complete a piece of work in 15 days. How many workers will do

it in 6 day?
3. After paying an income tax of 4 paisa in a rupee, a man has left 1600 left. What

was his income? find it.
4. A group of 1200 people had provisions for 50days. Some of them left the group

after 10 days and the remaining food lasted for 60days. How many people had
left the group?
5. The cost of 3 chairs and 4 tables is Rs. 7540. If the cost of 1chair is 22, find the
cost 1 table.
6. 12 workers can do a piece of work in 14 days. If how many days 21 workers can
complete the same work?
7. 20 men reap 30 hectares of the field in 6 days. How long will it take for 12 men
to reap 24 hectares?

Answers

Exercise 19.1
1. Rs. 880 2. Rs. 840 3. Rs. 714 4. 14 5. Rs. 2500 6. 100
7. 300 8. 25 9. Rs. 8100 10. Rs. 90 11. 6 days 12. Rs. 303
13. Rs. 7800 14. 28 days 15. 250kg. 16. 144kg. 17. 2hrs.

226 9 Unitary Method / Answers

At the end of this unit the student will be able to Estimated periods -7
Objectives:

● understand what is interest, principal, amount and rate of interest.
● calculate simple interest, amount, time , rate of interest and principal by using formula.

Teaching Materials:

Chart of current interest rates of different banks and finance companies.
Sample of loan agreements.
Chart of formula to calculate, interest, principal, amount, time and rate of interest.

Activities:

It is better to

● discuss about interest and process of providing loan.

● PxRxT is obtained. ( (A x 100
show how the formula I = 100
and P = 100 +TR is obtained.
● show how the formula T = I x 100 , R = I x 100 , P = I x 100
PxR PxT TxR

● say the students to calculate interest, time, rate of interest and principal by using the formula.

Prime Mathematics Book - 8

Unit - 1 Simple Interest Estimated periods : 7

1.1 Review

In the previous classes we have already discussed about the simple interest. Let
us revise it in brief. When a person borrows or deposits some of money from a money
lender or a bank for a certain period of time, then the takers have to return the money
at the end of the time with an additional money. That additional money is called simple
interest. For example, A man borrowed a sum of Rs. 3000 from a bank and returned Rs.
1
3405 to the bank after 1 2 years. Here, he returned Rs. 405 extra money. So, Rs. 405 is
called simple interest.

The sum of money which is borrowed is called the principal. It is denoted by P.

So, in the above example, Principal (P) = Rs. 3000

The total money which is returned at the end of time is called the Amount.

It is denoted by A. So, Amount(A) = Rs. 3405

The additional money to be paid with the borrowed money is called Simple Interest.

It is denoted by I. So, Interest (I) = Rs. 3405 - Rs.3000 = Rs. 405.

The specified time period for the loan is called time. It is denoted by T.

So, Time (T) = 1 1 years. Time is always in year.
2
The extra money that is paid for every Rs. 100 for a year is called the rate of
interest. It is denoted by R. It is always calculated in percentage. The rate of interest in
the above example is :

R= 405 × 100% = 405 x 2 × 100% = 9%
1 3 x 3000
1 2 x 3000

Formula for the simple Interest:

Calculation of the simple interest on the sum of money Rs. P for T years at the rate
of R% per annum.

We known that, the rate R % per annum means.

The interest on Rs. 100 for 1 year = Rs. R

The interest on Re. 1 for 1 year = Rs.1R00

R
The interest on Rs.P for 1 year = Rs. 100 × P

The interest on Rs.P for T years = Rs.R10x0P × T ∴ Interest (I) = PxRxT
We know that, 100

Amount(A) = Principal + Interest ∴ A = P + I

228 10 Simple Interest

Prime Mathematics Book - 8

Example 1: Find the interest on Rs.2500 at the rate of 8% per annum for 2 years. Also

calculate the amount.

