ML Aggarwal Class 7 Maths Solutions Chapter 07 Percentage and Its Applications

Exercise 7.1

Question 1. Convert the following percents into fractions in simplest form:
(i) 25%
(ii) 150%
(iii) \( 7\frac{1}{2} \)%
(iv) \( 33\frac{1}{3} \)%
Answer: When converting a percentage to a fraction, drop the % sign and replace it with \( \frac{1}{100} \), then simplify the result to lowest terms.

(i) \( 25 \times \frac{1}{100} = \frac{25}{100} = \frac{1}{4} \)

Hence, 25% = \( \frac{1}{4} \)

(ii) \( 150 \times \frac{1}{100} = \frac{150}{100} = \frac{3}{2} \)

Hence, 150% = \( \frac{3}{2} \)

(iii) \( 7\frac{1}{2}\)% = \( \frac{15}{2} \) × \( \frac{1}{100} = \frac{15}{200} = \frac{3}{40} \)

Hence, \( 7\frac{1}{2} \)% = \( \frac{3}{40} \)

(iv) \( 33\frac{1}{3} \)% = \( \frac{100}{3} \) × \( \frac{1}{100} = \frac{100}{300} = \frac{1}{3} \)

Hence, \( 33\frac{1}{3} \)% = \( \frac{1}{3} \)
In simple words: To change a percent to a fraction, write the number over 100, then reduce it to its simplest form by dividing both top and bottom by their greatest common factor.

Exam Tip: Always reduce the final fraction to simplest form - examiners will mark it wrong if you leave it unreduced. For mixed number percentages, first convert to an improper fraction before multiplying by \( \frac{1}{100} \).

Question 2. Convert the following fractions into percents:
(i) \( \frac{1}{8} \)
(ii) \( \frac{5}{4} \)
(iii) \( \frac{9}{16} \)
(iv) \( \frac{3}{7} \)
(v) \( \frac{11}{15} \)
(vi) \( 1\frac{3}{8} \)
Answer: To change a fraction into a percentage, multiply the fraction by 100 and add the % sign.

(i) \( \frac{1}{8} \times 100 = \frac{100}{8} = 12.5 \)

Hence, \( \frac{1}{8} \) = 12.5%

(ii) \( \frac{5}{4} \times 100 = \frac{500}{4} = 125 \)

Hence, \( \frac{5}{4} \) = 125%

(iii) \( \frac{9}{16} \times 100 = \frac{900}{16} = \frac{225}{4} = 56\frac{1}{4} \)

Hence, \( \frac{9}{16} \) = \( 56\frac{1}{4} \)%

(iv) \( \frac{3}{7} \times 100 = \frac{300}{7} = 42\frac{6}{7} \)

Hence, \( \frac{3}{7} \) = \( 42\frac{6}{7} \)%

(v) \( \frac{11}{15} \times 100 = \frac{1100}{15} = \frac{220}{3} = 73\frac{1}{3} \)

Hence, \( \frac{11}{15} \) = \( 73\frac{1}{3} \)%

(vi) \( 1\frac{3}{8} = \frac{11}{8} \)

\( \frac{11}{8} \times 100 = \frac{1100}{8} = 137.5 = 137\frac{1}{2} \)

Hence, \( 1\frac{3}{8} \) = \( 137\frac{1}{2} \)%
In simple words: Take your fraction and multiply the top number by 100, then divide by the bottom number. The result is your percentage - simple as that.

Exam Tip: Check if the fraction can be simplified before multiplying by 100 - this may make your calculation faster. Always express your final answer with the % symbol.

Question 3(i). 6 students out of 40 students in a class are absent. What percentage of the students are absent?
Answer: The number of absent students is 6 and the total count is 40.

Percentage of absent students = \( \left(\frac{6}{40} \times 100\right) \)% = \( \frac{600}{40} \)% = 15%

Therefore, the percentage of absent students equals 15%.
In simple words: Divide the number of absent students by the total, then multiply by 100 to get the percentage.

Exam Tip: Always set up the fraction as (part/whole) × 100. Double-check that you have the correct values in numerator and denominator before calculating.

Question 3(ii). Antony secured 384 marks out of 500 marks. Find the percentage of marks secured by Antony.
Answer: Antony's marks total 384 out of a maximum of 500.

Percentage of marks = \( \left(\frac{384}{500} \times 100\right) \)% = \( \frac{38400}{500} \)% = 76.8%

Therefore, the percentage of marks earned by Antony is 76.8%.
In simple words: Divide his marks by the total marks, then multiply by 100 to find what percentage he scored.

Exam Tip: For percentage calculations with larger numbers, simplify the fraction first if possible to avoid arithmetic errors. Show all steps clearly.

Question 3(iii). A shop has 500 shirts, out of which 15 are defective. What percentage of shirts are defective?
Answer: The number of defective shirts is 15 and the total number is 500.

Percentage of defective shirts = \( \left(\frac{15}{500} \times 100\right) \)% = \( \frac{1500}{500} \)% = 3%

Therefore, the percentage of defective shirts equals 3%.
In simple words: Divide the number of defective shirts by the total number of shirts, then multiply by 100.

