Percentage and its Applications

Convert the following percents into fractions in simplest form:

(i) 25%

(ii) 150%

(iii) $7\dfrac{1}{2}$%

(iv) $33\dfrac{1}{3}$%

Answer

To convert a percentage into a fraction, replace the % sign with $\dfrac{1}{100}$ and reduce the fraction to simplest form.

(i) 25%

$\Rightarrow 25 \times \dfrac{1}{100}\\[1em] \Rightarrow \dfrac{25}{100}\\[1em] \Rightarrow \dfrac{1}{4}$

Hence, 25% = $\dfrac{1}{4}$.

(ii) 150%

$\Rightarrow 150 \times \dfrac{1}{100}\\[1em] \Rightarrow \dfrac{150}{100}\\[1em] \Rightarrow \dfrac{3}{2}$

Hence, 150% = $\dfrac{3}{2}$.

(iii) $7\dfrac{1}{2}$%

$\Rightarrow \dfrac{15}{2}\\[1em] \Rightarrow \dfrac{15}{2} \times \dfrac{1}{100}\\[1em] \Rightarrow \dfrac{15}{200}\\[1em] \Rightarrow \dfrac{3}{40}$

Hence, $7\dfrac{1}{2}$% $= \dfrac{3}{40}$.

(iv) $33\dfrac{1}{3}$%

$\Rightarrow \dfrac{100}{3}\\[1em] \Rightarrow \dfrac{100}{3} \times \dfrac{1}{100}\\[1em] \Rightarrow \dfrac{100}{300}\\[1em] \Rightarrow \dfrac{1}{3}$

Hence, $33\dfrac{1}{3}$% $= \dfrac{1}{3}$.

Convert the following fractions into percents:

(i) $\dfrac{1}{8}$

(ii) $\dfrac{5}{4}$

(iii) $\dfrac{9}{16}$

(iv) $\dfrac{3}{7}$

(v) $\dfrac{11}{15}$

(vi) $1\dfrac{3}{8}$

Answer

To convert a fraction into a percentage, multiply the fraction by 100 and put the % sign.

(i) $\dfrac{1}{8}$

$\Rightarrow \dfrac{100}{8}$%

$\Rightarrow \left(\dfrac{1}{8} \times 100\right)$%

$\Rightarrow 12.5$%

Hence, $\dfrac{1}{8} = 12.5$%.

(ii) $\dfrac{5}{4}$

$\Rightarrow \left(\dfrac{5}{4} \times 100\right)$%

$\Rightarrow \dfrac{500}{4}$%

$\Rightarrow 125$%

Hence, $\dfrac{5}{4} = 125$%.

(iii) $\dfrac{9}{16}$

$\Rightarrow \left(\dfrac{9}{16} \times 100\right)$%

$\Rightarrow \dfrac{900}{16}$%

$\Rightarrow \dfrac{225}{4}$%

$\Rightarrow 56\dfrac{1}{4}$%

Hence, $\dfrac{9}{16} = 56\dfrac{1}{4}$%.

(iv) $\dfrac{3}{7}$

$\Rightarrow \left(\dfrac{3}{7} \times 100\right)$%

$\Rightarrow \dfrac{300}{7}$%

$\Rightarrow 42\dfrac{6}{7}$%

Hence, $\dfrac{3}{7} = 42\dfrac{6}{7}$%.

(v) $\dfrac{11}{15}$

$\Rightarrow \left(\dfrac{11}{15} \times 100\right)$%

$\Rightarrow \dfrac{1100}{15}$%

$\Rightarrow \dfrac{220}{3}$%

$\Rightarrow 73\dfrac{1}{3}$%

Hence, $\dfrac{11}{15} = 73\dfrac{1}{3}$%.

(vi) $1\dfrac{3}{8}$

$\Rightarrow \dfrac{11}{8}$

$\Rightarrow \left(\dfrac{11}{8} \times 100\right)$%

$\Rightarrow \dfrac{1100}{8}$%

$\Rightarrow 137.5$%

$\Rightarrow 137\dfrac{1}{2}$%

Hence, $1\dfrac{3}{8} = 137\dfrac{1}{2}$%.

6 students out of 40 students in a class are absent. What percentage of the students are absent?

Answer

Number of students absent = 6 and total number of students = 40.

Percentage of students absent = $\left(\dfrac{6}{40} \times 100\right)$%

$\Rightarrow \dfrac{600}{40}$%

$\Rightarrow 15$%

Hence, the percentage of students absent = 15%.

Antony secured 384 marks out of 500 marks. Find the percentage of marks secured by Antony.

Answer

Marks secured by Antony = 384 out of 500.

Percentage of marks = $\left(\dfrac{384}{500} \times 100\right)$%

$\Rightarrow \dfrac{38400}{500}$%

$\Rightarrow 76.8$%

Hence, the percentage of marks secured by Antony = 76.8%.

A shop has 500 shirts, out of which 15 are defective. What percentage of shirts are defective?

Answer

Number of defective shirts = 15 and total number of shirts = 500.

Percentage of defective shirts = $\left(\dfrac{15}{500} \times 100\right)$%

$\Rightarrow \dfrac{1500}{500}$%

$\Rightarrow 3$%

Hence, the percentage of defective shirts = 3%.

Vani has a collection of bangles. She has 20 gold bangles and 10 silver bangles. What is the percentage of each type of bangles?

Answer

Number of gold bangles = 20 and number of silver bangles = 10.

Total number of bangles = 20 + 10 = 30.

Percentage of gold bangles = $\left(\dfrac{20}{30} \times 100\right)$%

$\Rightarrow \dfrac{2000}{30}$%

$\Rightarrow \dfrac{200}{3}$%

$\Rightarrow 66\dfrac{2}{3}$%

Percentage of silver bangles = $\left(\dfrac{10}{30} \times 100\right)$%

$\Rightarrow \dfrac{1000}{30}$%

$\Rightarrow \dfrac{100}{3}$%

$\Rightarrow 33\dfrac{1}{3}$%

Hence, gold bangles = $66\dfrac{2}{3}$% and silver bangles = $33\dfrac{1}{3}$%.

There are 120 voters, 90 of them voted. What percent did not vote?

Answer

Total number of voters = 120 and number who voted = 90.

Number who did not vote = 120 − 90 = 30.

Percentage who did not vote = $\left(\dfrac{30}{120} \times 100\right)$%

$\Rightarrow \dfrac{3000}{120}$%

$\Rightarrow 25$%

Hence, the percentage of voters who did not vote = 25%.

Estimate the part of the figure which is shaded and hence find the percentage of the part which is shaded.

Answer

(i) From the figure, shaded part = $\dfrac{3}{4}$.

Percentage of shaded part = $\left(\dfrac{3}{4} \times 100\right)$% = 75%.

Hence, shaded part = $\dfrac{3}{4}$ and percentage of shaded part = 75%.

(ii) From the figure, shaded part = $\dfrac{2}{6} = \dfrac{1}{3}$.

Percentage of shaded part = $\left(\dfrac{1}{3} \times 100\right)$% $= 33\dfrac{1}{3}$%.

Hence, shaded part = $\dfrac{1}{3}$ and percentage of shaded part = $33\dfrac{1}{3}$%.

(iii) From the figure, shaded part = $\dfrac{5}{8}$.

Percentage of shaded part = $\left(\dfrac{5}{8} \times 100\right)$% = 62.5%.

Hence, shaded part = $\dfrac{5}{8}$ and percentage of shaded part = 62.5%.

Convert the following percentages into ratios in simplest form:

(i) 14%

(ii) $1\dfrac{3}{4}$%

(iii) $33\dfrac{1}{3}$%

(iv) 37.5%

Answer

To convert a percentage into a ratio, first convert the percentage into a fraction in simplest form and then write it as a ratio.

(i) 14%

$\Rightarrow \dfrac{14}{100}\\[1em] \Rightarrow \dfrac{7}{50}\\[1em] \Rightarrow 7 : 50$

Hence, 14% = 7 : 50.

(ii) $1\dfrac{3}{4}$%

$\Rightarrow \dfrac{7}{4}$%

$\Rightarrow \dfrac{7}{4} \times \dfrac{1}{100}$

$\Rightarrow \dfrac{7}{400}$

$\Rightarrow 7 : 400$

Hence, $1\dfrac{3}{4}$% = 7 : 400.