Solution: Here, Rate (R) = 8% Time(T) = 2 years
Principal (P) = Rs.2500 Amount(A) = ?
Interest( I) = ? and

Interest (I) = P xRx T = Rs. 2500 x 8 x 2 = Rs.400
100 100

Amount(A) = P + I = Rs.2500 + Rs.400 = Rs.2900

Example 2: A money lender lends Rs.6000 to the shopkeeper at the rate of 3 paisa per
rupee per 2 months. How much should the shopkeeper pay to clear the debt

at the end of 2 years 4 months ?

Solution: Here,
( (Principal (P) = Rs. 6000
Time(T) = 2 years 4 months = 2 years + 4 years = 2 + 1 years = 7 years
12 3 3

Rate(R) = 3 paisa per rupee per 2 months

= 3 x 100 x 12 paisa per Rs.100 per year
2
= Rs.18 per Rs.100 per year = 18%

Amount(A) = ?

Interest(I) = PxRxT = Rs.6000 x 7 x 18 = Rs.2520
100 100 x 3

Amount(A) = P + I = Rs.6000 + Rs.2520 = Rs.8520

Hence, he should pay Rs. 8520 to clear the debt.

Exercise 1.1

1. Find the simple interest on

(a) Rs. 450 for 2 1 years at the rate of 9% per annum.
2
(b) Rs. 1500 for 9 months at the rate of 12% per annum.

(c) Rs. 2500 for 3 years 6 months at the rate of 6% per annum.

2. Find the simple interest and amount on:

(a) Rs. 350 for 2 years at 4% per annum.
a13t%8p%eprearnannunmu.m.
(b) Rs. 868.50 for 16 months at 3
(c) Rs. 1800 for 5 years 4 months

Simple Interest 229

Prime Mathematics Book - 8

3. If Rs.3000 is lent for 6 years at the rate of 10% per annum, What will be the interest
and amount ?

4. Mohan deposited Rs. 25000 in Himalayan Bank which gives 12 1 % per annum
interest. What amount will he get after 4 years 6 months ? 2

5. A money lender lends Rs. 7000 to a man at the rate of 2 paisa per rupee per month.
How much should the man pay to clear the debt at the end of 2 years 6 months ?

6. Bir Bahadur borrowed Rs.8000 from a bank at 9% per annum for 2 years. On the
same day, he lent this money to the shopkeeper at 12% per annum for 2 years. How
muck did Bir Bahadur earn in this transaction ?

7. RamSharan deposited Rs.14000 in a bank at 8% simple interest. He withdrew Rs.8000
after 6 months and the remaining money after another 6 months. How much interest
did he get in all ?

1.2 Finding the time(T) and the Rate of Interest(R):

The formula for the calculation of simple interest is I = PxRxT
100

P × T × R = I × 100 T= I x 100 and R = I x 100
PxR PxT

By using the above formulae, to calculate the time and the rate of interest.
Let us discuss about that in the following examples.

Example 3: In what time will interest on Rs.900 at the rate of 8% per annum be Rs.216 ?

Solution: Here,

Principal(P) = Rs. 900, Rate(R) = 8%, Interest(I) = Rs. 216, Time(T) = ?

24 3
Rs.216 x 100
We know that, Time(T) = I x 100 = Rs.900 x 8 = 3 years
PxR

Example 4: In what time will the amount on Rs.2400 at the rate of 6% per annum be Rs.3000 ?

Solution: Here,

Principal(P) = Rs.2400, Rate(R) = 6%, Amount(A) = Rs.3000, Time(T) = ?

Interest(I) = A - P = Rs.3000 - Rs.2400 = Rs.600

Now, I x 100 = Rs.600 x 100 = 100 25 6 1 years = 4 years 2 months.
Time(T) = PxR Rs.2400 x 6 24 6= 6

230 10 Simple Interest

Prime Mathematics Book - 8

Example 5: In what time will a sum of money treble itself at the rate of 10% per annum ?