Exam Tip: Always confirm that the part being compared is correctly identified. Here, 15 (the defective ones) goes in the numerator and 500 (the total) in the denominator.

Question 3(iv). 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: Vani owns 20 gold bangles and 10 silver bangles, making a total of 30.

Percentage of gold bangles = \( \left(\frac{20}{30} \times 100\right) \)% = \( \frac{2000}{30} \)% = \( \frac{200}{3} \)% = \( 66\frac{2}{3} \)%

Percentage of silver bangles = \( \left(\frac{10}{30} \times 100\right) \)% = \( \frac{1000}{30} \)% = \( \frac{100}{3} \)% = \( 33\frac{1}{3} \)%

Therefore, gold bangles make up \( 66\frac{2}{3} \)% and silver bangles make up \( 33\frac{1}{3} \)% of the collection.
In simple words: Add the two types together to find the total. Then divide each type by the total and multiply by 100 to get their percentages.

Exam Tip: When finding percentages of multiple parts of a whole, verify that all percentages add up to 100% as a check on your work.

Question 3(v). There are 120 voters, 90 of them voted. What percent did not vote?
Answer: There are 120 voters total, and 90 actually cast their votes.

Number who did not vote = 120 - 90 = 30

Percentage who did not vote = \( \left(\frac{30}{120} \times 100\right) \)% = \( \frac{3000}{120} \)% = 25%

Therefore, the percentage of voters who did not vote is 25%.
In simple words: First find how many didn't vote by subtracting. Then divide that number by the total and multiply by 100.

Exam Tip: Read carefully - the question asks for those who did NOT vote, not those who did. Always identify what the question is asking for before setting up your fraction.

Question 4. Estimate the part of the figure which is shaded and hence find the percentage of the part which is shaded.
Answer:

(i) Looking at the square figure, the shaded part = \( \frac{3}{4} \)

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

Hence, shaded part = \( \frac{3}{4} \) and percentage of shaded part = 75%

(ii) Observing the rectangle, the shaded part = \( \frac{2}{6} = \frac{1}{3} \)

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

Hence, shaded part = \( \frac{1}{3} \) and percentage of shaded part = \( 33\frac{1}{3} \)%

(iii) From the circular figure, the shaded part = \( \frac{5}{8} \)

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

Hence, shaded part = \( \frac{5}{8} \) and percentage of shaded part = 62.5%
In simple words: Count the shaded sections and divide by the total sections to get a fraction. Multiply that fraction by 100 to find the percentage.

Exam Tip: When estimating shaded regions, identify the total number of equal parts in the figure first, then count the shaded ones. This approach prevents counting errors.

Question 5. Convert the following percentages into ratios in simplest form:
(i) 14%
(ii) \( 1\frac{3}{4} \)%
(iii) \( 33\frac{1}{3} \)%
(iv) 37.5%
Answer: To convert a percentage to a ratio, first change it to a fraction in simplest form, then express that fraction as a ratio.

(i) 14% = \( \frac{14}{100} = \frac{7}{50} \) = 7 : 50

Hence, 14% = 7 : 50

(ii) \( 1\frac{3}{4} \)% = \( \frac{7}{4} \)% = \( \frac{7}{4} \times \frac{1}{100} = \frac{7}{400} \) = 7 : 400

Hence, \( 1\frac{3}{4} \)% = 7 : 400

(iii) \( 33\frac{1}{3} \)% = \( \frac{100}{3} \)% = \( \frac{100}{3} \times \frac{1}{100} = \frac{1}{3} \) = 1 : 3

Hence, \( 33\frac{1}{3} \)% = 1 : 3

(iv) 37.5% = \( \frac{37.5}{100} = \frac{375}{1000} = \frac{3}{8} \) = 3 : 8

Hence, 37.5% = 3 : 8
In simple words: Write the percentage as a fraction with 100 in the denominator, reduce it as far as possible, then write it as a ratio using a colon.

Exam Tip: Always reduce your fraction completely before writing it as a ratio. The numbers in the final ratio should have no common factors other than 1.

Question 6. Express the following ratios as percentages:
(i) 5 : 4
(ii) 1 : 1
(iii) 2 : 3
(iv) 9 : 16
Answer: To change a ratio into a percentage, first convert the ratio to a fraction, then multiply by 100.

(i) 5 : 4 = \( \frac{5}{4} \)

\( \frac{5}{4} \times 100 \)% = 125%

Hence, 5 : 4 = 125%

(ii) 1 : 1 = \( \frac{1}{1} \)

\( \frac{1}{1} \times 100 \)% = 100%

Hence, 1 : 1 = 100%

(iii) 2 : 3 = \( \frac{2}{3} \)

\( \frac{2}{3} \times 100 \)% = \( \frac{200}{3} \)% = \( 66\frac{2}{3} \)%

Hence, 2 : 3 = \( 66\frac{2}{3} \)%

(iv) 9 : 16 = \( \frac{9}{16} \)

\( \frac{9}{16} \times 100 \)% = \( \frac{900}{16} \)% = \( 56\frac{1}{4} \)%

Hence, 9 : 16 = \( 56\frac{1}{4} \)%
In simple words: When you see a ratio like 2 : 3, write it as a fraction \( \frac{2}{3} \), then multiply by 100 and add the % sign.

