## Concepts for solving percentage problems

Percentage, symbolically %, means what out of 100.

Say 5% is 5 out of 100.

Percentage can be expressed in fraction or decimal too.

5\% = \displaystyle \frac{5}{100} = 0.05

More generally, x\% = \displaystyle \frac{x}{100} = 0.01x

Again if percentage is required for a fraction or decimal just multiply the value with 100.

## Solved problems on percentage

**Question**: If 20% of 30% of x = 3.6, x =?

(a) 36

(b) 600

(c) 30

(d) 60

**Ans**: (d) 60

**Explanation**:

20% of 30% of x = 3.6

\Rightarrow 0.2 × 0.3 × x = 3.6

\Rightarrow 0.06x = 3.6

\Rightarrow x = \displaystyle \frac{3.6}{0.06} = \displaystyle \frac{36\times 100}{6\times 10} = 60

**Question**: If 20% of a = b, then b% of 20 is same as

(a) 20% of a

(b) 10% of a

(c) 4% of a

(d) 40% of a

**Ans**: (c) 4% of a

**Explanation**:

20% of a = b

\Rightarrow b% = 0.2a%

\Rightarrow b% of 20 = 20 × 0.2a% = 4a%

**Question**: What percentage of numbers from 1 to 70 have 1 or 9 in the unit digit?

(a) \displaystyle 20\frac{2}{3}\%

(b) 20\%

(c) 21\%

(d) \displaystyle 22\frac{2}{3}\%

**Ans**: (b) 20%

**Explanation**:

Numbers with unit digit 1 \rightarrow 1, 11, 21, 31, 41, 51, 61 = 7 numbers

Numbers with unit digit 9 \rightarrow 9, 19, 29, 39, 49, 59, 69 = 7 numbers

Total numbers with unit digit 1 or 9 = 14

∴ Required percentage = \displaystyle \frac{14}{70}\times 100 = 20\%

**Question**: A’s income is 70% of B’s. B’s income is 50% of C. If C’s income is 100000. What is A’s income?

(a) Rs. 3500

(b) Rs. 3000

(c) Rs. 35000

(d) Rs. 350

**Ans**: (c) Rs. 35000

**Explanation**:

A = 70% of B and B = 50% of C

∴ A = 70% of 50% of C = 0.7 × 0.5 × 100000 = 35000

**Question**: A batsman scored 110 runs which included 3 boundaries and 8 sixes. What percent of his total score did he make by running between the wickets?

(a) \displaystyle 45\frac{4}{11}\%

(b) \displaystyle 45\%

(c) \displaystyle 45\frac{5}{11}\%

(d) \displaystyle 44\frac{5}{11}\%

**Ans**: (c) \displaystyle 45\frac{5}{11}\%

**Explanation**:

Runs between wickets = 110 – (3 × 4 + 8 × 6) = 50

∴ Required percentage = \displaystyle \frac{50}{110}\times 100 = \displaystyle 45\frac{5}{11}\%

**Question**: In an election between two candidates, one got 55% of total valid votes, 20% of the votes were invalid. If the total number of votes was 7500, what was the number of valid votes that the other candidates got?

(a) 2800

(b) 2700

(c) 2100

(d) 2500

**Ans**: (d) 2700

**Explanation**:

Valid votes = 80% of 7500 = \displaystyle \frac{80}{100} \times 7500 = 6000

Winner gets 55% of valid votes

So, loser gets 45% of valid votes

∴ Vote by loser =\displaystyle \frac{45}{100} \times 6000 = 2700

**Question**: In a competitive examination in state A, 6% candidates got selected from the total appeared candidates. State B had an equal number of candidates appeared and 7% candidates got selected with 80 more candidates got selected than A. What was the number of candidates appeared from each state?

(a) 8200

(b) 7500

(c) 7000

(d) 8000

**Ans**: (d) 8000

**Explanation**:

1% = 80

∴ 100% = 8000

**Question**: Arun got 30% of the maximum marks in an examination and failed by 10 marks. However, Sujith who took the same examination got 40% of the total marks and got 15 marks more than the passing marks. What were the passing marks in the examination?

(a) 90

(b) 250

(c) 75

(d) 85

**Ans**: (b) 250

**Explanation**:

10% = 25

100% = 250

**Question**: If 58% of workers in an organization are men and the number of men exceeds that of women by 288, find the number of women.

(a) 800

(b) 746

(c) 756

(d) 656

**Ans**: (c) 756

**Explanation**:

M = 58%

W = 42%

Difference = 16%

16% = 288

\Rightarrow 42% = \displaystyle \frac{288}{16} \times 42 = 756

**Question**: A ‘s salary is 10% less than that of the salary of B. B’s salary is what percentage more than A?

(a) 9%

(b) 9.09%

(c) 10%

(d) 11.11%

**Ans**: (d) 11.11%

**Explanation**:

**Question**: The price of onion is increased by 30%, by what percentage must a family reduce the consumption of onion so as to keep same expenditure?

(a) 30%

(b) 23.07%

(c) 29.27%

(d) 40%

**Ans**: (b) 23.07%

**Explanation**:

**Question**: Due to rise in price of salt by 20%, a housekeeper decreases his consumption. But by what percent should he do it so as to keep expense on this account same?