Solution: Let the sum of money be Rs.x. Then,

Principal(P) = Rs.x, Amount(A) = Rs.3x Interest(I) = A - P = 3x - x = 2x

Rate(R) = 10% 2
2x × 100
Now, Time (T) = I x 100 = x × 10 = 20 years.
PxR

Example 6: At what rate percent per annum will the amount on Rs.650 for 2 years be Rs.754 ?

Solution: Here, Principal (P) = Rs. 650 Amount (A) = Rs. 754

∴ Interest(I) = A - P = Rs. 754 - Rs. 650 = Rs. 104

Time(T) = 2 years, Rate(R) = ?

Now, Rate(R) = I x 100 = Rs.104 × 100 = 8%
PxT Rs.650 × 2

Example 7: At what rate per annum will the interest be 1 of the principal in 5 years?
4

Solution: Let the principal be Rs. x.

Then, the interest (I) = 1 of the principal = 1 ×x= x .
4 4 4

Time (T)= 5 years, Rate (R) = ?

Now, Rate(R) = I x 100 = x × 100 = 5%
PxT 4 × x ×5

Exercise 1.2

1. Find the time in which the interest on (b) Rs. 650 at 6% per annum is Rs. 195
(a) Rs. 2000 at 8% per annum is Rs. 480
(c) Rs. 2500 at 24% per annum is Rs. 1500

2. In how many years will Rs. 4000 yield an interest of Rs. 1500 at 8% per annum ?

3. In what time will the amount on Rs. 6250 be Rs. 7500 at 10% per annum ?

4. In what time will the sum of money double itself as the rate of 10% per annum ?

5. In what time will the interest on a sum of money at 4% rate of interest be 1 of the sum ?
5

6. Find the rate of interest per annum if:

(a) the interest on Rs. 440 for 2 years is Rs. 44.

(b) the interest on Rs. 1250 for 3 years is Rs. 225.

(c) the amount on Rs. 2400 for 2 years 6 months is Rs. 3120.

Simple Interest 231

Prime Mathematics Book - 8

7. At what rate percent per annum will a sum of money double itself in 8 years ?
1
8. At what rate percent per annum will the interest be 5 of the principal in 5 years ?
9. 1
A sum of Rs. 5000 was lent at simple interest. At the end of 2 2 years the total

amount received was Rs. 5812.50. Find the rate of interest.

10. The interest on a sum of money is 0.5 times of the money itself in 4 years. What is
the rate of interest per annum ?

1.3 Finding the principal (P):

The formula for the calculation of simple interest is;

I= PxRxT or, P × T × R = I × 100 or, P = I x 100
100 TxR
( (
We know that, A = P + I or, A = P + PRT or, A = P 1 + TR
100 100



( ( ( ( ( ( or, A = P
100 +TR or, P = A x 100 A x 100
100 100 +TR ∴ P = 100 +TR

Example 8: What sum should Anima deposit in a bank that will yield an interest of Rs. 90
in 2 years at the rate of 6% per annum ?

Solution: Here,
Principal(P) = ? Interest(I) = Rs. 90, Time(T) = 2 years, Rate(R) = 6%

Now, Principal(P) = I x 100 = Rs. 90 × 100 = Rs. 750
TxR 2×6

Hence, She deposited Rs. 750 at the bank.

Example 9: Find the principal which will amount to Rs.1488 in 3 years at 8% per annum.

Solution: Here,

Principal(P) = ? Amount(A) = Rs. 1488, Time(T) = 3 years, Rate(R) = 8%
( (
By the formula, Principal(P) = A x 100 = Rs.1488 × 100 = Rs. 148800 =
100 +TR 100 + 3 × 8 124
Rs.1200

Example 10: A certain sum amounts to Rs. 3600 in 2 years 6 months and Rs. 4080 in 4
years 6 months. Find the sum and the rate of interest per annum.

Solution: Let the sum of money and rate of interest be Rs. x and y % respectively.