(iii) $33\dfrac{1}{3}$%

$\Rightarrow \dfrac{100}{3}$%

$\Rightarrow \dfrac{100}{3} \times \dfrac{1}{100}$

$\Rightarrow \dfrac{1}{3}$

$\Rightarrow 1 : 3$

Hence, $33\dfrac{1}{3}$% = 1 : 3.

(iv) 37.5%

$\Rightarrow \dfrac{37.5}{100}\\[1em] \Rightarrow \dfrac{375}{1000}\\[1em] \Rightarrow \dfrac{3}{8}\\[1em] \Rightarrow 3 : 8$

Hence, 37.5% = 3 : 8.

Express the following ratios as percentages:

(i) 5 : 4

(ii) 1 : 1

(iii) 2 : 3

(iv) 9 : 16

Answer

To convert a ratio into a percentage, first convert the ratio into a fraction and then to a percentage.

(i) 5 : 4

$\Rightarrow \dfrac{5}{4}$

$\Rightarrow \left(\dfrac{5}{4} \times 100\right)$%

$\Rightarrow 125$%

Hence, 5 : 4 = 125%.

(ii) 1 : 1

$\Rightarrow \dfrac{1}{1}$

$\Rightarrow \left(\dfrac{1}{1} \times 100\right)$%

$\Rightarrow 100$%

Hence, 1 : 1 = 100%.

(iii) 2 : 3

$\Rightarrow \dfrac{2}{3}$

$\Rightarrow \left(\dfrac{2}{3} \times 100\right)$%

$\Rightarrow \dfrac{200}{3}$%

$\Rightarrow 66\dfrac{2}{3}$%

Hence, 2 : 3 = $66\dfrac{2}{3}$%.

(iv) 9 : 16

$\Rightarrow \dfrac{9}{16}$

$\Rightarrow \left(\dfrac{9}{16} \times 100\right)$%

$\Rightarrow \dfrac{900}{16}$%

$\Rightarrow 56\dfrac{1}{4}$%

Hence, 9 : 16 = $56\dfrac{1}{4}$%.

An alloy consists of 7 parts of zinc and 33 parts of copper. Find the percentage of copper in the alloy.

Answer

Parts of zinc = 7 and parts of copper = 33.

Total number of parts = 7 + 33 = 40.

Percentage of copper = $\left(\dfrac{33}{40} \times 100\right)$%

$\Rightarrow \dfrac{3300}{40}$%

$\Rightarrow 82.5$%

Hence, the percentage of copper in the alloy = 82.5%.

Chalk contains calcium, carbon and sand in the ratio 12 : 3 : 10. Find the percentage of carbon in the chalk.

Answer

Ratio of calcium : carbon : sand = 12 : 3 : 10.

Total number of parts = 12 + 3 + 10 = 25.

Percentage of carbon = $\left(\dfrac{3}{25} \times 100\right)$%

$\Rightarrow \dfrac{300}{25}$%

$\Rightarrow 12$%

Hence, the percentage of carbon in the chalk = 12%.

If ₹ 2500 is to be divided among Ravi, Raju and Roy, so that Ravi gets two parts, Raju three parts and Roy five parts. How much money will each get? What will it be in percentages?

Answer

Total amount = ₹ 2500.

Ravi : Raju : Roy = 2 : 3 : 5.

Total number of parts = 2 + 3 + 5 = 10.

Ravi's share = $\dfrac{2}{10} \times 2500 = ₹\ 500$.

Raju's share = $\dfrac{3}{10} \times 2500 = ₹\ 750$.

Roy's share = $\dfrac{5}{10} \times 2500 = ₹\ 1250$.

Ravi's percentage = $\left(\dfrac{2}{10} \times 100\right)$% = 20%.

Raju's percentage = $\left(\dfrac{3}{10} \times 100\right)$% = 30%.

Roy's percentage = $\left(\dfrac{5}{10} \times 100\right)$% = 50%.

Hence, Ravi gets ₹ 500 (20%), Raju gets ₹ 750 (30%) and Roy gets ₹ 1250 (50%).

Convert the following percentages to decimals:

(i) 28%

(ii) 3%

(iii) 0.44%

(iv) $37\dfrac{1}{2}$%

Answer

To convert a percentage into a decimal, replace the % sign with $\dfrac{1}{100}$ and then express the fraction as a decimal.

(i) 28%

$\Rightarrow \dfrac{28}{100}$

$\Rightarrow 0.28$

Hence, 28% = 0.28.

(ii) 3%

$\Rightarrow \dfrac{3}{100}$

$\Rightarrow 0.03$

Hence, 3% = 0.03.

(iii) 0.44%

$\Rightarrow \dfrac{0.44}{100}$

$\Rightarrow 0.0044$

Hence, 0.44% = 0.0044.

(iv) $37\dfrac{1}{2}$%

$\Rightarrow \dfrac{75}{2}$%

$\Rightarrow \dfrac{75}{2} \times \dfrac{1}{100}$

$\Rightarrow \dfrac{75}{200}$

$\Rightarrow 0.375$

Hence, $37\dfrac{1}{2}$% = 0.375.

Convert the following decimals to percents:

(i) 0.65

(ii) 0.9

(iii) 2.1

(iv) 0.02

Answer

To convert a decimal into a percentage, multiply the decimal by 100 and put the % sign.

(i) 0.65

$\Rightarrow (0.65 \times 100)$%

$\Rightarrow 65$%

Hence, 0.65 = 65%.

(ii) 0.9

$\Rightarrow (0.9 \times 100)$%

$\Rightarrow 90$%

Hence, 0.9 = 90%.

(iii) 2.1

$\Rightarrow (2.1 \times 100)$%

$\Rightarrow 210$%

Hence, 2.1 = 210%.

(iv) 0.02

$\Rightarrow (0.02 \times 100)$%

$\Rightarrow 2$%

Hence, 0.02 = 2%.

Find:

15% of 250

Answer

15% of 250

$\Rightarrow \dfrac{15}{100} \times 250\\[1em] \Rightarrow \dfrac{3750}{100}\\[1em] \Rightarrow 37.5$

Hence, 15% of 250 = 37.5.

Find:

25% of 120 litres

Answer

25% of 120 litres

$\Rightarrow \dfrac{25}{100} \times 120\\[1em] \Rightarrow \dfrac{3000}{100}\\[1em] \Rightarrow 30 \text{ litres}$

Hence, 25% of 120 litres = 30 litres.

Find:

1% of 1 hour

Answer

1% of 1 hour

We know, 1 hour = 60 minutes.

$\Rightarrow \dfrac{1}{100} \times 60 \text{ minutes}\\[1em] \Rightarrow \dfrac{60}{100} \text{ minutes}\\[1em] \Rightarrow 0.6 \text{ minutes}\\[1em] \Rightarrow 0.6 \times 60 \text{ seconds}\\[1em] \Rightarrow 36 \text{ seconds}$

Hence, 1% of 1 hour = 36 seconds.

Find:

75% of 1 kg

Answer

75% of 1 kg

We know, 1 kg = 1000 g.

$\Rightarrow \dfrac{75}{100} \times 1000\\[1em] \Rightarrow \dfrac{75000}{100}\\[1em] \Rightarrow 750 \text{ g}$

Hence, 75% of 1 kg = 750 g.

Find:

120% of ₹ 250

Answer

120% of ₹ 250

$\Rightarrow \dfrac{120}{100} \times 250\\[1em] \Rightarrow \dfrac{30000}{100}\\[1em] \Rightarrow ₹\ 300$

Hence, 120% of ₹ 250 = ₹ 300.

Find:

0.6% of 2 km

Answer

0.6% of 2 km

We know, 1 km = 1000 m, so 2 km = 2000 m.

$\Rightarrow \dfrac{0.6}{100} \times 2000\\[1em] \Rightarrow \dfrac{1200}{100}\\[1em] \Rightarrow 12 \text{ m}$

Hence, 0.6% of 2 km = 12 m.

8% children of a class of 25 like getting wet in the rain. How many children like getting wet in the rain?

Answer

Total number of children = 25.

