Percentage- Uses and Applications

228 Mathematics

10

Percentage and its Applications

10.1 INTRODUCTION

In every day life, you come across situations in which the word ‘percent’ is made use of veryfrequently. For example, you see a banner in the market.

“Sale upto 50 percent off”

You read news in the newspaper

“Votes turnout in the poll was over 65 percent”

“Banks have lowered the rate of interest on fixed deposits from 7.5 percent to 6.5 percent”

There are many such situations in different walks of life where the concept of percentage findsits use. In this lesson we shall study percent as a fraction or a decimal and its application insolving problems of profit and loss, discount, sales tax, instalments etc.

10.2 OBJECTIVES

After studying the lesson, the learner will be able to :

write a fraction and a decimal as a percent and vice-versa

calculate specified percent of a given number or a quantity

solve problems based on percentage

solve problems on profit and loss

calculate simple interest and amount when a given sum of money is interested for aspecified time period on a given rate of interest.

state the need for given discount

define discount and discount series (successive discounts, no. more than three)

find a single discount equivalent to a given discount series

calculate the discount and the selling price of an article, given marked price of the article,and the rate of discount

Percentage and its Applications 229

solve inverse problems pertaining to discount

calculate the sales tax on commission and the selling price of an article, given the markedprice of the article and the rate of sales tax or commission

solve inverse problems pertaining to sales tax

solve inverse problems on commission determine the amount of each instalment whengoods are purchased under investment plan (case of equal instalments only)

determine the rate of interest when equal instalments are given

10.3 EXPECTED BACKGROUND KNOWLEDGE

Four fundamental operations on whole numbers, fractions and decimals.’

10.4 PERCENT

You have learnt a lot about fractions. A fraction denotes part of a whole. For instances 34 means

3 out of 4 equal parts. Similarly, 710 means 7 out of ten equal parts and 17

100 means 17 out

of 100 equal parts.

A fraction whose denominator is 100 is read as a percent, for example 5100 is read as five

percent.

In Fig. 10.1, 33 out of 100 small squares are shaded.

This means 33100 of the larger square is shaded.

The word ‘percent’ is abbreviated form of the Latin word “percentum” which means “per hundred” or “hundredths”.

The symbol ‘%’ is used for the term percent.

A ratio whose second term is 100 is also called a percent.

When we say Anita has secured 80% marks in mathematics. This means that she has secured80 marks out of 100 or 40 marks out of 50. Similarly, when we say a man has spent 20% ofhis income on food, it means that out of every hundred rupees of his income, he has spentRs 20 on food.

Suppose, we wish to compare two fractions 34 and 4

5 .

Since these are fractions having different denominators we need to convert them into fractionswith common denominator.

Fig. 10.1

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34 = 3 5

4 51520

×× = ; 4

5 = 4 45 4

1620

×× =

Since 16 > 15,

∴ 1620

1520> ; and so 4

534>

Here, we have converted each of the two fractions into fractions with least commondenominator.

As a convention, we convert if possible each fraction into a fraction with denominator 100.In the above example,

34 = 3 25

4 2575

100×× = ; 4

5 = 4 205 20

80100

×× =

Since 80 > 75. we get 45

34

> .

10.5 CONVERSION OF A DECIMAL INTO A PERCENT AND VICE VERSA

Let us consider the following examples :

0.37 = 37100 = 37%, 0.7 = 7

1070

100= = 70%; 1.46 = 146100 = 146%

Thus, to write a decimal as a percent, we move the decimal point two places to the right andput the % sign.

0.25 = 25% 0.61 = 61% 0.37 = 37%

0.1 = 10% 0.07 = 7% 1.4 = 140%

Conversely,

To write a percent as a decimal, we drop the % sign and insert or move the decimal point twoplaces to the left. For example,

37% = 0.37 89% = 0.89 35% = 0.35

99% = 0.99 100% = 1.00 3% = 0.03

110% = 1.10 212% = 2.12 0.1% = 0.001

10.6 CONVERSION OF A PERCENT INTO A FRACTION AND VICE VERSA

To write a percent as fraction, we drop the % sign and divide the number by 100. For example,

69% = 69100 13% = 13

100 3% = 3100

4.5% = 4 5100

451000

. = 170% = 170100 216% = 216

100


In general, x% = x

100 .

Conversely,

To write a fraction as a percent, we multiply the fraction by 100, simplify it and suffix the% sign. For example

14 = 1

4 100×FH IK% = 25%

32 = 3

2 100×FH IK% = 150%

35 = 3

5 100×FH IK% = 60%

CHECK YOUR PROGRESS 10.1

1. Write each of the following decimals as percent :

(a) 0.56 (b) 0.03 (c) 0.75 (d) 0.02

(e) 0.97 (f) 0.8 (g) 0.04 (h) 1.4

2. Write each of the following percents as decimal :

(a) 75% (b) 14% (c) 3% (d) 115%

(d) 2% (f) 25% (g) 400% (h) 350%

3. Write each of the following percents as fraction :

(a) 35% (b) 40% (c) 70% (d) 85%

4. Convert each of the following fractions into percent :

(a) 34 (b) 1

5 (c) 310 (d) 4

25

5. Aruna obtained 19 marks in a test of 25 marks. What was her percentage of marks ?

6. Gurmeet got half the answers correct. What percent of their answers were correct ?

7. A suit piece consists of cotton and rayon fibre in which cotton is 3 out of 8 parts. Whatis the percentage of cotton in the suit piece ?

8. Kavita read 84 pages of 100-page book. What percent of the book did she read ?

9. A class of a school had 45% girls. What percent of the class were boys ?

10. One-fourth of the shoes in a shop were on sale. What percent of the shoes were thereon normal price ?

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11. In the word PERCENTAGE, what percent of the letters are E’s ?

