Chapter 8 PERCENTAGE

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PERCENTAGE INTRODUCTION:

A fraction whose denominator is 100 is called a percentage and the numerator of the fraction is called rate percent. Percentage is one of the methods of expressing fractions or parts of quantity. The word ‘percent’ is the combination of two words ‘per’ and ‘cent’.

‘Per’ means every and ‘cent’ means one hundred. It is denoted by the symbol %.

GENERAL CONVERSIONS:

1. Conversion of Percentage to Fraction:

To convert a rate percent to a fraction, write the rate percent in the numerator and write denominator equal to 100, then remove the % symbol. Further reduce this into the lowest terms.

For eg; 25% = ²⁵

100 =¹

4

20% = ²⁰

100=¹

5 30% = ³⁰

100= ³

10 140% = ¹⁴⁰

100=¹⁴

10 =⁷

5 and so on.

2. Conversion of Decimal to Percentage:

For converting a decimal into a percentage, the decimal number is multiplied by 100, and the symbol % added to it.

For eg; 0.42 = 0.42 × 100 = 42%

0.3 = 0.3 × 100 = 30%

0.07 = 0.07 × 100 = 7%

0.128 = 0.128 × 100 = 12.8% and so on.

3. Conversion of Percentage to decimal:

Percentage can be expressed as a fraction with numerator equal to rate percent and the denominator equal to 100. Its decimal equivalent can be easily written by putting in the rate percent a decimal point after leaving two extreme right digits.

For eg; 7% = ⁷

100= 0.07

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|www.myexamcoaching.com 2 43% = ⁴³

100= 0.43 135% = ¹³⁵

100= 1.35 and so on.

4. Conversion of Fraction to Percentage

A fraction can be converted into percentage by multiplying by 100.

For eg; ⁷

10× 100 = 70%

³

5× 100 = 60%

¹²

25× 100 = 48% and so on.

5. Some frequently used prime fractions:

1% = ¹

100= 0.01 5% = ⁵

100= 0.05 10% = ¹⁰

100 = ¹

10 = 0.1 12¹

2% = ⁵⁵

2 % = ²⁵

200= ¹

8= 0.125 20% = ²⁰

100 =¹

5= 0.2 25% = ²⁵

100 =¹

4= 0.25 33¹

3% = ¹⁰⁰

3 % =¹⁰⁰

300

50% = ⁵⁰

100 =¹

2= 0.5 75% = ⁷⁵

100 = 0.75 100% =¹⁰⁰

100= 1 EXAMPLES:

Example 1: In an organisation, monthly salary of clerks and officers are in the ratio of 3 : 5. Each clerk contributes 2% and each officer contributes 3% of his salary to the welfare fund. If each officer’s contribution is Rs. 210, what will be the clerk’s salary.

Solution:

Let the salary of a clerk be Rs. 3𝑥 and the salary of an officer be Rs. 5𝑥.

∴ 3% 𝑜𝑓 5𝑥 = ³

100× 5𝑥 =^3𝑥

20 But ^3𝑥

20 = 210

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∴ 3𝑥 = 210 × 20 = 4200

= 𝑅𝑠. 4200

Example 2: The income of Suresh is 40% more than the income of Ramesh. How much percent is Ramesh’s income less than that of Suresh?

Solution:

The required percentage = ^40×100

140 = 28.57%.

Example 3: With an increase of Rs. 2000, Raju’s monthly salary became Rs. 12000.

What is the percent increase in his salary?

Solution:

Initial salary = 12000 − 2000 = 𝑅𝑠. 10000 Therefore, Percentage increase = ^2000×100

10000 = 20%.

Example 4: If the price of milk is increased by 20%, find out how much percent a house-hold must reduce his consumptions so as not to increase his expenditure?

Solution:

Let the original price of milk = Re. 1 Now, increase in price = 20%.