Exam Tip: Remember that a ratio a : b becomes the fraction \( \frac{a}{b} \). If your answer is a fraction with 100 in the denominator, simplify it first before expressing as a percentage.

Question 7. The price of an article decreased from Rs 80 to Rs 60, find the percentage of decrease in the price of the article.
Answer: The starting price was Rs 80 and the new price became Rs 60. The amount that dropped is 80 - 60 = Rs 20. To find the percentage drop, divide the amount lost by the starting price and multiply by 100.

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

\( \Rightarrow \frac{2000}{80}\% \)

\( \Rightarrow 25\% \)

The percentage decrease in the price of the article is 25%.
In simple words: The price dropped by Rs 20 out of the original Rs 80. This means the price went down by 25%.

Exam Tip: Always find the decrease or increase amount first, then divide it by the original value (not the new value) to get the percentage change.

Question 8. A watch which was bought for Rs 300 is sold for Rs 336. Find the percentage of profit.
Answer: The cost price of the watch is Rs 300 and the selling price is Rs 336. The profit earned is 336 - 300 = Rs 36. To find the profit percentage, divide the profit by the cost price and multiply by 100.

\( \Rightarrow \left( \frac{36}{300} \times 100 \right)\% \)

\( \Rightarrow \frac{3600}{300}\% \)

\( \Rightarrow 12\% \)

The profit percentage is 12%.
In simple words: The watch cost Rs 300 and was sold for Rs 336, giving a gain of Rs 36. This gain is 12% of the cost.

Exam Tip: For profit percentage, always divide the profit by the cost price, not the selling price.

Question 9. Find the simple interest on Rs 2000 for 3 years at 5% per annum.
Answer: Principal (P) = Rs 2000, Rate (R) = 5% per annum, and Time (T) = 3 years. The simple interest formula is:

\( \text{Simple Interest} = \frac{P \times R \times T}{100} \)

\( \Rightarrow \frac{2000 \times 5 \times 3}{100} \)

\( \Rightarrow \frac{30000}{100} \)

\( \Rightarrow \text{Rs } 300 \)

The simple interest is Rs 300.
In simple words: If you keep Rs 2000 in the bank for 3 years at 5% interest each year, you earn Rs 300 as interest.

Exam Tip: Remember the simple interest formula and make sure you identify the principal, rate, and time correctly before substituting values.

Question 8. 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: The original cost was Rs. 1 per litre, and today it costs Rs. 95 per litre. The amount of increase is 95 - 1 = Rs. 94. To find the percentage increase, we calculate: \( \frac{94}{1} \times 100 \) = 9400%. Therefore, petrol prices have risen by 9400%.
In simple words: The price went up from Rs. 1 to Rs. 95. We divide the increase (Rs. 94) by the original price (Rs. 1) and multiply by 100 to get 9400%.

Exam Tip: Always identify the original value and the new value first, then calculate the increase. Use the formula: Percentage increase = (Increase / Original) × 100.

Question 9. 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 was Rs. 40 per kg. The rise in price = 20% of 40 = \( \frac{20}{100} \times 40 \) = Rs. 8. The current price = 40 + 8 = Rs. 48 per kg. So, tomatoes cost Rs. 48 per kg this year.
In simple words: Find 20% of the old price (Rs. 8), then add it to the old price (Rs. 40) to get the new price (Rs. 48).

Exam Tip: When a price increases by a certain percentage, always add the increase to the original price. Do not replace the original with only the percentage amount.

Question 10. 300 students took an exam. 28% failed. Calculate the number of students who passed the exam.
Answer: The total number of students = 300. The percentage who failed = 28%. This means the percentage who passed = 100 - 28 = 72%. Number of students who passed = 72% of 300 = \( \frac{72}{100} \times 300 \) = \( \frac{21600}{100} \) = 216. Therefore, 216 students passed the exam.
In simple words: If 28% failed, then 72% passed. Find 72% of 300, which gives 216 students.

Exam Tip: Remember that if a percentage fails, the remaining percentage passes. Always subtract from 100 to find the complement before calculating.

Question 11. Out of 15000 voters in a constituency, 60% voted. Find the number of voters who did not vote.
Answer: The total number of voters = 15000. The percentage who voted = 60%. The percentage who did not vote = 100 - 60 = 40%. Number who did not vote = 40% of 15000 = \( \frac{40}{100} \times 15000 \) = \( \frac{600000}{100} \) = 6000. So, 6000 voters did not vote.
In simple words: If 60% voted, then 40% did not vote. Calculate 40% of 15000 to get 6000 non-voters.

Exam Tip: Always find the complementary percentage first (the part that is not mentioned), then calculate based on the total.

Question 12. 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: The total length is 18 m. Green portion = 20%, yellow portion = 45%. The red portion = 100 - (20 + 45) = 100 - 65 = 35%. Length painted red = 35% of 18 m = \( \frac{35}{100} \times 18 \) = \( \frac{630}{100} \) = 6.3 m. Thus, 6.3 m of the pole is painted red.
In simple words: Add the green and yellow percentages (65%), then subtract from 100 to find the red portion (35%). Calculate 35% of 18 m to get 6.3 m.