12. If three fourths of students of a class wear glasses, what per cent of students of the classdo not wear glasses ?

13. There are 20 eggs in a fridge and 6 of them are brown. What percent of eggs arebrown ?

14. 60 candidates appeared in an examination and 45 of them passed. What percent ofcandidate passed ?

15. Mr. X spends Rs 310 out of Rs 500 and Mr. Y spent Rs 500 out of Rs 800 every week.Compute their spending as percentages and state who spends higher percentage.

16. In a class of 40 students, 10 secured first division, 15 secured second division, and 13just qualified. What is the percentage of students that failed ?

10.7 CALCULATION OF PERCENT OF A QUANTITY

To determine the percent of a number or quantity, we first change the percent to a fractionor a decimal and then multiply with the number.

For example, 45% of 90 = 0.45 × 90 = 40.50

or 45% of 90 =45

10090× = 40.50

60% of 120 = 0.60 × 120 = 72.0018% of 215 = 0.18 × 215 = 38.70135% of 80 = 1.35 × 80 = 108

Example 10.1 : A family spends 35% of its monthly budget of Rs 7500 on food. How muchdo they spend on food ?

Solution : Expenditure on food = 35% of Rs 7500

= 0.35 × Rs 7500= Rs (0.35 × 7500)= Rs 2625.00 or Rs 2625.

Example 10.2 : In a garden, there are 500 plants of which 35% are trees, 20% are shrubs and25% are herbs. The rest are creepers. Find out the number of trees, shrubs, herbs and creepers.

Solution : Number of trees = 35% of 500 = 0.35 × 500 = 175Number of shrubs = 20% of 500 = 0.20 × 500 = 100Number of herbs = 25% of 500 = 0.25 × 500 = 125

Since the remaining plants are creepers,Number of Creepers = 500 – (175 + 100 + 125)

= 500 – 400 = 100


Example 10.3 : 35% of students in a school are girls. If the total number of students is 1240,find the number of boys in the school.

Solution : Number of girls in the school = 35% of 1240

= 0.35 × 1240

= 434

∴ No. of boys in the school = 1240 – 434 = 806

Aliter

Since 35% of the students in the school are girls, (100% – 35%) i.e., 65% of the students inthe school are boys.

∴ Number of boys = 65% of 1240

= 0.65 × 1240

= 806

Example 10.4 : What percent of 240 is 96 ?

Solution : Percent = 96240 100× %

= 40%

Example 10.5 : If 27% of ‘a’ is 54, then find a.

Solution : We have 27% of a = 54

⇒ 27100 × a = 54

⇒ a = 54 10027× = 200

Thus, the value of a is 200.

Example 10.6 : 60 is reduced to 45. What is the reduction percent ?

Solution : Let 45 is less than 60 by x%, then

Reduction = 60 – 45 = 15

Reduction percent = 1560 100× % = 25%.

Example 10.7 : If 80 is increased to 125, what is the increase percent ?

Solution : Increase = 125 – 80 = 45

Increase percent = 4580 100× % = 56.25%.

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Example 10.8 : A voluntary organisation was collecting money for a relief camp. Theirtarget was Rs 20000, but they exceeded their target by 45%. How much money did theycollect ?

Solution : The money collected = 20000 + 45% of 20000= Rs 20000 + Rs (0.45 × 20000)= Rs 20000 + Rs (9000.00)= Rs 20000 + Rs 9000= Rs 29000

Hence, they collected Rs 29000.

Example 10.9 : 44% of the students of a class are girls. If the number of girls is 6 less thanthe number of boys, how many students are there in the class ?

Solution : Given that 44% of the students are girls. So, (100 – 44)% i.e., 56% of the studentsare boys.

Thus, there are 12% less girls than boys.

By the given condition,

12100 of the number of students = 6

∴ Number of students = 6 10012× = 50

Thus, there are 50 students in the class.

Example 10.10 : Raman has to secure 40% marks for passing. He gets 178 marks and failsby 22 marks. Find the maximum marks.

Solution : Since Ramesh secures 178 marks and fails by 22 marks, the pass marks are 178+ 22 = 200.

Let x be maximum marks, then

40% of x = 200

or 40100 × x = 200

or x = 200 10040× = 500

Thus, the maximum marks are 500.


1. Find :

(a) 15% of 440 (b) 16% of 1250

(c) 47% of Rs 1200 (d) 39% of 1700 metres.


2. At a school, 40% of the students come on foot to the school. There are 600 students inthe school. How many students come on foot to the school ?

3. There are 36 children in a class 25% of them are boys. How many are boys ? How manyare girls ?

4. An alloy is a combination of zinc and copper with 30% zinc and 70% copper. If a pieceof this alloy weighs 150 kg , how much zinc it contains ?

5. Naresh earns Rs 15400 per month. He keeps 50% for household expenses, 15% for hispersonal expenses, 25% for expenditure on his children and the rest he saves. What amountdoes he save per month ?

6. There are 32 boys in a class. On a particular day 12.5% of them were absent. How manyboys were absent on that day ?

7. It takes me 45 minutes to go to school and I spend 80% of the time travelling by bus.How long does the bus journey last ?

8. During a general election, 70% of the population voted. If 70000 people cast their votes,what is the population of the town ?

9. The cost of a saree was Rs 450. Its cost has increased to Rs 495. By what percent didthe cost increase ?

10. What percent of 160 is 64 ?

11. If 120 is reduced to 96, what is the percentage reduction ?

12. In an election, 25% voters did not cast their votes. A candidate secured 40% of the votespolled and was defeated by 900 votes. Find the total number of voters.

13. A’s income is 25% more than B’s and B’s income is 8% more than C’s. If A’s incomeis Rs 4050, then find the C’s income.

14. A reduction of 10% in the price of tea enables a dealer to purchase 25 kg more tea forRs 22500. What is the reduced price per kg of tea ? Also, find the original price per kg.

15. A rise of 25% in the price of sugar compels a person to buy 1.5 kg of sugar less forRs 240. Find the increased price as well as the original price per kg of the sugar.