In the first case, Amount(A) = Rs. 3600 6
12
Time(T) = 2 years 6 months = 2 years + years

232 10 Simple Interest

Prime Mathematics Book - 8

( ( = 2 1 years = 5 years,
+2 2

A x 100 or, x = 3600 × 100 ................(i)

( ( Principal(P) = 100 + TR 100 + 5 ×y
2

In the second case,



( (
Amount(A) = Rs. 4080 6 1 9
12 4 +2 2
Time(T) = 4 years 6 months = 4 years + years = years = years

( ( Principal(P) =
A x 100 or, x = 4080 × 100 ..........(ii)
100 +TR 100 + 9 ×y
2

From equn (i) and equn (ii)

3600 × 100 = 4080 × 100
100 + 5 × y
100 + 9 ×y
2 2

( ( ( ( or, 3600
9y = 4080 5y
100 + 2 100 + 2

or, 360000 +16200y = 408000 + 10200y

or, 16200y - 10200y = 408000 - 360000

or, 6000y = 48000

∴y=8

Substituting the value of y in equn(i), we get

x = 3600 × 100 3000 = 3000
= 360000
100 + 52× 8
120

Hence, the sum = Rs. 3000 and Rate = 8%

Exercise 1.3

1. Find the principal which earns an interest of
(a) Rs. 1250 in 2 years at 10% per annum.
(b) Rs. 650 in 3 years 4 months at 8% per annum.
(c) Rs. 450 in 3 years at 5% per annum.
2. Bijaya deposits a sum of money in Prabhu Bank at 6.5% per annum. After 5 years she

receives Rs.15900 in total. How much does she deposit in the bank ? Find it.

Simple Interest 233

Prime Mathematics Book - 8

3. Sonam Sherpa lent some money at 12% per annum. If he received Rs. 708 after 1
years 6 months, What sum did he lend ? Find it.

4. Find the principal which will amount to Rs. 2000 in 6 years 8 months at 5% per annum.
5. Mina Kumari borrowed sum of money from a bank at 9% simple interest for a year.

After 4 years she had to refund Rs. 10880, what was her loan ?
6. A certain sum amounts to Rs. 4720 in 3 years and to Rs. 5200 in 5 years. Find the

sum and the rate of interest.
7. Krishna borrowed Rs. 40,000 from a Financial Institution for 5 years at the rate

of 12% per annum simple interest. Then he paid half of the Principal and interest
earned at the end of 5 years. In how many years can he clear the debt at the same
rate of interest on which he paid interest of Rs. 28800 altogether ?

Area Revision Test (Simple Interest)

1. Find the interest and amount on Rs. 2000 for 3 years at the rate of interest 8%

per annum.

2. Ram Bahadur borrowed Rs. 3000 at 4% per annum from an Agricultural

Development Bank. What was the amount paid by him to the bank at the end of

3 years 6 month?

3. In what time will the amount on Rs. 4000 at the rate of 6% per annum be Rs.

6000.

4. At what rate percent per annum will the amount on Rs. 1000 for 2 years be Rs.

1500. 2 of principle in 8 years.
5. At what rate per annum will the interest be of the
6. What sum should Adarsh deposit in a bank that will y3ield an interest of Rs. 180 in

2 years at the rate of 6% per annum.

7. The difference between the simple interest and the sum of money in 2 years at

8% per annum is Rs. 2240. Find the sum lent out.

Answers

Exercise 1.1

1. a) Rs. 101.25 b) Rs. 135 c) Rs. 525

2. a) Rs. 28, Rs. 378 b) Rs. 38.6, Rs. 907.10 c) Rs. 768, Rs. 2568
6. Rs. 480 7. Rs. 800
3. Rs. 1800, Rs. 4800 4. Rs. 39062.50 5. Rs. 11200

1. a) 3 years b) 5 years Exercise 1.2
c) 2 years, 6months

2. 4 11 years 3. 2 years 4. 10 years 5. 5 years 6. a) 5% b) 6% c) 12%
16
7. a) 12.5% 8. 4% 9. 6.5% 10. 12.5%