Number of children who like getting wet in the rain = 8% of 25

$\Rightarrow \dfrac{8}{100} \times 25\\[1em] \Rightarrow \dfrac{200}{100}\\[1em] \Rightarrow 2$

Hence, 2 children like getting wet in the rain.

Vasundara ate 3 ice cream cups out of 20 kept in the fridge. What percent did she eat?

Answer

Number of ice cream cups eaten = 3 out of 20.

Required percentage = $\left(\dfrac{3}{20} \times 100\right)$%

$= \dfrac{300}{20}$%

$= 15$%

Hence, Vasundara ate 15% of the ice cream cups.

Express:

(i) 20 as a percentage of 50

(ii) 60 litres as a percentage of 40 litres

(iii) 90 cm as a percentage of 4.5 m

(iv) 350 g as a percentage of 5.6 kg

Answer

To express one quantity as a percentage of another quantity, use percentage = $\left(\dfrac{\text{one quantity}}{\text{second quantity}} \times 100\right)$%, taking both quantities in the same units.

(i) 20 as a percentage of 50

$\Rightarrow \left(\dfrac{20}{50} \times 100\right)$%

$\Rightarrow \dfrac{2000}{50}$%

$\Rightarrow 40$%

Hence, 20 is 40% of 50.

(ii) 60 litres as a percentage of 40 litres

$\Rightarrow \left(\dfrac{60}{40} \times 100\right)$%

$\Rightarrow \dfrac{6000}{40}$%

$\Rightarrow 150$%

Hence, 60 litres is 150% of 40 litres.

(iii) 90 cm as a percentage of 4.5 m

We know, 1 m = 100 cm, so 4.5 m = 450 cm.

$\Rightarrow \left(\dfrac{90}{450} \times 100\right)$%

$\Rightarrow \dfrac{9000}{450}$%

$\Rightarrow 20$%

Hence, 90 cm is 20% of 4.5 m.

(iv) 350 g as a percentage of 5.6 kg

We know, 1 kg = 1000 g, so 5.6 kg = 5600 g.

$\Rightarrow \left(\dfrac{350}{5600} \times 100\right)$%

$\Rightarrow \dfrac{35000}{5600}$%

$\Rightarrow \dfrac{25}{4}$%

$\Rightarrow 6\dfrac{1}{4}$%

Hence, 350 g is $6\dfrac{1}{4}$% of 5.6 kg.

What percent is:

(i) 12 of 80

(ii) 25 paise of 4 rupees

(iii) 300 g of 2 kg

Answer

(i) 12 of 80

$\Rightarrow \left(\dfrac{12}{80} \times 100\right)$%

$\Rightarrow \dfrac{1200}{80}$%

$\Rightarrow 15$%

Hence, 12 is 15% of 80.

(ii) 25 paise of 4 rupees

We know, 1 rupee = 100 paise, so 4 rupees = 400 paise.

$\Rightarrow \left(\dfrac{25}{400} \times 100\right)$%

$\Rightarrow \dfrac{2500}{400}$%

$\Rightarrow \dfrac{25}{4}$%

$\Rightarrow 6\dfrac{1}{4}$%

Hence, 25 paise is $6\dfrac{1}{4}$% of 4 rupees.

(iii) 300 g of 2 kg

We know, 1 kg = 1000 g, so 2 kg = 2000 g.

$\Rightarrow \left(\dfrac{300}{2000} \times 100\right)$%

$\Rightarrow \dfrac{30000}{2000}$%

$\Rightarrow 15$%

Hence, 300 g is 15% of 2 kg.

A school team won 6 games this year against 4 games won last year. What is the percent increase?

Answer

Games won last year = 4 and games won this year = 6.

Increase in games won = 6 − 4 = 2.

Percentage increase = $\left(\dfrac{\text{increase in value}}{\text{original value}} \times 100\right)$%

$\Rightarrow \left(\dfrac{2}{4} \times 100\right)$%

$\Rightarrow \dfrac{200}{4}$%

$\Rightarrow 50$%

Hence, the percent increase = 50%.

The price of an article decreased from ₹ 80 to ₹ 60, find the percentage of decrease in the price of the article.

Answer

Original price = ₹ 80 and new price = ₹ 60.

Decrease in price = 80 − 60 = ₹ 20.

Percentage decrease = $\left(\dfrac{\text{decrease in value}}{\text{original value}} \times 100\right)$%

$\Rightarrow \left(\dfrac{20}{80} \times 100\right)$%

$\Rightarrow \dfrac{2000}{80}$%

$\Rightarrow 25$%

Hence, the percentage decrease in price = 25%.

My grandmother says, in her childhood petrol was ₹ 1 per litre. It is ₹ 95 per litre today. By what percentage has the prices of petrol gone up?

Answer

Original price = ₹ 1 per litre and present price = ₹ 95 per litre.

Increase in price = 95 − 1 = ₹ 94.

Percentage increase = $\left(\dfrac{94}{1} \times 100\right)$%

= 9400%

Hence, the price of petrol has gone up by 9400%.

The price of tomatoes last year was ₹ 40 per kg. This year they are costly by 20%. What is the price this year?

Answer

Last year's price = ₹ 40 per kg.

Increase in price = 20% of 40 = $\dfrac{20}{100} \times 40 = ₹\ 8$.

Price this year = 40 + 8 = ₹ 48.

Hence, the price of tomatoes this year = ₹ 48 per kg.

300 students took an exam. 28% failed. Calculate the number of students who passed the exam.

Answer

Total number of students = 300.

Percentage of students who failed = 28%.

Percentage of students who passed = (100 − 28)% = 72%.

Number of students who passed = 72% of 300

$\Rightarrow \dfrac{72}{100} \times 300\\[1em] \Rightarrow \dfrac{21600}{100}\\[1em] \Rightarrow 216$

Hence, 216 students passed the exam.

Out of 15000 voters in a constituency, 60% voted. Find the number of voters who did not vote.

Answer

Total number of voters = 15000.

Percentage of voters who voted = 60%.

Percentage of voters who did not vote = (100 − 60)% = 40%.

Number of voters who did not vote = 40% of 15000

$\Rightarrow \dfrac{40}{100} \times 15000\\[1em] \Rightarrow \dfrac{600000}{100}\\[1em] \Rightarrow 6000$

Hence, 6000 voters did not vote.

20% of length of a flagpole is painted green, 45% is painted yellow and the remaining red. If the length of the pole is 18 m, what length of it is painted red?

Answer

Length of the flagpole = 18 m.

Percentage painted green = 20% and percentage painted yellow = 45%.

Percentage painted red = 100% − (20% + 45%) = 100% − 65% = 35%.

Length painted red = 35% of 18 m

$\Rightarrow \dfrac{35}{100} \times 18\\[1em] \Rightarrow \dfrac{630}{100}\\[1em] \Rightarrow 6.3 \text{ m}$

Hence, the length painted red = 6.3 m.

Chalk contains 10% calcium, 3% carbon, 12% oxygen and the remaining sand. Find the amount of carbon and calcium (in grams) in $2\dfrac{1}{2}$ kg of chalk. Also find the amount of sand (in kg).

Answer

Total quantity of chalk = $2\dfrac{1}{2}$ kg = 2.5 kg = 2500 g.

Amount of carbon = 3% of 2500 g

$\Rightarrow \dfrac{3}{100} \times 2500\\[1em] \Rightarrow 75 \text{ g}$

Amount of calcium = 10% of 2500 g

$\Rightarrow \dfrac{10}{100} \times 2500\\[1em] \Rightarrow 250 \text{ g}$

Percentage of sand = 100% − (10% + 3% + 12%) = 100% − 25% = 75%.

Amount of sand = 75% of 2500 g

$\Rightarrow \dfrac{75}{100} \times 2500\\[1em] \Rightarrow 1875 \text{ g}\\[1em] \Rightarrow 1.875 \text{ kg}$

Hence, carbon = 75 g, calcium = 250 g and sand = 1.875 kg.

Find the whole quantity if:

(i) 25% of it is 9

(ii) 75% of it is 15

(iii) 12% of it is ₹ 1080

(iv) 8% of it is 40 litres

Answer

(i) 25% of it is 9

Let the whole quantity be x.

Given,

25% of x = 9

$\Rightarrow \dfrac{25}{100} \times x = 9 \\[1em] \Rightarrow x = \dfrac{9 \times 100}{25} \\[1em] \Rightarrow x = 36.$

Hence, the whole quantity = 36.