16. A number is first increased by 10% and then decreased by 10%. What is the net increaseor decrease percent ?

17. A man donated 5% of his monthly income to a charity and deposited 12% of the restin a bank. If he has Rs 11704 with him now, what was his monthly income ?

10.8 APPLICATIONS OF PERCENTAGE

We come across a number of situations in our day to day life wherein we use the concept ofpercent. In the following, we discuss applications of the concept of percentage in differentfields.

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10.8.1 PROFIT AND LOSS

Let us recall the terms and formulae related to profit and loss that we have learnt earlier.

Cost Price (C.P.) : The price at which an article is purchased, is called its cost price.

Selling Price (S.P.) : The price at which an article is sold, is called its selling price.

Profit (Gain) : When S.P.>C.P., then there is a profit.

Profit = S.P. – C.P.

Loss : When C.P. > S.P, then there is a loss.

Loss = C.P. – S.P.

Note : Gain or Loss is always calculated on C.P.

Formulae Profit% = Pr. .ofit

C P ×FH IK100 % ...(1)

and Loss% = LossC P. . ×FH IK100 % ...(2)

Formula (1) gives

Profit = C P ofit. . Pr %×100

or S.P. – C.P. = C P ofit. . Pr %×100

or S.P. = C P C P ofit. . . . Pr %+ ×100

or S.P. = 100100+FH IK ×Pr % . .ofit C P ...(3)

Similarly, Formula (2) gives

Loss =C P Loss. . %×

100

or C.P. – S.P =C P Loss. . %×

100

or S.P. = C P C P Loss. . . . %− ×100

or S.P. =100

100−F

HGIKJ ×

Loss C P% . . ....(4)

Let us now consider some examples to illustrate the applications of these formulae in solvingproblems related to profit and loss.


Examples 10.11 : A shopkeeper bought an almirah from a wholesale dealer for Rs 4500 andsold it for Rs 6000. Find his profit or loss percent.

Solution : Here C.P. of the almirah = Rs 4500

S.P. of the almirah = Rs 6000

Since S.P. > C.P., there is a profit

Profit = S.P. – C.P.

= Rs 6000 – Rs 4500

= Rs 1500

∴ Profit % = Pr. . %ofit

C P ×FH IK100

=15004500

100×FHG

IKJ%

=100

3% i.e. 33 1

3% .

Example 10.12 : A retailer buys a cooler for Rs 3800 and overhead expenses on it areRs 50. If he sells the cooler for Rs 4400, determine his profit percent.

Solution : Here, C.P. of the cooler = Rs (3800 + 50) = Rs 3850

S.P. of the cooler = Rs 4400

Since S.P. > C.P., there is a profit

Profit = Rs 4400 – Rs 3850

= Rs 550

∴ Profit % = Pr. . %ofit

C P ×FH IK100

= 5503850 100×FH IK%

= 1007 % i.e. 14 2

7 % .

Example 10.13 : By selling a scooter to a customer for Rs 22400 an auto-dealer makes a profitof 12%. Find the cost price of the scooter.

Solution : Here, S.P. of the scooter = Rs 22400, Profit % = 12%

Using formula (3), we have

S.P. =100

100+F

HGIKJ ×

Pr % . .ofit C P

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or C.P. =S P

ofit. .

Pr %×

+100

100

= Rs 22400 100

100 12×+

= Rs 22400 100

112×

= Rs 20000

Thus, the cost price of the scooter is Rs 20000.

Example 10.14 : By selling a cycle for Rs 2024, a cycle merchant loses 12%. If he wishesto make a gain of 12%, what will be the selling price of the cycle ?

Solution : First Part : S.P. = Rs 2024 and Loss % = 12%

From formula (4), we have

C.P. =S P

Loss. .

%×

−100

100

= Rs 2024 100100 12

×−

= Rs 2024 100

88×

= Rs 2300

Second Part : C.P. of the cycle = Rs 2300 and Gain (Profit)% = 12%

Using formula (3), we have

S.P. = 100100+FH IK ×Pr % . .ofit C P

= 100 12100 2300+ ×

= Rs 112 2300100×

= Rs 2576

Thus, the selling price of the cycle will be Rs 2576.

Example 10.15 : If the cost price of 15 articles is the same as the selling price of 12 articles,find the gain or loss percent in the transaction.

Solution : Let C.P. of an article be Rs x, Then

C.P. of 15 articles = Rs 15x ...(i)


or S.P. of 12 articles = Rs 15x

or S.P. of 1 articles = Rs 1512 xFH IK ...(ii)

Since S.P. > C.P., there is a profit (gain) in the transaction

Gain = Rs 1512

x x−FH IK

= Rs 312

xFH IK or Rs x4FH IK

or Gain % = GainC P. . %×FH IK100

=xx4 100×FHG

IKJ%

= 14 100×FH IK% = 25%

Thus, the gain in the transaction is 25%.

Example 10.16 : By selling 45 oranges for Rs 160, a women loses 20%. How many orangesshould she sell for Rs 112 to gain 20% on the whole ?

Solution : First Part : S.P. of 45 oranges = Rs 160

Loss% = 20%

So, by formula (4), we have

C.P. of 45 oranges = S PLoss

. .%

×−

100100

= Rs 160 100100 20

×−

= Rs 160 10080× = Rs 200.

Second Part : C.P. of 45 oranges = Rs 200

Gain% = 20%

∴ S.P. of 45 oranges = 100+ ×Gain%100 C.P.

= 100 20100 200+ × Rs

240 Mathematics

= Rs 120 200

100

×

= Rs 240

Now, number of oranges for Rs 240 = 45

Number of oranges for Rs 112 = 45

240112× = 21

Thus, the women should sell 21 oranges for Rs 112.


1. A shopkeeper buys an article for Rs 320 and sells it for Rs 240. Find his gain or losspercent.

2. A dealer buys a wrist watch for Rs 450 and spends Rs 30 on its to repairs. If he sellsthe same for Rs 600, find the profit per cent.