Exercise 1.3

1. a) Rs. 6250 b) Rs. 2437.5 c) Rs. 3000 2. Rs. 12000 3. Rs. 600

4. Rs. 1500 5. Rs. 8000 6. Rs. 4000, 6% 7. 2 years

234 10 Simple Interest / Answers

Estimated periods - 10

At the end of this unit the student will be able to Objectives:

● Present the data in line graph and pie chart.
● Calculate central tendency (mean, median and mode) and qualities of data.
● Get introduction of dispersion and calculate range of the given data.

Teaching Materials:

Square papers, stat paper, attractive diagrams (simple bars, multiple bars, subdivided bars).

Activities:

It is better to
● Discuss about the four steps in statistics.
● Discuss about presenting in line graph & pie chart with a suitable data.
● Discuss what analysis of data.
● Explain about measure of dispersion & introduce about range only.

Prime Mathematics Book - 8

Unit - 1 Statistics Estimated periods : 10

Introduction:

Origin of the word statistics: The word statistics has been derived from the Latin word
"status" which means a political state.

Meaning of statistics: In the singular sense, statistics means the science which deals
with the collection, analysis and interpretation of numerical data. In the plural
sense,statistics means numerical facts (data)systematically collected for some definite
purpose.

Definition of statistics: According to Croxton and Cowden "statistics may be defined as
the collection, presentation, analysis and interpretation of numerical data."

Four stages in statistics:

1. Collection of data:
a) primary data: Data collected by the investigator to meet the objectives of his/her

own enquiry.
b) Secondary data: Data already collected by some other investigator for related study.

2. Presentation of data : The data freshly collected are raw and need to be presented
to make more convincing and appealing. Presentation includes editing, classifying,
arranging, tabulation, presenting in diagrams and graphs.

3. Analysis of data: Under this stage comes the calculations like measure of central
tendency, dispersion, skewness kurtosis etc.

4. Interpretation of data: After necessary calculations, the next step and the final step is
to draw a logical inference and give valid decision about the objective of the enquiry.

Terminologies:

Statistical data: In statistics, the observations (numerical facts) collected is called data.

Variables: A quantity which takes different values at stated conditions is called variables
and the different values taken by the variable are called varieties. For example ages
of teachers in a school is variable under which 23 years, 30 years, 32 years... etc are
the varieties.

Continuous variable: A variable which can assume any value (integral or factional)
between two given values is called a continuous variable. For example the height of
persons may be 145cm,147.2cm ,149.8cm, 150cm... etc.

236 11 Statistics

Prime Mathematics Book - 8

Discrete variable: A variable which assume integral values only is called desecrate
variable. for example the number of students in different classes in a school are 20,
21, 25, 30... etc.

Frequency: The number of times an observation (variety) occurs is called the
frequency.

Frequency distribution: A tabular arrangement of data showing the frequency of each
observation is called a frequency distribution.

1.1 Presentation of data:

In classes VI and VII you learned about presenting the data in frequency distribution
table, cumulative frequency distribution table, simple bar diagram and multiple bar
diagrams. In this class you will learn about line graph and pie chart.

Line graph or time series graph:

A series formed by the variates (values) of a variable at different periods of time
is called a time series. Plotting the variates (values) taking values along y-axis and
time periods along x-axis and joining the points with straight lines in order, we obtain
a line graph or a time series graph or a histogram. A line graph shows the changes in
the values of variables with the changes in time.

Example - 1: The data given below are production of a sugar factory.

Year 2007 2008 2009 2010 2011 2012 2013
Product (in 18 20 24 28 25 27 30
thousand tones)

Draw line graph representing the g3i5ven data.

Solution: Product(in thousand tonnes) 30
The line graph of the data is as shown 25
20

15

10

5

0 2007 2008 2009 201 0 201 1 201 2 201 3

Years

N ote:
• All the diagrams and graphs are not suitable for all the data.