(ii) 75% of it is 15

Let the whole quantity be x.

$\Rightarrow \dfrac{75}{100} \times x = 15\\[1em] \Rightarrow x = \dfrac{15 \times 100}{75}\\[1em] \Rightarrow x = 20$

Hence, the whole quantity = 20.

(iii) 12% of it is ₹ 1080

Let the whole quantity be ₹ x.

$\Rightarrow \dfrac{12}{100} \times x = 1080\\[1em] \Rightarrow x = \dfrac{1080 \times 100}{12}\\[1em] \Rightarrow x = 9000$

Hence, the whole quantity = ₹ 9000.

(iv) 8% of it is 40 litres

Let the whole quantity be x litres.

$\Rightarrow \dfrac{8}{100} \times x = 40\\[1em] \Rightarrow x = \dfrac{40 \times 100}{8}\\[1em] \Rightarrow x = 500$

Hence, the whole quantity = 500 litres.

Mohini saves ₹ 4000 from her salary. If this is 10% of her salary, then what is her salary?

Answer

Let Mohini's salary be ₹ x.

Given, 10% of her salary = ₹ 4000.

$\Rightarrow \dfrac{10}{100} \times x = 4000\\[1em] \Rightarrow x = \dfrac{4000 \times 100}{10}\\[1em] \Rightarrow x = 40000$

Hence, Mohini's salary = ₹ 40,000.

16% of the apples in a basket go bad. If there are 42 good apples in the basket, find the total number of apples in the basket.

Answer

Percentage of apples that go bad = 16%.

Percentage of good apples = (100 − 16)% = 84%.

Let the total number of apples be x.

84% of x = 42

$\Rightarrow \dfrac{84}{100} \times x = 42 \\[1em] \Rightarrow x = \dfrac{42 \times 100}{84} \\[1em] \Rightarrow x = 50$

Hence, the total number of apples in the basket = 50.

In an examination, a student has to secure 45% marks to pass the exam. If Varun got 251 marks and failed by 19 marks, what are the maximum marks?

Answer

Marks scored by Varun = 251 and marks needed to pass = 19 more.

Pass marks = 251 + 19 = 270.

Let the maximum marks be x. Since 45% of the maximum marks are needed to pass,

45% of x = 270

$\Rightarrow \dfrac{45}{100} \times x = 270\\[1em] \Rightarrow x = \dfrac{270 \times 100}{45}\\[1em] \Rightarrow x = 600$

Hence, the maximum marks = 600.

On a rainy day, 94% of the students were present in a school. If the number of students absent on that day was 174, find the total strength of the school.

Answer

Percentage of students present = 94%.

Percentage of students absent = (100 − 94)% = 6%.

Let the total strength of the school be x.

6% of x = 174

$\Rightarrow \dfrac{6}{100} \times x = 174\\[1em] \Rightarrow x = \dfrac{174 \times 100}{6}\\[1em] \Rightarrow x = 2900$

Hence, the total strength of the school = 2900.

40% of the population of a town are men and 39% are women. If the number of children is 12600, find the number of men.

Answer

Percentage of men = 40% and percentage of women = 39%.

Percentage of children = 100% − (40% + 39%) = 100% − 79% = 21%.

Let the total population be x.

21% of x = 12600

$\Rightarrow \dfrac{21}{100} \times x = 12600\\[1em] \Rightarrow x = \dfrac{12600 \times 100}{21}\\[1em] \Rightarrow x = 60000$

Number of men = 40% of 60000

$\Rightarrow \dfrac{40}{100} \times 60000\\[1em] \Rightarrow 24000$

Hence, the number of men = 24000.

If the price of a watch is increased by 15%, the increase in the price is ₹ 90. What was the price of watch earlier?

Answer

Let the earlier price of the watch be ₹ x.

Given, 15% of the price = ₹ 90.

$\Rightarrow \dfrac{15}{100} \times x = 90\\[1em] \Rightarrow x = \dfrac{90 \times 100}{15}\\[1em] \Rightarrow x = 600$

Hence, the earlier price of the watch = ₹ 600.

(i) Find the number which when increased by 30% becomes 39.

(ii) Find the number which when decreased by 8% becomes 506.

Answer

(i) Let the number be x.

When increased by 30%, the number becomes 130% of x.

130% of x = 39

$\Rightarrow \dfrac{130}{100} \times x = 39\\[1em] \Rightarrow x = \dfrac{39 \times 100}{130}\\[1em] \Rightarrow x = 30$

Hence, the required number = 30.

(ii) Let the number be x.

When decreased by 8%, the number becomes (100 − 8)% = 92% of x.

92% of x = 506

$\Rightarrow \dfrac{92}{100} \times x = 506\\[1em] \Rightarrow x = \dfrac{506 \times 100}{92}\\[1em] \Rightarrow x = 550$

Hence, the required number = 550.

The price of a shirt is reduced by 7% to ₹ 465. What is its original price?

Answer

Let the original price of the shirt be ₹ x.

When reduced by 7%, the price becomes (100 − 7)% = 93% of x.

93% of x = 465

$\Rightarrow \dfrac{93}{100} \times x = 465\\[1em] \Rightarrow x = \dfrac{465 \times 100}{93}\\[1em] \Rightarrow x = 500$

Hence, the original price of the shirt = ₹ 500.

If 15% of 60 is greater than 25% of a number by 3, then find the number.

Answer

15% of 60 = $\dfrac{15}{100} \times 60 = 9$.

Let the number be x.

Given, 15% of 60 is greater than 25% of x by 3.

$\Rightarrow 9 = 25$% of x + 3

$\Rightarrow 25$% of x = 9 - 3

$\Rightarrow \dfrac{25}{100} \times x = 6$

$\Rightarrow x = \dfrac{6 \times 100}{25}$

$\Rightarrow x = 24$

Hence, the required number = 24.

A 60 litre tank was full of petrol. Peter used 30% of it and poured the rest into a 50 litre tank.

(i) What percent of 50 litre tank was filled with petrol?

(ii) If Peter used 2.8 litres of petrol daily, what percent of petrol in the 50 litre tank would be used in 10 days?

Answer

Petrol in the full tank = 60 litres.

Petrol used = 30% of 60 = $\dfrac{30}{100} \times 60 = 18$ litres.

Petrol poured into the 50 litre tank = 60 − 18 = 42 litres.

(i) Percentage of 50 litre tank filled = $\left(\dfrac{42}{50} \times 100\right)$%

$\Rightarrow \dfrac{4200}{50}$%

$\Rightarrow 84$%

Hence, 84% of the 50 litre tank was filled with petrol.

(ii) Petrol used in 10 days = 2.8 × 10 = 28 litres.

Petrol present in the 50 litre tank = 42 litres.

Percentage of petrol used = $\left(\dfrac{28}{42} \times 100\right)$%

$\Rightarrow \dfrac{2800}{42}$%

$\Rightarrow \dfrac{200}{3}$%

$\Rightarrow 66\dfrac{2}{3}$%

Hence, $66\dfrac{2}{3}$% of the petrol in the 50 litre tank would be used in 10 days.

Rohan bought a calculator for ₹ 760 and sold it for ₹ 874. Find his profit and profit percentage.

Answer

C.P. of the calculator = ₹ 760 and S.P. of the calculator = ₹ 874.

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

Profit = S.P. − C.P. = 874 − 760 = ₹ 114.

Profit percentage = $\left(\dfrac{\text{profit}}{\text{cost price}} \times 100\right)$%

$\Rightarrow \left(\dfrac{114}{760} \times 100\right)$%

$\Rightarrow \dfrac{11400}{760}$%

$\Rightarrow 15$%

Hence, profit = ₹ 114 and profit percentage = 15%.

Kirti bought a saree for ₹ 2500 and sold it for ₹ 2300. Find her loss and loss percent.

Answer

C.P. of the saree = ₹ 2500 and S.P. of the saree = ₹ 2300.

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

Loss = C.P. − S.P. = 2500 − 2300 = ₹ 200.

Loss percentage = $\left(\dfrac{\text{loss}}{\text{cost price}} \times 100\right)$%

$\Rightarrow \left(\dfrac{200}{2500} \times 100\right)$%

$\Rightarrow \dfrac{20000}{2500}$%

$\Rightarrow 8$%

Hence, loss = ₹ 200 and loss percent = 8%.