3. A dealer sold two machines at Rs 2400 each. On selling one machine he gained 20% andon selling the other he lost 20%. Find the dealer’s gain or loss percent.

4. A sells an article costing Rs 1000 to B and earns a profit of 6%. B, in turn sells it toC at a loss of 5%. At what price did C purchase the article ?

5. By selling 90 ball pens for Rs 160, a person loses 20%. How many ball pens should hesell for Rs 96, so as to have a gain of 20% ?

6. If the selling price of 20 articles is equal to the cost price of 23 articles, find the lossor gain percent.

7. A watch was sold at a profit of 12%. Had it been sold for Rs 33 more, the profit wouldhave been 14%. Find the cost price of the watch.

8. By selling a book for Rs 258, a publisher gains 20%. For how much should he sell itto gain 30% ?

9. A vendor bought bananas at 6 for 5 rupees and sold them at 4 for 3 rupees. Find his gainor loss percent.

10.8.2 Simple Interest

All the transactions that take place around us involve money. Sometimes, a person has to borrowsome money as a loan from his friends, relatives, bank etc. He promises to return it after aspecified time period. So, he has to give back not only the money he borrows but also someextra money to the lender for using his money.

The money borrowed is called the principal, usually denoted by P. The extra money paid iscalled interest, usually denoted by I.


The sum of principal and the interest is called the amount usually denoted by A.

A = P + I

Interest is calculated on the principal

Interest is mostly expressed as a rate percent per year (per annum).

Interest depends on how much money (P) has been borrowed and the duration of the time (T)for which it is borrowed. Interest is calculated according to an agreement, which specifies acertain percent of the principal for each year’s use, called the rate of interest.

I = P × R × T

Interest as calculated above is called simple interest.

Example 10.17 : A man borrowed Rs 50000 from a finance company for buying a motor bicyclefor a period of 2 years. If the finance company charged simple interest at the rate of 15% perannum, how much interest was paid by the man to the finance company.

Solution : Here, Principal (P) = Rs 50000

Time (T) = 2 years

Rate (R%) = 15% = 0.15

We have I = P × R × T

= Rs (50000 × 0.15 × 2)

= Rs 15000

Hence, the man paid Rs 15000 as the interest to the finance company.

Example 10.18 : A certain sum of money was deposited for 5 years. Simple interest at therate of 12% was paid. Calculated the sum deposited if the simple interest received by thedepositor is Rs 1200.

Solution : Let the sum deposited be Rs P.

Given that I = Rs 1200; T = 5 years; R = 12%

We have I = P × R × T

or P = IR T×

= Rs 1200012 5. × = Rs 2000

Thus, the sum deposited was Rs 2000.

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Example 10.19 : At what rate of simple interest will a sum of Rs 3000 become Rs 4920 atsimple interest in 4 years ?

Solution : Here, P = Rs 3000; A = Rs 4920; T = 4 years

We have I = A – P = Rs 4920 – Rs 3000 = Rs 1920

Now, I = P × R × T

⇒ R = IP T× = Rs 1920

3000 4×

= 16100 = 16%

Thus, at the rate of 16%, Rs 3000 will become Rs 4920 in 4 years.

Example 10.20 : In what time will Rs 8000 amount to Rs 12000, if simple interest is chargedat the rate of 6% per annum ?

Solution : Here, P = Rs 8000; A = Rs 12000; R = 6% = 0.06

We have I = A – P

= Rs 12000 – Rs 8000 = Rs 4000

Now, I = P × R × T

⇒ T = IP R× = 4000

8000 0 06× .

= 10012 = 8 1

3 years

= 8 years 4 months

Thus, in 8 years 4 months, Rs 8000 will amount to Rs 12000 at the rate of interest 6% perannum.


1. Ramesh borrowed Rs 7000 from his friend at 8% per annum simple interest. He returnedthe money after 2 years. How much did he pay back altogether ?

2. Jaya deposited Rs 15600 in a bank. The bank pays interest at 8% per annum. Find theinterest she will receive at the end of 3 years.

3. Subnam lent Rs 25000 to her friend. She gave Rs 10000 at 10% per annum and theremaining at 12% per annum. How much interest did she receive in 2 years ?

4. Nalini borrowed Rs 5000 from her friend at 8% per annum. She returned the money after6 months. How much amount did she pay to her friend ?


5. Find the interest received by Anil if he deposits Rs 16000 for 8 months at the rate of9% per annum. Also, find the amount.

6. Shalini deposited Rs 14500 in a finance company for 3 years and received Rs 4785 assimple interest. What was the rate of interest per annum ?

7. In how many years will Rs 8000 amount to Rs 16000, if simple interest is earned at therate of 12% per annum ?

8. In how much time will simple interest be 14 th of the principal at the rate of 10% per

annum ?

9. In which case, is interest earned more : (a) Rs 5000 deposited for 5 years at 4% per annum(b) Rs 4000 deposited for 6 years at 5% per annum ?

10. At what rate of interest will simple interest be half the principal in 5 years ?

10.9 DISCOUNT

You must have seen all around advertisements of the following types, especially during thefestival seasons.

SALE

Discount upto 50%

A discount is a reduction in the marked (or list) price.

“25% discount” means a reduction of 25% in the marked price of an article. For instance, ifthe marked price of an article is Rs 100, it is sold for Rs 75, i.e., Rs 25 less than the markedprice.

Note. Discount is always calculated on Marked Price.

Marked Price (or list price) : The marked price (M.P) of an article is the price at which itis listed for sale.

Discount : The discount is the reduction from the marked price of the article.

Net Selling price : In case of discount sale, the price of the article obtained by subtractingdiscount from the list price is called the Net selling Price.

Let us consider the following example to illustrate.

Example 10.21 : A shirt with marked price Rs 165 is sold at a discount of 10%. Find its netselling price.