Exam Tip: When multiple parts make up a whole, always check that all percentages sum to 100% before calculating the final part.

Question 13. Chalk contains 10% calcium, 3% carbon, 12% oxygen and the remaining sand. Find the amount of carbon and calcium (in grams) in \( 2\frac{1}{2} \) kg of chalk. Also find the amount of sand (in kg).
Answer: Total chalk = \( 2\frac{1}{2} \) kg = 2.5 kg = 2500 g. Amount of carbon = 3% of 2500 g = \( \frac{3}{100} \times 2500 \) = 75 g. Amount of calcium = 10% of 2500 g = \( \frac{10}{100} \times 2500 \) = 250 g. Percentage of sand = 100 - (10 + 3 + 12) = 100 - 25 = 75%. Amount of sand = 75% of 2500 g = \( \frac{75}{100} \times 2500 \) = 1875 g = 1.875 kg. Therefore, carbon = 75 g, calcium = 250 g, and sand = 1.875 kg.
In simple words: Calculate each component separately by finding its percentage of the total. First convert kg to grams, then work with the numbers.

Exam Tip: When a question asks for different units (some in grams, some in kg), ensure you convert to the requested unit before giving your final answer. Always verify that all percentages sum to 100%.

Question 14. 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) Let the whole quantity be x. Given, 25% of x = 9. So \( \frac{25}{100} \times x = 9 \)
\( \implies x = \frac{9 \times 100}{25} \)
\( \implies x = 36 \). The whole quantity = 36.

(ii) Let the whole quantity be x. Given, 75% of x = 15. So \( \frac{75}{100} \times x = 15 \)
\( \implies x = \frac{15 \times 100}{75} \)
\( \implies x = 20 \). The whole quantity = 20.

(iii) Let the whole quantity be Rs. x. Given, 12% of x = 1080. So \( \frac{12}{100} \times x = 1080 \)
\( \implies x = \frac{1080 \times 100}{12} \)
\( \implies x = 9000 \). The whole quantity = Rs. 9000.

(iv) Let the whole quantity be x litres. Given, 8% of x = 40. So \( \frac{8}{100} \times x = 40 \)
\( \implies x = \frac{40 \times 100}{8} \)
\( \implies x = 500 \). The whole quantity = 500 litres.
In simple words: To find the whole when a percentage is given, set up an equation where the percentage equals the given amount, then solve for the unknown whole by multiplying the given amount by 100 and dividing by the percentage.

Exam Tip: Use the method of reverse calculation: if you know a percentage of a number, divide the given value by the percentage and multiply by 100 to recover the whole.

Question 15. Mohini saves ₹ 4000 from her salary. If this is 10% of her salary, then what is her salary?
Answer: Let Mohini's salary be Rs. x. We know 10% of her salary = Rs. 4000. So \( \frac{10}{100} \times x = 4000 \)
\( \implies x = \frac{4000 \times 100}{10} \)
\( \implies x = 40000 \). Mohini's salary is Rs. 40,000.
In simple words: If Rs. 4000 is 10% of her salary, then her total salary is 10 times Rs. 4000, which is Rs. 40,000.

Exam Tip: When a savings or portion is given as a percentage of the total, always use the reverse formula to find the total amount. Double-check by verifying: 10% of 40,000 = 4,000.

Question 16. 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. We have 84% of x = 42. So \( \frac{84}{100} \times x = 42 \)
\( \implies x = \frac{42 \times 100}{84} \)
\( \implies x = 50 \). The total number of apples in the basket = 50.
In simple words: Find what percentage is good (84%), then work backwards: if 84% equals 42 apples, divide 42 by 84 and multiply by 100 to find the total (50 apples).

Exam Tip: When dealing with a part that goes bad or is lost, always calculate the remaining percentage first, then use reverse calculation to find the original total.

Question 17. 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: Varun scored 251 marks and fell short by 19 marks to pass. So the passing marks = 251 + 19 = 270. Let the maximum marks be x. Since 45% of the maximum marks equals the passing marks: \( \frac{45}{100} \times x = 270 \)
\( \implies x = \frac{270 \times 100}{45} \)
\( \implies x = 600 \). The maximum marks = 600.
In simple words: Add the marks Varun got to the marks he fell short by to find the passing mark (270). Then use 45% of the total equals 270 to find the total marks (600).

Exam Tip: Always find the actual passing mark first by adding the shortfall to the score obtained, then solve for the maximum using the passing percentage.

Question 18. 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 be x. We have 6% of x = 174. So \( \frac{6}{100} \times x = 174 \)
\( \implies x = \frac{174 \times 100}{6} \)
\( \implies x = 2900 \). The total strength of the school = 2900.
In simple words: If 94% were present, then 6% were absent. Use the absence percentage and the number absent to find the total school strength.

Exam Tip: When attendance or presence is given as a percentage, find the absence or complementary percentage first, then use that percentage to calculate the total.

Question 19. 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%, percentage of women = 39%. Percentage of children = 100 - (40 + 39) = 100 - 79 = 21%. Let the total population be x. We have 21% of x = 12600. So \( \frac{21}{100} \times x = 12600 \)
\( \implies x = \frac{12600 \times 100}{21} \)
\( \implies x = 60000 \). Number of men = 40% of 60000 = \( \frac{40}{100} \times 60000 \) = 24000. The number of men = 24000.
In simple words: Find the children's percentage (21%), use it to calculate the total population (60,000), then find 40% of the total to get the number of men (24,000).