• Continuous variables are generally represented by line graph.

• The graph shows general trend of the data. However the line

between the points do not have real representation.

Statistics 237

Prime Mathematics Book - 8

Pie-Chart

A circle divided by radial lines into sectors that represent the component parts of the
total magnitude of the data is called a pie chart or a pie diagram or a circular diagram.

• When the data contains different categories under a topics then pie-chart is prefer-
able to represent the data.

• In a pie-chart the angle around the centre of a circle which is 360º is supposed to
be total magnitude of the data and the component values are expressed in term of
angles which are represented by the sectors.

Steps for drawing a pie chart:

• Since total = 360º

Angular value of a component = 3600 × value of the item
Total value

• Convert each value into angles.
• Draw a circle of suitable radius. Taking a radius as base draw sectors of corresponding

angles.
• Give different shades to distinguish the various sectors.
• Give suitable topic (heading) and index.

Example 2: The data given below is the budget of a family.

Rent Food Clothing Education Saving Miscellaneous
Rs.600
Rs.2400 Rs.4800 Rs.1800 Rs.1200 Rs.1200

Present the data in a pie-chart.

Solution : Items Expenditure (Rs.) Angular value (º)

Rent Rs. 2400 2400 × 360º = 72º
12000

Food Rs. 4800 4800 × 360º = 14º
12000

Clothing Rs. 1800 1800 × 360º = 54º
12000

Education Rs. 1200 1200 × 360º = 36º
12000

Saving Rs. 1200 1200 × 360º = 36º
12000

Miscellaneous Rs. 600 600 × 360º = 18º
12000

Rs. 12000 3600

238 11 Statistics

Prime Mathematics Book - 8

Note: i) Pie-chart is named so because MONTHLY BUDGET OF A FAMILY

ii) iPfielo-cohk as rltikies a sliced pie. to 18o 72o Rent
not preferable 36o Food
construct if there are more Clothing
than 6 characteristics. 36o Education
Saving
iii) It is better finding percentage of Miscellaneous
each item and write percentage
values in each sector. 54o
144o

Example 3: Study the chart (diagram) PEOPLE WITH DIFFERENT BLOOD
given in the figure and answer GROUPS
the following. A
B
i) Name the diagram. AB
ii) Calculate the percentage of O

the people with each blood
group.
iii) If the total number of people
is 60, find the number of
people with each blood
group shown in table.

Solution: i) The name of the diagram is pie-chart. (A) = 1080 × 100% = 30%
ii) Percentage of the people with blood group 3600

Percentage of the people with blood group (B) = 720 × 100% = 20%
3600

Percentage of the people with blood group (AB) = 360 × 100% = 10%
3600

Percentage of the people with blood group (O) = 1440 × 100% = 40%
3600

iii) No. of people with blood group (A) = 30% of 60 = 18
No. of people with blood group (B) = 20% of 60 = 12
No. of people with blood group (AB) = 10% of 60 = 6
No. of people with blood group (O) = 40% of 60 = 24
The data is tabulated as;

Blood group A B AB O
No. of Persons 18 12 6 24

Statistics 239

Prime Mathematics Book - 8

Exercise 1.1

1. Daily wages of 100 workers are given below. Construct a line graph to represent the data.

Wages(Rs.) 400 500 600 700 800 900 1000
18
No.of workers 8 12 22 15 14 11

2. The data given below shows the production of cars by a moter company. Draw a line
graph representing the data.

Years 2009 2010 2011 2012 2013
No. of produced 10,200 12,400 11,200 15,100 18,000

3. Monthly average temperatures of a certain locality are given in the table given
below. Draw the line graph representing the data.

Months Baishakh Jestha Ashadh Sharawan Bhadra Aswin
Temperature 22º 30º 33º 28º 31º 20º

4. Draw a line graph of the data:

Years 2006 2007 2008 2009 2010 2011 2012 2013
Profit(Rs. Lakhs) 10 14 18 20 14 16 18 22

5. The table given below shows the number of students of a school. Represent the data
in a pie-chart.

Classes I II III IV V
No. of students 30 40 60 30 20

6. Monthly expenditure of a family is shown in the table given below. Construct a pie-
chart to represent the above data.