Solution : Here, Marked Price (M.P.) of the shirt = Rs 165, Discount = 10%

∴ Net selling price = Marked price – Discount

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= Rs 165 – 10% of Rs 165

= Rs 165 – Rs 10100 165×FH IK

= Rs 165 – Rs 16.50

= Rs 148.50

Thus, the net selling price of the shirt is Rs 148.50.

Aliter

Since the discount offered is 10%

S.P. = M.P. – 10% of M.P.

= 90% of M.P.

or S.P. = 90100 × Rs 165

= Rs 148.50

Example 10.22 : A pair of socks is marked at Rs 40 and is being offered at Rs 32. Find thediscount per cent being offered.

Solution : Here, M.P. = Rs 40; and S.P. = Rs 32

So, Discount = M.P. – S.P.= Rs 40 – Rs 32 = Rs 8

∴ Discount % = DiscountM P. . ×100

= 840 100×

= 20

Hence, the discount being offered is 20%.

10.9.1 Discount Series

Sometimes a manufacturer, offers a discount of 10% on discounted price in addition to aprevious discount of 20%, because suddenly he gets a supply of cloth at a very low price. Hemay allow another discount of 5% on the discounted price, to some of his customers for promptpayments. In other words, he allows a discount series.

In a discount series, the first figure denotes the discount on the list price, the seconddenotes the discount on the discounted price and so on.

If a shirt is marked for Rs 120 and a discount series 20%, 10% and 5% is offered, thencomputation for calculating net selling price is as under :


Marked price Rs 120 with a discount series 20%, 10% and 5%.

20% discount on Rs 120 = Rs 120 20100× = Rs 24

∴ Discounted price = Rs (120 – 24) = Rs 96

10% discount on Rs 96 = Rs 96 10100× = Rs 9.60

∴ Discounted price = Rs (96 – 9.60) = Rs 86.40

5% discount on Rs 86.40 = Rs 86.40 × 5100 = Rs 4.32

Net selling price = Rs (86.40 – 4.32) = Rs 82.08.

10.9.2 Conversion of Discount Series to a Single Discount

Instead of computing a series of discounts one by one, it is sometimes more convenient to reducethe series to a single discount.

Let us take some examples to illustrate :

Example 10.23 : Convert the discount series 20%, 10% and 5% to an equivalent singlediscount.

Solution : Let the list price = Rs 100


Discounted price = Rs (100 – 20) = Rs 80


Discounted price = Rs (80 – 8) = Rs 72

and 5% discount on Rs 72 = Rs 72 5100× = Rs 3.60

Discounted price = Rs (72 – 3.60) = Rs 68.40

∴ Single discount on Rs 100 = Rs (100 – 68.40) = 31.60 or 31.6%

Example 10.24 : An old scooter is sold at three successive discounts of 10%, 5% and 2%.If the marked price of the scooter is Rs 18000, find the selling price of the scooter.

Solution : Here, list price = Rs 18000

First discount of 10% = Rs 18000 × 10100 = Rs 1800

Price after first discount = Rs (18000 – 1800) = Rs 16200

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Second discount of 5% = Rs 16200 × 5100 = Rs 810

Price after second discount= Rs (16200 – 810) = Rs 15390

Third discount of 2% = Rs 15390 × 2100 = Rs 307.80

Price after third discount = Rs (15390 – 307.80) = Rs 15082.20Thus the net selling price of the scooter is Rs 15080.20AliterNet selling price of the scooter = (100 – 2)% of (100 – 5)% of (100 – 10)% of Rs 18000

= Rs 18000 98100

95100

90100× × ×FH IK

= Rs 15082.20Example 10.25 : Find the single discount equivalent to the discount series of 20%, 15%and 10%.

Solution : Let the marked price = Rs 100

Price after the given discount series

= (100 – 10)% of (100 – 15)% of (100 – 20)% of Rs 100= 0.90 × 0.85 × 0.80 × Rs 100= Rs 51.20

Hence, the total discount = M.P. – S.P.= Rs 100 – Rs 51.20= Rs 48.80

Hence, the equivalent single discount= Rs 48.80 on M.P. of Rs 100= 48.8%

Example 10.26 : A dealer buys a table listed at Rs 1500 and gets successive discounts of 20%and 10%. He spends Rs 20 on transportation and sells it at a profit of 10%. Find the sellingprice of the table.

Solution : Here, list price of the table = Rs 1500

Price after a discount series of 20% and 10% = (100 – 10)% of (100 – 20)% of Rs 1500

= 90100

80100× × Rs 1500

= Rs 1080


Since the dealer spends Rs 20 on transportationC.P. of the table = Rs 1080 + Rs 20 = Rs 1100

Profit = 10%

∴ S.P. of the table = 100100+FH IK ×Pr % . .ofit C P

= 100 10100 1100+ × Rs

= 110100 1100×Rs

= Rs 1210Thus, the selling price of the table is Rs 1210.


1. A coat is marked at Rs 1200. Find its selling price if a discount of 15% is offered.

2. A man pays Rs 2100 for a machine listed at Rs 2800. Find the rate of discount offered.

3. An article listed at Rs 2650 is sold at a discount of 10%. Due to festival season, theshopkeeper allows a further discount of 5%. Find the selling price of article.

4. Find a single discount equivalent to a discount series given in each of the followingdiscount series :

(a) 25%, 20% and 10%

(b) 20%, 15% and 10%

(c) 20%, 10% and 5%.

5. Which of the following discount series is better for a customer ?

20%, 10% and 5% OR 10%, 5% and 20%.

6. The list price of a table fan is Rs 840 and it is available to a retailer at 25% discount.For how much should the retailer sell it to earn a profit of 15% ?

7. The marked price of a TV set is Rs 25000. A discount series of 20%, 10%, 5% is allowedon it. How much money does one have to pay for the TV set ?