Exam Tip: When a population is divided into multiple groups given as percentages, always ensure all percentages sum to 100%. Use the known group to find the total, then calculate other groups.

Question 20. 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 be Rs. x. We know 15% of the price = Rs. 90. So \( \frac{15}{100} \times x = 90 \)
\( \implies x = \frac{90 \times 100}{15} \)
\( \implies x = 600 \). The earlier price of the watch = Rs. 600.
In simple words: If a 15% increase equals Rs. 90, divide 90 by 15 and multiply by 100 to find the original price (Rs. 600).

Exam Tip: When the increase or decrease amount is given along with its percentage, use direct division to find the original value: Original = (Given Amount / Percentage) × 100.

Question 21. (i) Find the number which when increased by 30% becomes 39.
Answer: Let the number be x. When increased by 30%, the number becomes 130% of x. So 130% of x = 39. \( \frac{130}{100} \times x = 39 \)
\( \implies x = \frac{39 \times 100}{130} \)
\( \implies x = 30 \). The required number = 30.

(ii) Find the number which when decreased by 8% becomes 506.
Answer: Let the number be x. When decreased by 8%, the number becomes (100 - 8)% = 92% of x. So 92% of x = 506. \( \frac{92}{100} \times x = 506 \)
\( \implies x = \frac{506 \times 100}{92} \)
\( \implies x = 550 \). The required number = 550.
In simple words: For an increase, add the increase percentage to 100. For a decrease, subtract the decrease percentage from 100. Then solve the equation.

Exam Tip: When a number is changed by a percentage, remember: increased by 30% means 130%, decreased by 8% means 92%. Use these multipliers to form and solve the equation.

Question 22. The price of a shirt is reduced by 7% to ₹ 465. What is its original price?
Answer: Let the original price be Rs. x. When reduced by 7%, the price becomes (100 - 7)% = 93% of x. So 93% of x = 465. \( \frac{93}{100} \times x = 465 \)
\( \implies x = \frac{465 \times 100}{93} \)
\( \implies x = 500 \). The original price of the shirt = Rs. 500.
In simple words: If the price after a 7% reduction is Rs. 465, then this is 93% of the original price. Divide 465 by 93 and multiply by 100 to find the original price (Rs. 500).

Exam Tip: For any reduced price, always use (100 - reduction percentage) to find what percentage the final price represents, then work backwards.

Question 23. If 15% of 60 is greater than 25% of a number by 3, then find the number.
Answer: First, calculate 15% of 60 = \( \frac{15}{100} \times 60 \) = 9. Let the number be x. According to the problem, 15% of 60 is greater than 25% of x by 3. This means: 9 = 25% of x + 3. So 25% of x = 9 - 3 = 6. Therefore, \( \frac{25}{100} \times x = 6 \)
\( \implies x = \frac{6 \times 100}{25} \)
\( \implies x = 24 \). The required number = 24.
In simple words: Calculate 15% of 60 first (which is 9). If this is 3 more than 25% of the unknown number, then 25% of the number is 6. Finally, find what number gives 25% as 6 (the answer is 24).

Exam Tip: Break down word problems step-by-step: first calculate any fixed percentages, then set up an equation using the comparison (greater/less by a certain amount), and solve.

Question 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:
(i) Petrol in the full tank = 60 litres. Petrol used = 30% of 60 = \( \frac{30}{100} \times 60 \) = 18 litres. Petrol poured into the 50 litre tank = 60 - 18 = 42 litres. Percentage of 50 litre tank filled = \( \frac{42}{50} \times 100 \) = \( \frac{4200}{50} \) = 84%. Therefore, 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 used = \( \frac{28}{42} \times 100 \) = \( \frac{2800}{42} \) = 66.67% (or approximately 66⅔%). Thus, approximately 66.67% of petrol in the 50 litre tank would be used in 10 days.
In simple words: First, find how much petrol remains after 30% is used (42 litres). Calculate what fraction of the 50-litre tank this fills (84%). Then find how much is consumed in 10 days and what percentage this is of the available 42 litres.

Exam Tip: For multi-part problems, solve each part independently and clearly. In part (ii), ensure you calculate the percentage based on the petrol in the smaller tank (42 litres), not the original 60 litres.

Exercise 7.3

Question 1. Rohan bought a calculator for Rs. 760 and sold it for Rs. 874. Find his profit and profit percentage.
Answer: The cost price of the calculator is Rs. 760 and the selling price is Rs. 874. Since the selling price exceeds the cost price, Rohan makes a profit. The profit earned is calculated as: Profit = 874 - 760 = Rs. 114. To find the profit percentage, we use the formula: Profit percentage = (Profit / Cost price) × 100 = (114 / 760) × 100 = 15%. Thus, Rohan's profit is Rs. 114 and his profit percentage is 15%.
In simple words: Rohan bought the calculator for Rs. 760 and sold it for Rs. 874, earning a profit of Rs. 114. This is 15% profit on what he paid.