Topics Food Clothing Education Rent Miscellaneous
Expenditure(in Rs.) 6000 4200 3000 4800 3600

7. The table below shows the number of teachers of ABC school in different levels.
Represent the data in a pie-chart.

Level Pre-Primary Primary L. Secondary Secondary
No. of Teachers 6 10 12 8

240 11 Statistics

Prime Mathematics Book - 8

8. Monthly income of a family is shown in the table given below. Present the data in

pie-chart.

Sources House rent Business Agricultures Pension Salary
Income (in Rs.) 8000 12000 7000 15000 18000

9. For the construction of a house a house owner spent as:

Items Bricks Cement Steel Timber Labour Miscellaneous
270
Expenditure (in Rs.000) 270 360 270 180 450

Construct a circular diagram to represent the data.

10. Present Energy consumption in the world is given below:

Source Oil Hydro electricity Natural gas coal Nuclear

Percentage consumption 41 12 21 20 6

Represent the above data in an angular diagram.

11. Study the diagram shown in the figure and answer the following questions.

i) What type of diagram is it?

ii) What is the diagram about? AREAS OF OCEANS OF THE
WORLD

iii) Which Ocean occupy the biggest 180 110 Indian
and the smallest area. 670

iv) Calculate the percentage area 167 0
covered by Indian Ocean. 97 0

v) If total area covered by Oceans is MONTHLY EXPENDITURE
396 million square km, calculate OF A FAMILY
the area covered by each Ocean.
90 0 1440 Food
12. Monthly expenditure of a family 56 0 Rent
is shown in the pie diagram given 340 Clothings
below. If the total expenditure is 360
Rs. 14400,

i) Calculate the expenditure under
each heading.

ii) Show the data in a table.

Statistics 241

Prime Mathematics Book - 8

1.2 ANALYSIS OF DATA:

After the presentation of data, the next step is to analyse it to get further
informations. Under analysis comes necessary calculations like central tendency,
dispersion, skewness, kurtosis, etc.

1. Central tendency

Characteristics of a group tends (goes) towards the central value (average) i.e
average represents the whole data. Such property of group is called central tendency. A
number or a quantity which is typical or representative of a set of data is called central

tendency. Measures of central tendency are the averages (mean, median and mode).

1.2.1 Mean (arithmetic mean)

Mean of a given set of observations is obtained by dividing the sum of all the observations

by their number. It is denoted by X (read as "x bar")

∴ Mean (X) = Sum of the observations
Number of observations

For example, let the data is 8, 22, 15, 6, 10.

Sum of the observations = 8 + 22 + 15 + 6 + 10 = 61

Number of observations = 5



Sum of the observations = 61 = 12.5
5
∴ Mean (X) = Number of observations


i) Mean of raw data (individual data):
Let there be n number of observations: x1, x2, x3, ........ , xn

Then their Mean (X) = x1+ x2+ x3+ ........ + xn
n

Which is also written as (X) = 1 n or simply written as (X) = ∑X ,
n n
n∑=1xn

Note: the symbol ∑ (Greek letter mediastinoscope') stands for "summation of".

Example 4: Find the mean of the data. 27, 36, 40, 51, 62, 68, 75, 82, 90, 95
Solution: Here, ∑X = 27 + 36 + 40 + 51 + 62 + 68 + 75 + 82 + 90 + 95 = 626

And (N) = 10 ∴ Mean (X) = ∑x = 626 = 62.6
N 10

242 11 Statistics

Prime Mathematics Book - 8

Example 5: If the mean of the data 27, 25, 37, 43, a, 67, 59, 29, 14, 32 is 38.8, find the
value of a.