8. If a shopkeeper marks his goods 50% more than their cost price and allows a discountof 40%, find his gain or loss per cent.

9. The list price of a watch is Rs 320. After two successive discounts it is sold for Rs 244.80.If the first discount is 10%, what is the rate of second discount ?

10. A retailer buys shirts from a manufacturer at the rate of Rs 75 per shirt and marked themat Rs 100 each. He allows some discount and gets a profit of 30% on the cost price. Whatpercentage discount does he allow to his customers ?

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10.9.3 Sales Tax

Government levies some taxes to have earning called revenue. One such tax which is leviedon the sale of goods is called sales tax. The rates of sales tax are different for differentcommodities. Some essential commodities are exempted from sales tax. This tax is chargedon the net selling price of commodities and its rate is expressed as a percentage.

For example, if an article is sold for Rs 750 and the rate of sales tax is 8%, then

Sales tax = Rs 750 8100× = Rs 60

Price inclusive of sales tax = Rs (750 + 60) = Rs 810

The customer will have to pay Rs 810.

Example 10.27 : The marked price of a pair of shoes is Rs 320. If the rate of sales tax is 4%,calculate the amount to be paid by a customer for the purchase of the shoes.

Solution : Marked price of shoes = Rs 320

Rate of sales tax = 4%

∴ Sales tax = 4% of Rs 320

= 4100 × Rs 320

= Rs 12.80

Thus, the customer has to pay (Rs 320 + Rs 12.80) = Rs 332.80 for purchasing the shoes.

Examples 10.28 : Anita purchased a shirt for Rs 594 including sales tax. If the rate of salestax is 8%, find the list price of the shirt.Solution : Let the list price of the shirt be Rs P

Then P + 8% of P = 594or 108% of P = 594

or108100

P= 594

or P = 594108. = 550

Thus, the list price of the shirt is Rs 550.

Example 10.29 : Hari Om bought a radio set for Rs 1870, after getting 15% discount on thelist price and then 10% sales tax on the reduced price. Find the list price of the radio set.

Solution : Let the list price or the radio set be Rs P.

Thus, selling price of the radio after discount= Rs P – 15% of Rs P

= 85% of Rs P


= Rs 85100

P

Sales tax = 10% of Rs 85100FH IK P

= 10100

85100× P

= Rs 851000 P

The net price to be paid for the purchase of the radio set is

Rs 85100

851000P P+FH IK = Rs 935

1000 P

Equating it to Rs 1870, we get

9351000 P = 1870

or P = 1870 1000935× = 2000

Thus, the list price of the radio set is Rs 2000.Example 10.30 : The list price of a washing machine is Rs 9000. The dealer allows a discountof 5% on the cash payment. How much money will a customer pay to the dealer in cash, ifthe rate of sales tax is 10% ?

Solution : Here, list price = Rs 9000 and discount = 5%

∴ Cash price of the washing machine

= Rs 9000 5100 9000− ×FH IK

= Rs (9000 – 450)

= Rs 8550

∴ Sales tax = 10% of Rs 8550

= 10100 × Rs 8550

= Rs 855

Hence, the customer has to pay Rs 8550 + Rs 855 = Rs 9405 for the purchase of the washingmachine.

Example 10.31 : The list price of the air-conditioner is Rs 25630. The rate of sales tax is 10%The customer requests the dealer to allow a discount to such an extent that the price of the

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air-conditioner amounts to Rs 25630 inclusive of sales tax. Find the discount in the price ofthe air-conditioner.

Solution : Let Rs P be the net price exclusive of sales tax.

Then, price + sales tax = P P+ 10100 = 110

100 P

This is given as Rs 25630

∴ 110100 P = 25630

or P = 23300

∴ Discount allowed = Rs (25630 – 23300) = Rs 2330


1. The marked price of a sewing machine is Rs 3500. If the sales tax on sewing machineis charged at the rate of 6%, find how much a customer has to pay for purchasing themachine.

2. Amita purchases a pair of socks whose list price is Rs 44. The shopkeeper charges salestax at the rate of 5%. Find how much money Amita has to pay for purchasing the socks.

3. Mrs. Mohini purchased a saree for Rs 1100 including sales tax. If the list price of thesaree is Rs 1000, find the rate of sales tax charged.

4. A refrigerator is available for Rs 13915 including sales tax. If the rate of sales tax is 10%,find the selling price of the refrigerator.

5. Radhika purchased a car with a marked price of Rs 2.1 lakhs at a discount of 5%. If thesales tax is charged at the rate of 12%, find the amount Radhika had to pay for purchasingthe car.

6. Dayakant bought a set of cosmetic items for Rs 345 including 15% sales tax and a pursefor Rs 110 including 10% sales tax. What percent is the sales tax charged on the wholetransaction ?

[Hint. C.P. of cosmetic items = Rs (345 ÷ 1.15) ;

and C.P. of purse = Rs (110 ÷ 1.1)]

7. Kamal wants to buy a suitcase whose list price is Rs 504. The rate of sales tax is 5%.He requests the shopkeeper to reduce the list price to such an extent that he has to payRs 504 only. Calculate the discount given in the price of the suit-case.

10.10 COMMISSION

Manufacturers of goods, farmers and owners of properties, frequently uses the services of amiddle man to find a buyer in order to sell their goods or properties. The middle man is calledan agent, who gets some money for the services rendered by him. This money paid is calledcommission. In general, commission is expressed in terms of percentage.


Example 10.32 : A book agent sold 140 books at Rs 20 each. His commission was 25%. Howmuch money did he earn as commission ?

Solution : Total price of books = Rs (20 × 140) = Rs 2800Amount of commission = 25% of Rs 2800

= 25100 2800× Rs

= Rs 700Thus, the book agent earns Rs 700 as commission.Example 10.33 : A salesman earns Rs 300 as commission at the rate of 8%. For what amountdid he sell the goods ?