Calculate the profit or loss in the following transactions. Also find profit percent or loss percent in each case:

(i) Gardening shears bought for ₹ 250 and sold for ₹ 325

(ii) A shirt bought for ₹ 250 and sold at ₹ 150

Answer

(i) C.P. of the shears = ₹ 250 and S.P. of the shears = ₹ 325.

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

Profit = S.P. − C.P. = 325 − 250 = ₹ 75.

Profit percentage = $\left(\dfrac{75}{250} \times 100\right)$%

$\Rightarrow \dfrac{7500}{250}$%

$\Rightarrow 30$%

Hence, profit = ₹ 75 and profit percent = 30%.

(ii) C.P. of the shirt = ₹ 250 and S.P. of the shirt = ₹ 150.

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

Loss = C.P. − S.P. = 250 − 150 = ₹ 100.

Loss percentage = $\left(\dfrac{100}{250} \times 100\right)$%

$\Rightarrow \dfrac{10000}{250}$%

$\Rightarrow 40$%

Hence, loss = ₹ 100 and loss percent = 40%.

Rajinder bought one almirah for ₹ 4800 and the other for ₹ 3640. He sold the first almirah at a gain of $13\dfrac{1}{3}$% and the other at a loss of 15%. How much did he gain or lose in the whole deal?

Answer

First almirah:

C.P. = ₹ 4800 and gain = $13\dfrac{1}{3}$% = $\dfrac{40}{3}$%.

Gain = $\dfrac{40}{3}$% of 4800 = $\dfrac{40}{3 \times 100} \times 4800 = ₹\ 640$.

S.P. of first almirah = 4800 + 640 = ₹ 5440.

Second almirah:

C.P. = ₹ 3640 and loss = 15%.

Loss = 15% of 3640 = $\dfrac{15}{100} \times 3640 = ₹\ 546$.

S.P. of second almirah = 3640 − 546 = ₹ 3094.

Whole deal:

Total C.P. = 4800 + 3640 = ₹ 8440.

Total S.P. = 5440 + 3094 = ₹ 8534.

Since total S.P. > total C.P., there is a gain.

Gain = 8534 − 8440 = ₹ 94.

Hence, Rajinder gained ₹ 94 in the whole deal.

In a furniture shop, 24 tables were bought at the rate of ₹ 4500 per table. The shopkeeper sold 16 of them at the rate of ₹ 6000 per table and the remaining at the rate of ₹ 4000 per table. Find his gain or loss percent.

Answer

Total C.P. = 24 × 4500 = ₹ 108000.

S.P. of 16 tables = 16 × 6000 = ₹ 96000.

Remaining tables = 24 − 16 = 8.

S.P. of 8 tables = 8 × 4000 = ₹ 32000.

Total S.P. = 96000 + 32000 = ₹ 128000.

Since total S.P. > total C.P., there is a gain.

Gain = 128000 − 108000 = ₹ 20000.

Gain percentage = $\left(\dfrac{20000}{108000} \times 100\right)$%

$\Rightarrow \dfrac{2000000}{108000}$%

$\Rightarrow \dfrac{500}{27}$%

$\Rightarrow 18\dfrac{14}{27}$%

Hence, the shopkeeper has a gain of $18\dfrac{14}{27}$%.

By selling a lamp for ₹ 810, a dealer makes a profit of ₹ 60. What is the cost price of the lamp? What is his profit percent?

Answer

S.P. of the lamp = ₹ 810 and profit = ₹ 60.

C.P. = S.P. − profit = 810 − 60 = ₹ 750.

Profit percentage = $\left(\dfrac{60}{750} \times 100\right)$%

$\Rightarrow \dfrac{6000}{750}$%

$\Rightarrow 8$%

Hence, the cost price of the lamp = ₹ 750 and profit percent = 8%.

By selling a jacket for ₹ 3906, a manufacturer suffers a loss of ₹ 294. Find the cost price of the jacket and his loss percentage.

Answer

S.P. of the jacket = ₹ 3906 and loss = ₹ 294.

C.P. = S.P. + loss = 3906 + 294 = ₹ 4200.

Loss percentage = $\left(\dfrac{294}{4200} \times 100\right)$%

$\Rightarrow \dfrac{29400}{4200}$%

$\Rightarrow 7$%

Hence, the cost price of the jacket = ₹ 4200 and loss percentage = 7%.

The cost price of a vase is ₹ 120. If the shopkeeper sells it at a loss of 10%, find the price at which it was sold.

Answer

C.P. of the vase = ₹ 120 and loss = 10%.

Loss = 10% of 120 = $\dfrac{10}{100} \times 120 = ₹\ 12$.

S.P. = C.P. − loss = 120 − 12 = ₹ 108.

Hence, the vase was sold for ₹ 108.

I buy a T.V. for ₹ 10000 and sell it at a profit of 20%. How much money do I get for it?

Answer

C.P. of the T.V. = ₹ 10000 and profit = 20%.

Profit = 20% of 10000 = $\dfrac{20}{100} \times 10000 = ₹\ 2000$.

S.P. = C.P. + profit = 10000 + 2000 = ₹ 12000.

Hence, I get ₹ 12000 for the T.V.

A shopkeeper sells an article for ₹ 300, thus earning a profit of 20%. Find the cost price of the article.

Answer

S.P. of the article = ₹ 300 and profit = 20%.

Let the C.P. be ₹ x.

S.P. = C.P. + 20% of C.P. = (100 + 20)% of C.P. = 120% of C.P.

$\Rightarrow \dfrac{120}{100} \times x = 300\\[1em] \Rightarrow x = \dfrac{300 \times 100}{120}\\[1em] \Rightarrow x = 250$

Hence, the cost price of the article = ₹ 250.

A shopkeeper sells an article for ₹ 320, thus suffering a loss of 20%. Find the cost price of the article.

Answer

S.P. of the article = ₹ 320 and loss = 20%.

Let the C.P. be ₹ x.

S.P. = C.P. − 20% of C.P. = (100 − 20)% of C.P. = 80% of C.P.

$\Rightarrow \dfrac{80}{100} \times x = 320\\[1em] \Rightarrow x = \dfrac{320 \times 100}{80}\\[1em] \Rightarrow x = 400$

Hence, the cost price of the article = ₹ 400.

By selling a chair for ₹ 522, a shopkeeper makes a profit of 16%. What is its cost price?

Answer

S.P. of the chair = ₹ 522 and profit = 16%.

Let the C.P. be ₹ x.

S.P. = (100 + 16)% of C.P. = 116% of C.P.

$\Rightarrow \dfrac{116}{100} \times x = 522\\[1em] \Rightarrow x = \dfrac{522 \times 100}{116}\\[1em] \Rightarrow x = 450$

Hence, the cost price of the chair = ₹ 450.

A trader sold some damaged garments for ₹ 7360 at a loss of 8%. Find the cost price of the garments.

Answer

S.P. of the garments = ₹ 7360 and loss = 8%.

Let the C.P. be ₹ x.

S.P. = (100 − 8)% of C.P. = 92% of C.P.

$\Rightarrow \dfrac{92}{100} \times x = 7360\\[1em] \Rightarrow x = \dfrac{7360 \times 100}{92}\\[1em] \Rightarrow x = 8000$

Hence, the cost price of the garments = ₹ 8000.

By selling a table for ₹ 3168, Rashid loses 12%. Find its cost price. What percent would he gain or lose by selling the table for ₹ 3870?

Answer

S.P. of the table = ₹ 3168 and loss = 12%.

Let the C.P. be ₹ x.

S.P. = (100 − 12)% of C.P. = 88% of C.P.

$\Rightarrow \dfrac{88}{100} \times x = 3168\\[1em] \Rightarrow x = \dfrac{3168 \times 100}{88}\\[1em] \Rightarrow x = 3600$

So, the cost price of the table = ₹ 3600.

Now, if the table is sold for ₹ 3870:

Since 3870 > 3600, there is a gain.

Gain = 3870 − 3600 = ₹ 270.