Exam Tip: Always compare selling price with cost price first to identify profit or loss, then apply the correct formula using the cost price as the denominator.

Question 2. Kirti bought a saree for Rs. 2500 and sold it for Rs. 2300. Find her loss and loss percent.
Answer: The cost price of the saree is Rs. 2500 and the selling price is Rs. 2300. Since the cost price is more than the selling price, Kirti incurs a loss. The loss is: Loss = 2500 - 2300 = Rs. 200. The loss percentage is calculated using: Loss percentage = (Loss / Cost price) × 100 = (200 / 2500) × 100 = 8%. Therefore, Kirti's loss is Rs. 200 and her loss percentage is 8%.
In simple words: Kirti paid Rs. 2500 for the saree but sold it for Rs. 2300, losing Rs. 200. This is an 8% loss.

Exam Tip: Remember that loss percentage is always calculated on the cost price, just like profit percentage - never on the selling price.

Question 3. Calculate the profit or loss in the following transactions. Also find profit percent or loss percent in each case:
(i) Gardening shears bought for Rs. 250 and sold for Rs. 325
(ii) A shirt bought for Rs. 250 and sold at Rs. 150
Answer:
(i) The cost price of the shears is Rs. 250 and the selling price is Rs. 325. Since selling price is higher than cost price, there is a profit. The profit is: Profit = 325 - 250 = Rs. 75. The profit percentage is: (75 / 250) × 100 = 30%. So the profit is Rs. 75 and profit percent is 30%.
(ii) The cost price of the shirt is Rs. 250 and the selling price is Rs. 150. Since cost price exceeds selling price, there is a loss. The loss is: Loss = 250 - 150 = Rs. 100. The loss percentage is: (100 / 250) × 100 = 40%. So the loss is Rs. 100 and loss percent is 40%.
In simple words: In the first case, the shears were bought for Rs. 250 and sold for Rs. 325, making a profit of Rs. 75 or 30%. In the second case, the shirt was bought for Rs. 250 but sold for only Rs. 150, causing a loss of Rs. 100 or 40%.

Exam Tip: When dealing with multiple transactions, handle each item separately first, then identify profit/loss for each before calculating the percentage.

Question 4. Rajinder bought one almirah for Rs. 4800 and the other for Rs. 3640. He sold the first almirah at a gain of 13 1/3 % and the other at a loss of 15%. How much did he gain or lose in the whole deal?
Answer: For the first almirah: The cost price is Rs. 4800 and the gain is 13 1/3 % = 40/3 %. The gain amount is: (40/3) % of 4800 = (40 / 300) × 4800 = Rs. 640. The selling price is: 4800 + 640 = Rs. 5440. For the second almirah: The cost price is Rs. 3640 and the loss is 15%. The loss amount is: 15% of 3640 = (15 / 100) × 3640 = Rs. 546. The selling price is: 3640 - 546 = Rs. 3094. For the whole deal: Total cost price = 4800 + 3640 = Rs. 8440. Total selling price = 5440 + 3094 = Rs. 8534. Since total selling price is more than total cost price, there is a gain. Gain = 8534 - 8440 = Rs. 94. Rajinder gained Rs. 94 in the whole transaction.
In simple words: Rajinder sold one almirah at a profit and the other at a loss. When we combine both deals, his total profit was Rs. 94.

Exam Tip: For transactions involving multiple items, calculate profit/loss for each item separately, then combine the total cost prices and selling prices to find the overall gain or loss.

Question 5. In a furniture shop, 24 tables were bought at the rate of Rs. 4500 per table. The shopkeeper sold 16 of them at the rate of Rs. 6000 per table and the remaining at the rate of Rs. 4000 per table. Find his gain or loss percent.
Answer: Total cost price = 24 × 4500 = Rs. 108,000. The selling price of 16 tables = 16 × 6000 = Rs. 96,000. The remaining number of tables = 24 - 16 = 8. The selling price of 8 tables = 8 × 4000 = Rs. 32,000. Total selling price = 96,000 + 32,000 = Rs. 128,000. Since total selling price is more than total cost price, there is a gain. Gain = 128,000 - 108,000 = Rs. 20,000. The gain percentage = (20,000 / 108,000) × 100 = 18 14/27 %. The shopkeeper achieves a gain of 18 14/27 %.
In simple words: The shopkeeper paid Rs. 108,000 for all the tables but received Rs. 128,000 from selling them, making a profit of Rs. 20,000 or about 18.5%.

Exam Tip: When items are sold at different rates, calculate the total selling price by finding the selling price for each group separately, then combine them before finding the percentage.

Question 6. By selling a lamp for Rs. 810, a dealer makes a profit of Rs. 60. What is the cost price of the lamp? What is his profit percent?
Answer: The selling price of the lamp is Rs. 810 and the profit is Rs. 60. We can find the cost price using the relationship: Cost price = Selling price - Profit = 810 - 60 = Rs. 750. The profit percentage is: (60 / 750) × 100 = 8%. Therefore, the cost price is Rs. 750 and the profit percent is 8%.
In simple words: When you know the selling price and the profit amount, subtract the profit from the selling price to get the cost price. The profit of Rs. 60 on a cost of Rs. 750 is 8%.