∴ Increased price = 1 ×¹²⁰

100= 𝑅𝑠.⁶

5

Now, to keep the expenses remain same, the hold must consume only ⁵

6 of the original amount (the reciprocal of ⁶

5 =⁵

6)

∴ The difference = 1 −⁵

6 =¹

6

Thus, ¹

6× 100 = ¹⁰⁰

6 = 16²

3 %.

Example 5: B’s monthly income is 30% more than that of A. A’s monthly income is 10%

less than that of C. If the monthly income of C is Rs. 3300, then what is the monthly income of A?

Solution:

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|www.myexamcoaching.com 4 Given that, monthly income of C = 𝑅𝑠. 3300.

Therefore monthly income of A = 3300 − (10% 𝑜𝑓 3300)

= 3300 − 330

= 𝑅𝑠. 2970 EXERCISE:

1. Rohit earns Rs. 13800 per month. He spends Rs. 4400 on food, Rs. 1840 on travelling and 10% of his monthly income on the education of his children. If he saves the remaining money, then how much does he save in a year?

(a) Rs. 70000 (b) Rs. 74160 (c) Rs. 80000 (d) Rs. 73560 (e) None of these

2. 33% marks are required to pass an examination. A candidate who gets 210 marks fails by 21 marks. The total marks in the examination are:

(a) 500 (b) 600 (c) 700 (d) 800 (e) None of these

3. A reduction of 20% in the price of apples enables a woman to buy 5 apples more for Rs. 100. The price of an apple before reduction is.

(a) Rs. 2 (b) Rs. 4 (c) Rs. 5 (d) Rs. 6 (e) None of these

4. The difference between one-fifth of 1000 and one-fifth percent of 1000 is (a) 0 (b) 18 (c) 180

(d) 198 (e) None of these

5. In an examination 65% of the students passed. The number of students failed was 400. Find the total number of students?

6. A village cost 12% of its goats in a flood and 5% of the remainder died from disease. If the number left over is 8360, what was the original number before the flood?

(a) 8800 (b)10000 (c) 10800 (d) 11200 (e) None of these

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|www.myexamcoaching.com 5 7. A student has to secure 40% marks to pass. He gets 178 marks and fails by 22

marks. Find the maximum marks.

8. The population in a city increases at the rate of 5% annually. Its present population is 1,85,220. What was its population 3 years ago.

(a) 1,60,000 (b) 1,62,000 (c) 1,63,500 (d) 1,65,000 (e) None of these

9. If A’s height is 10% more than B’s height, how much percent less is B’s height than that of A?

(a) 9 ¹

11 % (b) 8 ⁷

11 % (c) 9 ³

11 % (b) 8 ³

11 % (e) None of these

10. What will come in place of the question mark (?)

√5476+? = 16% 𝑜𝑓 1800

11. A litre of water is evaporated from 6 litre of salt solution containing 5% of salt.

What will be the percentage of salt in the remaining solutions?

(a) 4% (b) 5% (c) 6%

(d) 7% (e) None of these

12. A company engaged a sales person on Rs. 10000 per month. Besides this he was authorized to get 15% commission of the monthly sale. How much sale he should make in a year so that he may get Rs. 22000 per month.

13. 35% 𝑜𝑓 ⁹

21 is equal to

(a) 25 (b) 15 (c) 1.5 (d) 0.15 (e) None of these

14. 36 × 14 is what % of 1400

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|www.myexamcoaching.com 6 15. What is the percentage of whole numbers from 1 to 10, which are exactly

multiples of 3.

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|www.myexamcoaching.com 7 ANSWERS (HINTS)

1. Monthly income of Rohit = 𝑅𝑠. 13800 Monthly exp.= (4400 + 1840 + 13800 × ¹⁰

100)

= 7620

∴ Monthly saving = 13800 − 7620 = 6180

∴ Saving in a year = 6180 × 12 = 74160

∴ 𝑅𝑠 = 74160.

Thus correct answer is (b).