Solution : Let the amount be Rs x

Then x × 8100 = 300

⇒ x = 300 1008×

= 3750The sales man sold goods worth Rs 3750.

Example 10.34 : A commission merchant charged Rs 427.50 for selling 1500 packets of salt,at 3% commission. At what price per bag did he sell the salt ?

Solution : Let the selling price per packet be Rs x.

Then amount of commission

= 3% of Rs (1500 × x)

= Rs 1500 3100x ×FH IK = Rs 45x

But this is given to be Rs 427.50.Thus, 45x = 427.50

or x = 427 5045

.

or x = 9.5Thus, the selling price per packet of salt was Rs 9.50.


1. A commission merchant sells 1200 tins of oil at Rs 270 a tin on commission of 2 12 %.

Find the amount of the commission. Also find the net proceeds.

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2. An auctioneer sold a property worth Rs 17.8 lakhs. His commission was 1.25%. Howmuch did the auctioneer earn as commission ?

3. A commission merchant charged Rs 4212 as commission at rate of 3% for selling riceat Rs 450 per bag. How many bags of rice did he sell ?

4. A commission merchant sells a certain amount of goods at a commission of 10% in thefirst two weeks of a month. His commission is then raised to 12%, and he sells an equalamount of goods in the remaining part of the month. If his total earning (commission)were Rs 8000, how much did he sell altogether ?

10.11 INSTALMENT BUYING

With the cost of articles going up day by day it has become difficult for the common man tobuy some articles like scooter, fridge, colour TV etc. which are needed by him and his family.Such articles are available on easy instalments. We shall study about instalment purchasescheme.

10.11.1 What is Instalment Buying ?

Instalment purchase scheme, enables a person to buy costly goods like colour TV sets,refrigerator, video cameras, scooters etc. on convenient terms of payment. In this scheme, thecustomer does not make full payment of the cost of the article at the time of purchase, butmakes a partial payment in the beginning and takes away the article for use. The remainingpayment is made in easy monthly, quarterly or half yearly instalments, as per the agreementsigned between the customer and the shopkeeper.

Cash Price : It is the amount for which the article can be purchased on full payment i.e., theselling price of the article.

Cash down Payment : It is the partial payment made by the customer at the time of signingthe agreement and taking away the article for use. In fact, it is a part of the selling price.

Instalments : It is the amount which is paid by the customer at regular intervals towards theremaining part of the selling price of the article.

It may be noted that in the instalment plan only part payment of the total cost is paid by thecustomer at the time of purchase. The remaining part of the cost is paid on subsequent dates;and therefore the seller charges some extra amount for deferred payments. This extra amountis actually the interest charged on the amount of money which the customer owes to the sellerat different times of instalments paid.

In the following, we solve a few examples to illustrate the process.

Example 10.35 : Bimla buys a sewing machine, which is available for Rs 2600 cash paymentor under an instalment plan for Rs 1000 cash down payment and 3 monthly instalment ofRs 550 each. Find the rate of interest charged under the instalment plan.

Solution : Cash payment price = Rs 2600

Cash down payment = Rs 1000


Balance to be paid in instalments = Rs 1600

Amount paid in 3 instalments = Rs 550 × 3 = Rs 1650

Interest charged in the instalment plan = Rs (1650 – 1600) = Rs 50

The buyer owes to the seller for the 1st month = Rs 1600

The buyer owes to the seller for the 2nd month = Rs (1600 – 550) = Rs 1050

The buyer owes to the seller for the 3rd month = Rs (1050 – 550) = Rs 500

Total = Rs 3150

∴ She has to pay interest on Rs 3150 for 1 month.

But the interest she has to pay = Rs 50

If R% is the rate of interest p.a.

then, 3150 × 112 100× R = 50

∴ R = 50 12 1003150× × %

= 40021 %

= 19 121 %

∴ Rate of interest paid by her in the instalment plan is 19 121 %.

Example 10.36 : A computer is available for Rs 34000 cash or Rs 20000 cash down paymenttogether with 5 equal monthly instalments. If the rate of interest charged under the instalmentplan is 30% per annum, calculate the amount of each instalment.

Solution : Cash price = Rs 34000

Cash down payment = Rs 20000

Balance to paid in 5 equal installments = Rs 14000

Let each instalment be = Rs PInterest charged under instalment plan = Rs (5P – 14000)The buyer owes to the seller

For the 1st month = Rs 14000Fr the 2nd month = Rs (14000 – P)

For the 3rd month = Rs (14000 – 2P)For the 4th month = Rs (14000 – 3P)For the 5th month = Rs (14000 – 4P)

Total = Rs (70000 – 10P)

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Thus, he had to pay interest on Rs (70000 – 10P) for 1 month at the rate of 30% p.a.

∴ (70000 – 10P) × 112

30100× = 5P – 14000

or (70000 – 10P) × 140 = 5P – 14000

or 70000 – 10P = 40(5P – 14000)

or 70000 – 10P = 200P – 560000

or 210 P = 560000 + 70000

or P = 630000210

= 3000

∴ Amount of each instalment = Rs 3000.


1. A T.V. set is available for 21000 cash or for Rs 4000 cash down payment and 6 equalmonthly instalments of Rs 3000 each. Calculate the rate of interest charged under theinstalment plan.

2. Anil purchased a type writer priced at Rs 6800 cash payment under the instalment planby making a cash down payment of Rs 2000 and 5 monthly instalments of Rs 1000 each.Find the rate of interest charged under the instalment plan.

3. A scooter is available for Rs 30000 cash or for Rs 15000 cash down payment and 4 equal

monthly instalments. If the rate of interest charged under the instalment plan is 33 13 %,

find the amount of each instalment.

4. A microwave oven is available for Rs 9600 cash or for 4000 cash down payment and

3 equal monthly instalment. If the rate of interest charged is 22 29 % per annum, find the

amount of each instalment.

LET US SUM UP

Percent means ‘per hundred’.