Solution: Here, given data

27, 25, 37, 43, a, 67, 59, 29, 14, 32

Sum of observations (∑X) = 27 + 25 + 37 + 43 + a + 67 + 59 + 29 + 14 + 32

= 333 + a

Number of observations (N) = 10

∑x
∴ Mean (X) = n

or, 38.8 = 333 + a or, 388 = 333 + a or, a = 388 - 333 ∴ a = 55
10

ii) Mean of un grouped data (Discrete data):

In such data, items are repeated (have frequency).

Let x1, x2, x3, ..... , xn be the observation with frequencies f1, f2, f3, ...., fn respectively.
This means observation X1 occurs f1 times, X2 occurs f2 times and so on.

Then sum of all the observation = f1x1 + F2x2 + F3x3 + ..... Fnxn.
n
∑=i=1 fixi = ∑fx. (simply)


And the total number of observations = f1 + f2 + f3 + ........ + fn.

n

∑=i=1 fi = ∑f = N. (simply)

∴ The mean of the data = sum of observations
Total number of observations

n
∑ fixi
X = fi = ∑fx =.
i=1 ∑f

n

∑
i=1

For a desecrate data,

- Prepare a table with the columns of x, f and fx

- Sum of f values given N and sum of fx values given ∑fx.

- Then, X= ∑x
n

Statistics 243

Prime Mathematics Book - 8

Example 6: Find the mean of the data: x 5 10 15 20 25

Solution: Here, f 6 14 18 9 5

∑fx = 5 × 6 + 10×14 + 15 × 18 + 20 × 9 + 25 × 5

= 30 + 140 + 270 + 180 + 125 = 745 And N = ∑f = 6 + 14 + 18 + 9 + 5 = 50

∴ Mean (X ) = ∑x = 745 = 14.9
N 50

Alternate Method: xf fx
The given data can be tabulated as
56 30

10 14 140

Here, N = 50 and ∑fx = 745 15 18 270
Now we have,
20 9 180

25 5 125

X = ∑fx X = 745 = 14.9 N = 50 ∑fx = 745
N 50

Example 7: If the mean of the data given below is 14.92, find the missing frequency m.

X 12 13 14 15 16 17 18
F 4 7 9 m 12 5 3

Solution: The data can be tabulated as:

xf fx

12 4 48

13 7 91

14 9 126

15 m 15m

16 12 192

17 5 85

18 3 54

N = 40 + m ∑fx = 596 + 15m

Here, N = 40 + m, ∑fx = 596 + 15m and X = 14.92

Now we have, ∑fx
=N
X

or, 14.92 = 596 + 15m or, 596.8 + 14.92m = 596 + 15m
40 + m

or, 0.08m = 0.8 ∴ m = 10
Hence, the value of m is 10.

244 11 Statistics

Prime Mathematics Book - 8

iii) Mean of grouped data (continuous data) : If the data is grouped i.e. presented

in class interval form, mean of the data is X = ∑fm , where m is the mid value of
N

each class.

Steps:

- Prepare a table with the columns for class, f, m, fm

Lower limit + Upper limit
Mid value (m) = 2 in each class.

- Sum of f values gives ∑f = N and sum of the fm values gives ∑fm

∑fm
Then Mean (X) = N

Example 8: Calculate the mean of the following data:

Marks obtained (x) 20-30 30-40 40-50 50-60 60-70 70-80
No. of students (f) 3 7 10 12 6 2
Solution:

The data is tabulated as.

Marks obtained(x) No. of students (f) Mid value (m) fm

20-30 3 25 75

30-40 7 35 245

40-50 10 45 450

50-60 12 45 660

60-70 6 65 390

70-80 2 75 150

N=40 ∑fm = 1970

Here,

N = 40, fm = 1970

∴ Mean (X) = ∑fm = 1970 X = 49.25
N 40

Note: Mid value of inclusive and exclusive classes being same, limit correction for
inclusive class intervals is not necessary for the mean.



Statistics 245