Gain percentage = $\left(\dfrac{270}{3600} \times 100\right)$%

$\Rightarrow \dfrac{27000}{3600}$%

$\Rightarrow 7.5$%

Hence, the cost price = ₹ 3600 and he would gain 7.5% by selling it for ₹ 3870.

By selling an article for ₹ 4550, Tony incurs a loss of 9%. What percent would he gain or lose by selling it for ₹ 4825?

Answer

S.P. of the article = ₹ 4550 and loss = 9%.

Let the C.P. be ₹ x.

S.P. = (100 − 9)% of C.P. = 91% of C.P.

$\Rightarrow \dfrac{91}{100} \times x = 4550\\[1em] \Rightarrow x = \dfrac{4550 \times 100}{91}\\[1em] \Rightarrow x = 5000$

So, the cost price of the article = ₹ 5000.

Now, if the article is sold for ₹ 4825:

Since 5000 > 4825, there is a loss.

Loss = 5000 − 4825 = ₹ 175.

Loss percentage = $\left(\dfrac{175}{5000} \times 100\right)$%

$\Rightarrow \dfrac{17500}{5000}$%

$\Rightarrow 3.5$%

Hence, he would lose 3.5% by selling it for ₹ 4825.

Arif bought a second hand car for ₹ 80000 and spent 12.5% of the cost of the car on its repairs. At what price should he sell the car to make a profit of 15%?

Answer

Cost of the car = ₹ 80000.

Amount spent on repairs = 12.5% of 80000 = $\dfrac{12.5}{100} \times 80000 = ₹\ 10000$.

Total cost price = 80000 + 10000 = ₹ 90000.

To make a profit of 15%:

S.P. = (100 + 15)% of C.P. = 115% of 90000

$\Rightarrow \dfrac{115}{100} \times 90000\\[1em] \Rightarrow \dfrac{10350000}{100}\\[1em] \Rightarrow 103500$

Hence, Arif should sell the car for ₹ 103500.

Find the simple interest on:

(i) ₹ 350 for 2 years at 11% per annum

(ii) ₹ 20000 for $4\dfrac{1}{2}$ years at $8\dfrac{1}{2}$% per annum

(iii) ₹ 648 for 8 months at $16\dfrac{2}{3}$% per annum

Also find the amount in each case.

Answer

Simple interest is given by the formula $I = \dfrac{P \times R \times T}{100}$ and amount = principal + interest.

(i) Here, P = ₹ 350, R = 11% per annum and T = 2 years.

$\Rightarrow I = \dfrac{350 \times 11 \times 2}{100}\\[1em] \Rightarrow I = \dfrac{7700}{100}\\[1em] \Rightarrow I = ₹\ 77$

Amount = P + I = 350 + 77 = ₹ 427.

Hence, simple interest = ₹ 77 and amount = ₹ 427.

(ii) Here, P = ₹ 20000, R = $8\dfrac{1}{2} = \dfrac{17}{2}$% per annum and T = $4\dfrac{1}{2} = \dfrac{9}{2}$ years.

$\Rightarrow I = \dfrac{20000 \times \dfrac{17}{2} \times \dfrac{9}{2}}{100}\\[1em] \Rightarrow I = \dfrac{20000 \times 17 \times 9}{2 \times 2 \times 100}\\[1em] \Rightarrow I = \dfrac{3060000}{400}\\[1em] \Rightarrow I = ₹\ 7650$

Amount = P + I = 20000 + 7650 = ₹ 27650.

Hence, simple interest = ₹ 7650 and amount = ₹ 27650.

(iii) Here, P = ₹ 648, R = $16\dfrac{2}{3} = \dfrac{50}{3}$% per annum and T = 8 months = $\dfrac{8}{12} = \dfrac{2}{3}$ year.

$\Rightarrow I = \dfrac{648 \times \dfrac{50}{3} \times \dfrac{2}{3}}{100}\\[1em] \Rightarrow I = \dfrac{648 \times 50 \times 2}{3 \times 3 \times 100}\\[1em] \Rightarrow I = \dfrac{64800}{900}\\[1em] \Rightarrow I = ₹\ 72$

Amount = P + I = 648 + 72 = ₹ 720.

Hence, simple interest = ₹ 72 and amount = ₹ 720.

Find the time when:

(i) simple interest on ₹ 2500 at 4% per annum is ₹ 200

(ii) simple interest on ₹ 12000 at $6\dfrac{1}{2}$% per annum is ₹ 2730

Answer

Using $I = \dfrac{P \times R \times T}{100}$, we get $T = \dfrac{I \times 100}{P \times R}$.

(i) Here, P = ₹ 2500, R = 4% per annum and I = ₹ 200.

$\Rightarrow T = \dfrac{200 \times 100}{2500 \times 4}\\[1em] \Rightarrow T = \dfrac{20000}{10000}\\[1em] \Rightarrow T = 2 \text{ years}$

Hence, the time = 2 years.

(ii) Here, P = ₹ 12000, R = $6\dfrac{1}{2} = \dfrac{13}{2}$% per annum and I = ₹ 2730.

$\Rightarrow T = \dfrac{2730 \times 100}{12000 \times \dfrac{13}{2}}\\[1em] \Rightarrow T = \dfrac{2730 \times 100 \times 2}{12000 \times 13}\\[1em] \Rightarrow T = \dfrac{546000}{156000}\\[1em] \Rightarrow T = \dfrac{7}{2}\\[1em] \Rightarrow T = 3\dfrac{1}{2} \text{ years}$

Hence, the time = $3\dfrac{1}{2}$ years.

Find the rate of interest when:

(i) simple interest on ₹ 1560 in 3 years is ₹ 585

(ii) simple interest on ₹ 1625 in $2\dfrac{1}{2}$ years is ₹ 325

Answer

Using $I = \dfrac{P \times R \times T}{100}$, we get $R = \dfrac{I \times 100}{P \times T}$.

(i) Here, P = ₹ 1560, T = 3 years and I = ₹ 585.

$\Rightarrow R = \dfrac{585 \times 100}{1560 \times 3}$%

$\Rightarrow R = \dfrac{58500}{4680}$%

$\Rightarrow R = \dfrac{25}{2}$%

$\Rightarrow R = 12\dfrac{1}{2}$%

Hence, the rate of interest = $12\dfrac{1}{2}$% per annum.

(ii) Here, P = ₹ 1625, T = $2\dfrac{1}{2} = \dfrac{5}{2}$ years and I = ₹ 325.

$\Rightarrow R = \dfrac{325 \times 100}{1625 \times \dfrac{5}{2}}$%

$\Rightarrow R = \dfrac{325 \times 100 \times 2}{1625 \times 5}$%

$\Rightarrow R = \dfrac{65000}{8125}$%

$\Rightarrow R = 8$%

Hence, the rate of interest = 8% per annum.

Find the principal when:

(i) simple interest at 16% per annum for $2\dfrac{1}{2}$ years is ₹ 3840

(ii) simple interest at $7\dfrac{1}{2}$% per annum for 2 years 4 months is ₹ 2730

Answer

Using $I = \dfrac{P \times R \times T}{100}$, we get $P = \dfrac{I \times 100}{R \times T}$.

(i) Here,

    R = 16% per annum,

    T = $2\dfrac{1}{2} = \dfrac{5}{2}$ years and

     I = ₹ 3840.

$\Rightarrow P = \dfrac{3840 \times 100}{16 \times \dfrac{5}{2}}\\[1em] \Rightarrow P = \dfrac{3840 \times 100 \times 2}{16 \times 5}\\[1em] \Rightarrow P = \dfrac{768000}{80}\\[1em] \Rightarrow P = ₹\ 9600$

Hence, the principal = ₹ 9600.

(ii) Here,

     R = $7\dfrac{1}{2} = \dfrac{15}{2}$% per annum,

     T = 2 years 4 months = $2\dfrac{4}{12} = \dfrac{7}{3}$ years and

     I = ₹ 2730.

$\Rightarrow P = \dfrac{2730 \times 100}{\dfrac{15}{2} \times \dfrac{7}{3}}\\[1em] \Rightarrow P = \dfrac{2730 \times 100 \times 2 \times 3}{15 \times 7}\\[1em] \Rightarrow P = \dfrac{1638000}{105}\\[1em] \Rightarrow P = ₹\ 15600$

Hence, the principal = ₹ 15600.