Percents can be written as fractions as well as decimals and vice-versa.

To write a percent as a fraction, we drop the % sign and divide the number by 100.

To write a fraction as a percent, we multiply the fraction by 100, simplify it and suffixthe % sign.


To determine the specific percent of a number or quantity, we change the percent to afraction or a decimal and then multiply.

When the selling price is more than the cost price of the goods, there is a profit (or gain)

When the selling price is less than the cost price of the goods, there is a loss

Profit (Gain) = S.P. – C.P. ; Loss = C.P. – S.P.

Gain% = GainC P. . ×100 ; Loss = Loss

C P. . ×100

Further S.P. = 100100+ ×Gain C P% . . ; S.P. = 100

100− ×Loss C P% . .

The simple interest (S.I.) on a principal (P) at the rate of R% for a time T years, iscalculated, using the formula

S.I. = P × R × T

Discount is a reduction in the list price of goods.

Discount is always calculated on the marked price of the goods.

(Marked price – discount), gives the price, which a customer has to pay while buyingan article.

Two or more successive discounts are said to form a discount series.

A discount series can be reduced to a single discount.

Sales tax is charged on the sale price of goods.

Commission is paid to an agent for his services in arranging the sale or purchase of goodsfrom some one else.

An Instalment plan enables a person to buy costlier goods.

TERMINAL EXERCISE

1. Write each of the following as a per cent :

(a) 720 (b) 0.25 (c) 1.4 (d) 0.07

2. Write each of the following as a decimal :

(a) 63% (b) 13% (c) 3% (d) 0.3%

3. Write each of the following as fraction :

(a) 0.13% (b) 1.3% (c) 11.3% (d) 113%

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4. Find each of the following :

(a) 37% of 400 (b) 3.5% of 800

5. What percent of 700 is 294 ?

6. By what percent is 60 more than 45 ?

7. Find the number whose 15% is 270 ?

8. What number increased by 10% of itself is 352 ?

9. What number decreased by 7% of itself is 16.74 ?

10. Arun at the beginning of a year had a bank balance of its 17500 and at the end of theyear he had a balance of Rs 21350. By what percent did his balance increase ?

11. A man loses 25% by selling a scooter for Rs 8400. For what amount did he buy thescooter ?

12. A commission merchant charged Rs 700 as commission for selling 100 bags of cottonat 5%. Find the price of one bag at which it was sold ?

13. Shalim deposited Rs 14000 in a bank for 2 years and received Rs 4200 as simple interest.At what rate per annum, was interest paid to him ?

14. Simple interest on a sum of money is 13 rd of the sum itself and the number of years is

thrice the rate percent. Find the rate of interest.

15. Ahmad purchased a bicycle by making a cash down payment of Rs 400 and 3 monthlyinstalments of Rs 275 each. The bicycle was also available on cash payment of Rs 1200.Find the rate of interest per annum charged under the instalment plan.

16. Rita purchased a washing machine for Rs 4000 cash down payment and 4 equal monthlyinstalments. The washing machine was also available for Rs 15000 cash payment. If therate of interest charged in the instalment plan is 18% p.a. Find the amount of eachinstalment.


ANSWERS

Check Your Progress 10.1

1. (a) 56% (b) 3% (c) 75% (d) 2%

(e) 97% (f) 80% (g) 4% (h) 140%

2. (a) 0.75 (b) 0.14 (c) 0.03 (d) 1.15

(e) 0.02 (f) 0.25 (g) 4 (h) 3.5

3. (a)35

100 ; 35 : 100 (b)40

100 ; 40 : 100

(c)70

100 ; 70 : 100 (d)85

100 ; 85 : 100

4. (a) 75% (b) 20% (c) 30% (d) 16%

5. 76% 6. 50% 7. 37.5% 8. 84%

9. 55% 10. 75%` 11. 30% 12. 25%

13. 30% 14. 75% 15. 62% ; 62.5 %; Y16. 5%.


1. (a) 66 (b) 200 (c) Rs 564 (d) 663 metres

2. 240 students 3. 9 boys ; 27 girls 4. 45 kg 5. Rs 1540;

6. 4 boys 7. 36 minutes 8. 1 lakh 9. 10%

10. 40% 11. 20% 12. 6000 votes 13. Rs 3000

14. Rs 90 ; Rs100 15. Rs 40; Rs 32 16. 1% decrease 17. Rs 14000


1. Loss 25% 2. Profit 25% 3. 4% loss 4. Rs 1007

5. 36 ball pens 6. 15% gain 7. Rs 1650 8. Rs 279.50

9. 10% loss


1. Rs 8120 2. Rs 3744 3. Rs 5600 4. Rs 5200

5. Rs 960 ; Rs 16960 6. 11% 7. 8 years 4 months

8. 2 years 6 months 9. (b) 10. 10%

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1. Rs 1020 2. 25% 3. 2265.75

4. (a) 46% (b) 38.8% (c) 31.6%

5. Both same 6. Rs 724.50 7. Rs 17,100 8. Loss 10%

9. 15% 10. 2.5%


1. Rs 3710 2. 46.20 3. 10% 4. Rs 12650

5. Rs 2,23,440 6. 13 34 % 7. Rs 24


1. Rs 8100 2. Rs 22250 3. 312 bags 4. Rs 80,000


1. 21 119 % 2. 17 1

7 % 3. Rs 4000 4. Rs 2000

Terminal Exercise

1. (a) 35% (b) 25% (c) 140% (d) 7%

2. (a) 0.63 (b) 0.13 (c) 0.03 (d) 0.003

3. (a) 1310000 (b) 13

1000 (c) 1131000 (d) 113

100

4. (a) 148 (b) 28 5. 42% 6. 33 13 %

7. 1800 8. 320 9. 18 10. 22%

11. Rs 11200 12. Rs 140

13. 15% 14. 3 13 % 15. 19 1

21%

16. Rs 2850.86

Percentage- Uses and Applications

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