Exam Tip: Remember the relationships: if profit is given, then C.P. = S.P. - Profit; if loss is given, then C.P. = S.P. + Loss.

Question 7. By selling a jacket for Rs. 3906, a manufacturer suffers a loss of Rs. 294. Find the cost price of the jacket and his loss percentage.
Answer: The selling price of the jacket is Rs. 3906 and the loss is Rs. 294. The cost price is found using: Cost price = Selling price + Loss = 3906 + 294 = Rs. 4200. The loss percentage is: (294 / 4200) × 100 = 7%. Therefore, the cost price of the jacket is Rs. 4200 and the loss percentage is 7%.
In simple words: The manufacturer sold the jacket for Rs. 3906 but had paid Rs. 4200 for it, resulting in a loss of Rs. 294 or 7%.

Exam Tip: When loss is given along with selling price, add the loss to the selling price to find the original cost price.

Question 8. The cost price of a vase is Rs. 120. If the shopkeeper sells it at a loss of 10%, find the price at which it was sold.
Answer: The cost price of the vase is Rs. 120 and the loss is 10%. The loss amount is calculated as: Loss = 10% of 120 = (10 / 100) × 120 = Rs. 12. The selling price is: Selling price = Cost price - Loss = 120 - 12 = Rs. 108. Therefore, the vase was sold for Rs. 108.
In simple words: A 10% loss on Rs. 120 means losing Rs. 12, so the vase is sold for Rs. 120 - Rs. 12 = Rs. 108.

Exam Tip: Always calculate the loss or profit amount first, then add or subtract it from the cost price to find the selling price.

Question 9. I buy a T.V. for Rs. 10,000 and sell it at a profit of 20%. How much money do I get for it?
Answer: The cost price of the T.V. is Rs. 10,000 and the profit rate is 20%. The profit amount is: Profit = 20% of 10,000 = (20 / 100) × 10,000 = Rs. 2,000. The selling price is: Selling price = Cost price + Profit = 10,000 + 2,000 = Rs. 12,000. Therefore, I receive Rs. 12,000 for the T.V.
In simple words: A 20% profit on Rs. 10,000 gives a profit of Rs. 2,000, so the selling price is Rs. 12,000.

Exam Tip: When profit percentage is given with cost price, multiply the cost price by the percentage to get the profit amount, then add it to find the selling price.

Question 10. A shopkeeper sells an article for Rs. 300, thus earning a profit of 20%. Find the cost price of the article.
Answer: The selling price is Rs. 300 and the profit is 20%. Let the cost price be x. We know that Selling price = Cost price + 20% of Cost price = (100 + 20)% of Cost price = 120% of Cost price. Therefore: (120 / 100) × x = 300, which gives us x = (300 × 100) / 120 = 250. The cost price of the article is Rs. 250.
In simple words: If the shopkeeper makes 20% profit, the selling price is 120% of the cost price. So Rs. 300 is 120% of the cost price, meaning the cost price is Rs. 250.

Exam Tip: When selling price and profit percentage are given, set up an equation where S.P. = (100 + Profit%) / 100 × C.P., then solve for C.P.

Question 11. A shopkeeper sells an article for Rs. 320, thus suffering a loss of 20%. Find the cost price of the article.
Answer: The selling price is Rs. 320 and the loss is 20%. Let the cost price be x. We know that Selling price = Cost price - 20% of Cost price = (100 - 20)% of Cost price = 80% of Cost price. Therefore: (80 / 100) × x = 320, which gives us x = (320 × 100) / 80 = 400. The cost price of the article is Rs. 400.
In simple words: If there is a 20% loss, the selling price is 80% of the cost price. So Rs. 320 is 80% of the cost price, meaning the cost price is Rs. 400.

Exam Tip: When selling price and loss percentage are given, use the formula S.P. = (100 - Loss%) / 100 × C.P. to find the cost price.

Question 12. By selling a chair for Rs. 522, a shopkeeper makes a profit of 16%. What is its cost price?
Answer: The selling price of the chair is Rs. 522 and the profit is 16%. Let the cost price be x. The selling price equals (100 + 16)% of the cost price, which is 116% of the cost price. Setting up the equation: (116 / 100) × x = 522, we get x = (522 × 100) / 116 = 450. The cost price of the chair is Rs. 450.
In simple words: With a 16% profit, the selling price becomes 116% of the cost price. Since the selling price is Rs. 522, the cost price works out to Rs. 450.

Exam Tip: Always add profit percentage to 100% or subtract loss percentage from 100% before setting up the equation to find cost price.

Question 13. A trader sold some damaged garments for Rs. 7360 at a loss of 8%. Find the cost price of the garments.
Answer: The selling price of the garments is Rs. 7360 and the loss is 8%. Let the cost price be x. The selling price equals (100 - 8)% of the cost price, which is 92% of the cost price. Setting up the equation: (92 / 100) × x = 7360, we get x = (7360 × 100) / 92 = 8000. The cost price of the garments is Rs. 8000.
In simple words: An 8% loss means the selling price is 92% of the cost price. Since the selling price is Rs. 7360, the original cost price must have been Rs. 8000.

Exam Tip: Check your answer by calculating: 92% of 8000 should equal 7360, confirming your solution is correct.