(a) 15%

(b) 12%

(c) 16.66%

(d) 220%

**Ans**: (c) 16.66%

**Explanation**:

**Question**: The reduction of price of sugar by 20% enables a man to purchase 5 kg more sugar for Rs. 400. Find the original price per kg of sugar.

(a) Rs. 20

(b) Rs. 25

(c) Rs. 18

(d) Rs.30

**Ans**: (a) Rs. 20

**Explanation**:

Let original price = Rs. x/kg

Sugar in Rs 400 = \displaystyle \frac{400}{x} kg

New CP = Rs. 0.8x/kg

Sugar in Rs 400 now = \displaystyle \frac{400}{0.8x} kg

Now,

**Question**: John’s salary was decreased by 50% and subsequently increased by 50%. How much percent does he loss?

(a) 35%

(b) 25%

(c) 32%

(d) 28%

**Ans**: (b) 25%

**Explanation**:

By applying ab method,

(-50, 50) \rightarrow -50 + 50 + \displaystyle\frac{-50\times 50}{100} = – 25%

So, 25% loss.

**Question**: If the price of petrol increases by 25% and Benson intends to spend only an additional 15% on petrol, by how much % will he reduce the quantity of petrol purchased?

(a) 8%

(b) 7%

(c) 10%

(d) 6%

**Ans**: (a) 8%

**Explanation**:

Let Benson spends ₹ x for petrol at ₹ y/lit

So, Petrol purchased = \displaystyle \frac{x}{y} lit

Now, spent on petrol = ₹1.15x at ₹1.25y/lit

So, Petrol purchased = \displaystyle \frac{1.15x}{1.25y} = \displaystyle \frac{23x}{25y} lit

∴ Reduction in petrol purchase = \displaystyle \frac{\displaystyle\frac{x}{y}-\displaystyle\frac{23x}{25y}}{\displaystyle\frac{x}{y}}\times 100 = \displaystyle \frac{2}{8}\times 100 = 8%

**Aliter**:

Price needed is 125%

But he intends to spend 115%

So, Reduction required = \displaystyle \frac{125-115}{125}\times 100 = 8%

**Question**: The population of a town increases by 10% in the first year & 12% in the second year. At the beginning of third year what has been net increase of population?

(a) 23.1%

(b) 22%

(c) 23%

(d) 23.2%

**Ans**: (d) 23.2%

**Explanation**:

By applying ab method,

(10, 12) ⟶ 22 + 1.2 = 23.2%

**Question**: The area of a rectangle first increases by 10% & is then decreased by 15%. What was the net change in area?

(a) 6% increase

(b) 6.5% decrease

(c) 6.5% increase

(d) 5.5% increase

**Ans**: (b) 6.5% decrease

**Explanation**:

By applying ab method

(10, -15) ⟶ 10 – 15 – 1.5 = – 6.5%

**Question**: The salary of a person decreased by 10% and then increased by a certain % if the net percentage increase is 8%, what was the second increase?

(a) 10%

(b) 12%

(c) 20%

(d) 16%

**Ans**: (c) 20%

**Explanation**:

(-10, x) ⟶ 8%

\Rightarrow -10 + x – 0.1x = 8

\Rightarrow 0.9x = 18

\Rightarrow x = 20

**Question**: Rahul went to a shop and bought things worth Rs. 25, out of which 30 paisa went on sales tax on taxable purchases. If the tax rate was 6%, then what was the cost of the tax free items?

(a) Rs.15

(b) Rs.12.10

(c) Rs.19.70

(d) Rs.16.80

**Ans**: (c) Rs.19.70

**Explanation**:

Total amount paid = (cost of taxable item + tax) + cost of tax free item

Tax = 30 paisa = Rs. 0.3

Tax rate is 6% i.e., 0.3 is 6% of cost of taxable item

∴ 6% = 0.3

\Rightarrow 100% = \displaystyle \frac{0.3}{6}\times 100 = 5

∴ cost of taxable item = Rs. 5

Now, using the starting equation,

25 = (5 + 0.3) + cost of tax free item

\Rightarrow cost of tax free item = 19.7

**Question**: A man spends 10% of his income in home rent, 40% on food and 25% on miscellaneous. If he saves Rs. 4000, what is his income?

(a) Rs. 14000

(b) Rs. 15000

(c) Rs. 16000

(d) Rs. 17000

**Ans**: (c) Rs. 16000

**Explanation**:

Saving = 100 – (10 +40 +25) = 25%

∴ 25% = 4000

\Rightarrow 100% = 16000

**Question**: A man spends 20% of his income on house rent, 40% of the remainder on food and miscellaneous. If he saves Rs 9600, find his income.

(a) Rs 20000

(b) Rs 25000

(c) Rs 10000

(d) Rs 30000

**Ans**: (a) 20000

**Explanation**:

Let income = Rs. x

x \times \displaystyle \frac{80}{100}\times \displaystyle \frac{60}{100} = 9600

\Rightarrow x = \displaystyle \frac{9600\times 100\times 100}{80\times 60}= 20000

**Aliter**:

Applying ab method

(- 20, – 40) ⟶ – 60 + 8 = – 52%

Savings is 48%

48% = 9600

100% = 20000

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