Find the rate of interest when:

(i) ₹ 1200 amounts to ₹ 1320 in 2 years

(ii) ₹ 300 amounts to ₹ 400 in 2 years

Answer

(i) Here, P = ₹ 1200, A = ₹ 1320 and T = 2 years.

I = A − P = 1320 − 1200 = ₹ 120.

$\Rightarrow R = \dfrac{I \times 100}{P \times T}$%

$\Rightarrow R = \dfrac{120 \times 100}{1200 \times 2}$%

$\Rightarrow R = \dfrac{12000}{2400}$%

$\Rightarrow R = 5$%

Hence, the rate of interest = 5% per annum.

(ii) Here, P = ₹ 300, A = ₹ 400 and T = 2 years.

I = A − P = 400 − 300 = ₹ 100.

$\Rightarrow R = \dfrac{100 \times 100}{300 \times 2}$%

$\Rightarrow R = \dfrac{10000}{600}$%

$\Rightarrow R = \dfrac{50}{3}$%

$\Rightarrow R = 16\dfrac{2}{3}$%

Hence, the rate of interest = $16\dfrac{2}{3}$% per annum.

Find the time when:

(i) ₹ 1250 amounts to ₹ 1950 at 16% per annum

(ii) ₹ 6540 amounts to ₹ 8447.50 at $12\dfrac{1}{2}$% per annum

Answer

(i) Here,

P = ₹ 1250, A = ₹ 1950 and R = 16% per annum.

I = A − P = 1950 − 1250 = ₹ 700.

$\Rightarrow T = \dfrac{I \times 100}{P \times R}\\[1em] \Rightarrow T = \dfrac{700 \times 100}{1250 \times 16}\\[1em] \Rightarrow T = \dfrac{70000}{20000}\\[1em] \Rightarrow T = \dfrac{7}{2}\\[1em] \Rightarrow T = 3\dfrac{1}{2} \text{ years}$

Hence, the time = $3\dfrac{1}{2}$ years.

(ii) Here,

P = ₹ 6540, A = ₹ 8447.50 and R = $12\dfrac{1}{2} = \dfrac{25}{2}$% per annum.

I = A − P = 8447.50 − 6540 = ₹ 1907.50.

$\Rightarrow T = \dfrac{1907.50 \times 100}{6540 \times \dfrac{25}{2}}\\[1em] \Rightarrow T = \dfrac{1907.50 \times 100 \times 2}{6540 \times 25}\\[1em] \Rightarrow T = \dfrac{381500}{163500}\\[1em] \Rightarrow T = \dfrac{7}{3}\\[1em] \Rightarrow T = 2\dfrac{1}{3} \text{ years} = 2 \text{ years } 4 \text{ months}$

Hence, the time = 2 years 4 months.

₹ 14000 is invested at 4% per annum simple interest. How long will it take for the amount to reach ₹ 16240?

Answer

Here,

P = ₹ 14000, A = ₹ 16240 and R = 4% per annum.

I = A − P = 16240 − 14000 = ₹ 2240.

$\Rightarrow T = \dfrac{I \times 100}{P \times R}\\[1em] \Rightarrow T = \dfrac{2240 \times 100}{14000 \times 4}\\[1em] \Rightarrow T = \dfrac{224000}{56000}\\[1em] \Rightarrow T = 4 \text{ years}$

Hence, it will take 4 years.

An amount of money invested trebled in 6 years. Find the rate of interest earned.

Answer

Let the principal be ₹ P.

Since the money trebled, amount A = 3P.

Interest I = A − P = 3P − P = 2P.

Here, T = 6 years.

$\Rightarrow R = \dfrac{I \times 100}{P \times T}$%

$\Rightarrow R = \dfrac{2P \times 100}{P \times 6}$%

$\Rightarrow R = \dfrac{200}{6}$%

$\Rightarrow R = \dfrac{100}{3}$%

$\Rightarrow R = 33\dfrac{1}{3}$%

Hence, the rate of interest = $33\dfrac{1}{3}$% per annum.

Find the principal when:

(i) final amount is ₹ 4500 at 20% per annum for 5 years

(ii) final amount is ₹ 2420 at 4% per annum for $2\dfrac{1}{2}$ years

Answer

(i) Here,

A = ₹ 4500, R = 20% per annum and T = 5 years.

Let the principal be ₹ P.

$\Rightarrow I = \dfrac{P \times 20 \times 5}{100} = P\\[1em] \Rightarrow A = P + I = P + P = 2P\\[1em] \Rightarrow 2P = 4500\\[1em] \Rightarrow P = \dfrac{4500}{2}\\[1em] \Rightarrow P = ₹\ 2250$

Hence, the principal = ₹ 2250.

(ii) Here,

A = ₹ 2420, R = 4% per annum and T = $2\dfrac{1}{2} = \dfrac{5}{2}$ years.

Let the principal be ₹ P.

$\Rightarrow I = \dfrac{P \times 4 \times \dfrac{5}{2}}{100} = \dfrac{P \times 10}{100} = \dfrac{P}{10}\\[1em] \Rightarrow A = P + I = P + \dfrac{P}{10} = \dfrac{11P}{10}\\[1em] \Rightarrow \dfrac{11P}{10} = 2420\\[1em] \Rightarrow P = \dfrac{2420 \times 10}{11}\\[1em] \Rightarrow P = ₹\ 2200$

Hence, the principal = ₹ 2200.

If the simple interest on a certain sum of money for 3 years is three-tenth of the sum, then find the rate of interest per annum.

Answer

Let the sum (principal) be ₹ P and T = 3 years.

Given, simple interest I = $\dfrac{3}{10}$ of P = $\dfrac{3P}{10}$.

$\Rightarrow R = \dfrac{I \times 100}{P \times T}$%

$\Rightarrow R = \dfrac{\dfrac{3P}{10} \times 100}{P \times 3}$%

$\Rightarrow R = \dfrac{3P \times 100}{10 \times P \times 3}$%

$\Rightarrow R = \dfrac{300}{30}$%

$\Rightarrow R = 10$%

Hence, the rate of interest = 10% per annum.

What sum of money will amount to ₹ 2760 in 3 years at 5% per annum simple interest?

Answer

Here,

A = ₹ 2760, T = 3 years and R = 5% per annum.

Let the sum (principal) be ₹ P.

$\Rightarrow I = \dfrac{P \times 5 \times 3}{100} = \dfrac{15P}{100} = \dfrac{3P}{20}\\[1em] \Rightarrow A = P + I = P + \dfrac{3P}{20} = \dfrac{23P}{20}\\[1em] \Rightarrow \dfrac{23P}{20} = 2760\\[1em] \Rightarrow P = \dfrac{2760 \times 20}{23}\\[1em] \Rightarrow P = ₹\ 2400$

Hence, the required sum = ₹ 2400.

A sum of ₹ 6000 amounts to ₹ 6900 in 3 years. What will it amount to if the rate of interest is increased by 2%?

Answer

Here,

P = ₹ 6000, A = ₹ 6900 and T = 3 years.

I = A − P = 6900 − 6000 = ₹ 900.

$\Rightarrow R = \dfrac{I \times 100}{P \times T}$%

$\Rightarrow R = \dfrac{900 \times 100}{6000 \times 3}$%

$\Rightarrow R = \dfrac{90000}{18000}$%

$\Rightarrow R = 5$%

New rate of interest = 5% + 2% = 7% per annum.

New interest = $\dfrac{6000 \times 7 \times 3}{100} = \dfrac{126000}{100} = ₹\ 1260$.

New amount = P + new interest = 6000 + 1260 = ₹ 7260.

Hence, the sum will amount to ₹ 7260.

Fill in the blanks:

(i) 6% of ₹ 50 = ....

(ii) If 25% of a number is 12, then the number is ....

(iii) The mixed fraction $1\dfrac{3}{4}$ converted to percentage form is ....

(iv) If a number increases from 20 to 28, then the increase percentage is ....

(v) If cost price is ₹ 400 and loss is 15%, then selling price is ....

(vi) The profit or loss percentage is always calculated on .....

(vii) The simple interest on a sum of ₹ 5600 at 8% p.a. for one year is ....

(viii) 135% converted to decimals is ....

(ix) .... is 50% more than 60

(x) 25 mL is .... percent of 5 litres.