2. Let the total mark in the examination = 𝑥 Therefore pass mark = 𝑥 × ³³

100 =^33𝑥

100

Now, as per the question, ^33𝑥

100− 21 = 210

∴ 33𝑥 − 2100 = 21000 33𝑥 = 22100

∴ 𝑥 =²²¹⁰⁰

33 = 700.

Hence correct answer is (c).

3. Going to reduction in price of apples, there is a saving of 20% on 𝑅𝑠. 100, which is Rs. 20. Now with this Rs. 20, she can purchase 5 apples more.

∴ Reduced rate of an apple =²⁰

5 = 𝑅𝑠. 4.

Now, if the reduced price is Rs. 4, then the original price before reduction = ¹⁰⁰

80 × 4 = 5

= 𝑅𝑠. 5 .

∴ The correct answer is (c).

4. One-fifth of 1000 = ¹

5× 1000 = 200 One-fifth percent of 1000 = ¹

5% 𝑜𝑓 1000 = ^0.2

100× 1000 = 2

∴ The difference = 200 − 2 = 198

∴ The correct answer is (d).

5. Since 65% students passed, the failed students are 35%

Now, 35% = 400

∴ 100% =¹⁰⁰

35 × 400 = 1143

∴ correct answer is (e).

6. Let the total number of goats = 𝑥

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|www.myexamcoaching.com 8 𝑥 − [12% 𝑜𝑓 𝑥] − [5% 𝑜𝑓 88% 𝑜𝑓 𝑥] = 8360

𝑥 − ¹²

100𝑥 − ⁵

100× ⁸⁸

100𝑥 = 8360

10000𝑥 − 1200𝑥 − 440𝑥 = 83600000 8360𝑥 = 83600000

∴ 𝑥 = 10000

∴ Correct answer is (b).

7. The student got 178 marks and failed by 22.

∴ pass mark = 178 + 22 = 200

∴ 40% = 200

∴ Total mark = 100% =²⁰⁰

40 × 100 = 500

∴ Correct answer is (b).

8. Present population = 1,85,220 Rate of growth per annum = 5%

Hence, population 3 years ago 𝑃 [1 − ^𝑅

100]^𝑁 = 185200 [1 − ⁵

100]³ = 185200 [1 − ¹

20]³

= 185200 ×19 20×19

20×19 20

= 1,60,000 Correct answer is (a).

9. Let B’s height = 𝑥

∴ A’s height =¹¹⁰

100𝑥 =^11𝑥

10

∴ Height difference ^11𝑥

10 − 𝑥 = ^𝑥

10

∴ Required % =

𝑥 10 11𝑥

10

× 100 = ¹

11× 100

∴ = 9 ¹

11 %

Correct answer is (a).

10. √5476+? = 16% 1800 74+? = 288

∴ ? = 288 − 74 = 214 Correct answer is (b).

11. Let the percentage of salt in the remaining solution be 𝑥. When the water is evaporated, the quantity of salt remains the same.

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|www.myexamcoaching.com 9 Therefore, 6 × ⁵

100= 5 × ^𝑥

100

𝑥 = 6%

The correct answer is (c).

12. Monthly salary = 𝑅𝑠. 10000 Now monthly salary 𝑅𝑠. 22000

∴ commission = 22000 − 10000 = 12000 Now let 𝑥 be the monthly sale

∴ 𝑥 × ¹⁵

100= 12000 𝑥 = 12000 ×¹⁰⁰

15 = 𝑅𝑠. 80000.

𝐴𝑛𝑛𝑢𝑎𝑙𝑙𝑦 = 80000 × 12

∴ Annual sale = 960000.

The correct answer is (d).

13. 35% 𝑜𝑓 ⁹

21 = 35 × ⁹

21 = 15 Correct answer is (b).

14. ^36×14

1400 × 100 = 36

∴ Correct answer is (a).

15. From 1 to 10, there are 10 numbers, out of which 3, 6, 9 are multiples of 3. There are total 3 multiples

∴ ³

10× 100 = 30%.