Answer

(i) 6% of ₹ 50 = $\dfrac{6}{100} \times 50 = ₹\ 3$.

Hence, 6% of ₹ 50 = ₹ 3.

(ii) Let the number be x. Then 25% of x = 12, so $x = \dfrac{12 \times 100}{25} = 48$.

Hence, the number is 48.

(iii) $1\dfrac{3}{4} = \dfrac{7}{4} = \left(\dfrac{7}{4} \times 100\right)$% = 175%.

Hence, $1\dfrac{3}{4}$ = 175%.

(iv) Increase = 28 − 20 = 8. Increase percentage = $\left(\dfrac{8}{20} \times 100\right)$% = 40%.

Hence, the increase percentage is 40%.

(v) Loss = 15% of 400 = $\dfrac{15}{100} \times 400 = ₹\ 60$. Selling price = 400 − 60 = ₹ 340.

Hence, the selling price is ₹ 340.

(vi) Hence, the profit or loss percentage is always calculated on cost price (C.P.).

(vii) S.I. = $\dfrac{5600 \times 8 \times 1}{100} = ₹\ 448$.

Hence, the simple interest is ₹ 448.

(viii) 135% = $\dfrac{135}{100} = 1.35$.

Hence, 135% = 1.35.

(ix) 50% of 60 = 30. So the number = 60 + 30 = 90.

Hence, 90 is 50% more than 60.

(x) 5 litres = 5000 mL. Required percentage = $\left(\dfrac{25}{5000} \times 100\right)$% = 0.5%.

Hence, 25 mL is 0.5 percent of 5 litres.

State whether the following statements are true (T) or false (F):

(i) 20% more than 30 is 36

(ii) The ratio 2 : 5 converted to percentage is 60%

(iii) $6\dfrac{1}{4}$% expressed as a fraction is $\dfrac{1}{16}$

(iv) 80% of 450 m is equal to 360 m

(v) If a number decreases from 20 to 15, then the decrease is 25%

(vi) If Feroz obtains 336 marks out of 600 marks, then percentage of marks obtained by him is 33.6

(vii) 0.018 is equivalent to 8%

(viii) 250 cm is 4% of 1 km

(ix) If S.P. of an article is ₹ 540 and loss is ₹ 40, then its C.P. is ₹ 500

(x) By selling a book for ₹ 500, a shopkeeper suffers a loss of 10%. The cost price of the book is ₹ 600

Answer

(i) 20% of 30 = 6, so 20% more than 30 = 30 + 6 = 36. The statement is correct.

Hence, the statement is True.

(ii) 2 : 5 = $\dfrac{2}{5} = \left(\dfrac{2}{5} \times 100\right)$% = 40%, not 60%. The statement is incorrect.

Hence, the statement is False.

(iii) Solving,

$\Rightarrow 6\dfrac{1}{4}$%

$\Rightarrow \dfrac{25}{4}$%

$\Rightarrow \dfrac{25}{4} \times \dfrac{1}{100} = \dfrac{25}{400} = \dfrac{1}{16}$.

Thus, the statement is correct.

Hence, the statement is True.

(iv) 80% of 450 m = $\dfrac{80}{100} \times 450 = 360$ m. The statement is correct.

Hence, the statement is True.

(v) Decrease = 20 − 15 = 5. Decrease percentage = $\left(\dfrac{5}{20} \times 100\right)$% = 25%. The statement is correct.

Hence, the statement is True.

(vi) Percentage of marks = $\left(\dfrac{336}{600} \times 100\right)$% = 56%, not 33.6. The statement is incorrect.

Hence, the statement is False.

(vii) 0.018 = (0.018 × 100)% = 1.8%, not 8%. The statement is incorrect.

Hence, the statement is False.

(viii) 1 km = 100000 cm. Required percentage = $\left(\dfrac{250}{100000} \times 100\right)$% = 0.25%, not 4%. The statement is incorrect.

Hence, the statement is False.

(ix) C.P. = S.P. + loss = 540 + 40 = ₹ 580, not ₹ 500. The statement is incorrect.

Hence, the statement is False.

(x) S.P. = ₹ 500 at a loss of 10% means S.P. = 90% of C.P. So C.P. = $\dfrac{500 \times 100}{90} = ₹\ 555.56$ (approximately), not ₹ 600. The statement is incorrect.

Hence, the statement is False.

The ratio of Fatima's income to her saving is 4 : 1. The percentage of money saved by her is

1. 20%

2. 25%

3. 40%

4. 80%

Answer

Income : saving = 4 : 1.

Percentage of money saved = $\left(\dfrac{\text{saving}}{\text{income}} \times 100\right)$%

= $\left(\dfrac{1}{4} \times 100\right)$%

= 25%.

Hence, Option 2 is the correct option.

225% is equal to

1. 2 : 3

2. 3 : 2

3. 4 : 9

4. 9 : 4

Answer

225% = $\dfrac{225}{100} = \dfrac{9}{4} = 9 : 4$.

Hence, Option 4 is the correct option.

If 30% of x is 72, then x is equal to

1. 120

2. 240

3. 360

4. 480

Answer

30% of x = 72.

$\Rightarrow \dfrac{30}{100} \times x = 72\\[1em] \Rightarrow x = \dfrac{72 \times 100}{30}\\[1em] \Rightarrow x = 240$

Hence, Option 2 is the correct option.

If x% of 80 = 12, then x is equal to

1. 15

2. 20

3. 25

4. 30

Answer

x% of 80 = 12.

$\Rightarrow \dfrac{x}{100} \times 80 = 12\\[1em] \Rightarrow x = \dfrac{12 \times 100}{80}\\[1em] \Rightarrow x = 15$

Hence, Option 1 is the correct option.

0.025 when expressed as a percent is

1. 250%

2. 25%

3. 4%

4. 2.5%

Answer

0.025 = (0.025 × 100)% = 2.5%.

Hence, Option 4 is the correct answer.

In a class, 45% of students are girls. If there are 22 boys in the class, then the total number of students in the class is

1. 30

2. 36

3. 40

4. 44

Answer

Percentage of girls = 45%, so percentage of boys = (100 − 45)% = 55%.

Let the total number of students be x.

55% of x = 22

$\Rightarrow \dfrac{55}{100} \times x = 22 \Rightarrow x = \dfrac{22 \times 100}{55} \Rightarrow x = 40$

Hence, Option 3 is the correct option.

If a man buys an article for ₹ 80 and sells it for ₹ 100, then gain percentage is

1. 20%

2. 25%

3. 40%

4. 125%

Answer

C.P. = ₹ 80 and S.P. = ₹ 100.

Gain = 100 − 80 = ₹ 20.

Gain percentage = $\left(\dfrac{20}{80} \times 100\right)$% = 25%.

Hence, Option 2 is the correct option.

If a man buys an article for ₹ 120 and sells it for ₹ 100, then his loss percentage is

1. 10%

2. 20%

3. 25%

4. $16\dfrac{2}{3}$%

Answer

C.P. = ₹ 120 and S.P. = ₹ 100.

Loss = 120 − 100 = ₹ 20.

Loss percentage = $\left(\dfrac{20}{120} \times 100\right)$%

= $\dfrac{50}{3}$%

= $16\dfrac{2}{3}$%.

Hence, Option 4 is the correct answer.

The salary of a man is ₹ 24000 per month. If he gets an increase of 25% in the salary, then the new salary per month is

1. ₹ 2500

2. ₹ 28000

3. ₹ 30000

4. ₹ 36000

Answer

Increase = 25% of 24000 = $\dfrac{25}{100} \times 24000 = ₹\ 6000$.

New salary = 24000 + 6000 = ₹ 30000.

Hence, Option 3 is the correct option.

On selling an article for ₹ 100, Renu gains ₹ 20. Her gain percentage is

1. 25%

2. 20%

3. 15%

4. 40%

Answer

S.P. = ₹ 100 and gain = ₹ 20.

C.P. = S.P. − gain = 100 − 20 = ₹ 80.

Gain percentage = $\left(\dfrac{20}{80} \times 100\right)$% = 25%.

Hence, Option 1 is the correct option.

The simple interest on ₹ 6000 at 8% p.a. for one year is

1. ₹ 600

2. ₹ 480

3. ₹ 400

4. ₹ 240