Question 14. By selling a table for Rs. 3168, Rashid loses 12%. Find its cost price. What percent would he gain or lose by selling the table for Rs. 3870?
Answer: The selling price of the table is Rs. 3168 and the loss is 12%. Let the cost price be x. The selling price equals 88% of the cost price (since 100 - 12 = 88). Therefore: (88 / 100) × x = 3168, giving x = (3168 × 100) / 88 = 3600. So the cost price is Rs. 3600. Now, if the table is sold for Rs. 3870: Since 3870 is more than 3600, there is a gain. Gain = 3870 - 3600 = Rs. 270. Gain percentage = (270 / 3600) × 100 = 7.5%. Therefore, Rashid would gain 7.5% by selling it for Rs. 3870.
In simple words: First, we find that the table originally cost Rs. 3600. If Rashid sells it for Rs. 3870, he makes a gain of Rs. 270, which is 7.5% profit.

Exam Tip: This is a two-part question - find the cost price first from the given loss scenario, then use that cost price to calculate the new gain or loss for the different selling price.

Question 15. By selling an article for Rs. 4550, Tony incurs a loss of 9%. What percent would he gain or lose by selling it for Rs. 4825?
Answer: The selling price is Rs. 4550 and the loss is 9%. Let the cost price be x. The selling price equals 91% of the cost price (since 100 - 9 = 91). Therefore: (91 / 100) × x = 4550, giving x = (4550 × 100) / 91 = 5000. So the cost price is Rs. 5000. Now, if the article is sold for Rs. 4825: Since 5000 is more than 4825, there is a loss. Loss = 5000 - 4825 = Rs. 175. Loss percentage = (175 / 5000) × 100 = 3.5%. Therefore, Tony would lose 3.5% by selling it for Rs. 4825.
In simple words: The article's cost price is Rs. 5000. When sold for Rs. 4825, Tony loses Rs. 175, which equals 3.5% loss.

Exam Tip: Always find the cost price from the first given condition, then apply it to the new selling price to determine if there is a gain or loss.

Question 16. Arif bought a second hand car for Rs. 80,000 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: The purchase price of the car is Rs. 80,000. The amount spent on repairs is 12.5% of 80,000 = (12.5 / 100) × 80,000 = Rs. 10,000. The total cost price = 80,000 + 10,000 = Rs. 90,000. To make a profit of 15%, the selling price should be: Selling price = (100 + 15)% of 90,000 = 115% of 90,000 = (115 / 100) × 90,000 = Rs. 103,500. Arif should sell the car for Rs. 103,500.
In simple words: Arif spent Rs. 80,000 on the car and Rs. 10,000 on repairs, totaling Rs. 90,000. To earn 15% profit, he must sell it for Rs. 103,500.

Exam Tip: In such problems, always add all costs (purchase price plus any additional expenses) to find the total cost price before calculating the selling price for desired profit.

Exercise 7.4

Question 1. Find the simple interest on:
(i) Rs. 350 for 2 years at 11% per annum
(ii) Rs. 20,000 for 4 1/2 years at 8 1/2 % per annum
(iii) Rs. 648 for 8 months at 16 2/3 % per annum
Also find the amount in each case.
Answer: Simple interest is calculated using the formula: I = (P × R × T) / 100, where P is principal, R is rate per annum, and T is time in years. The amount is found by adding the principal and interest together.
(i) Here, P = Rs. 350, R = 11% per annum, and T = 2 years. Interest = (350 × 11 × 2) / 100 = 7700 / 100 = Rs. 77. Amount = P + I = 350 + 77 = Rs. 427. So the simple interest is Rs. 77 and the amount is Rs. 427.
(ii) Here, P = Rs. 20,000, R = 8 1/2 % = 17/2 % per annum, and T = 4 1/2 = 9/2 years. Interest = (20,000 × 17/2 × 9/2) / 100. Calculating: (20,000 × 17 × 9) / (2 × 2 × 100) = 3,060,000 / 400 = Rs. 7,650. Amount = 20,000 + 7,650 = Rs. 27,650. So the simple interest is Rs. 7,650 and the amount is Rs. 27,650.
(iii) Here, P = Rs. 648, R = 16 2/3 % = 50/3 % per annum, and T = 8 months = 8/12 = 2/3 years. Interest = (648 × 50/3 × 2/3) / 100. Calculating: (648 × 50 × 2) / (3 × 3 × 100) = 64,800 / 900 = Rs. 72. Amount = 648 + 72 = Rs. 720. So the simple interest is Rs. 72 and the amount is Rs. 720.
In simple words: Simple interest depends on three things - how much money you lend (principal), what rate of interest is charged, and for how long. Multiply all three and divide by 100 to find the interest. Add this interest to the principal to get the total amount.

Exam Tip: Always convert mixed fractions and mixed percentages to improper fractions, and convert months to years by dividing by 12 before using the simple interest formula.

Question 1. Find the simple interest and amount when: (i) principal is Rs. 20000, rate is 8.5% per annum for 9 months
Answer: P = Rs. 20000, R = 8.5% per annum, T = 9 months = \( \frac{9}{12} = \frac{3}{4} \) year

\( \implies I = \frac{P \times R \times T}{100} = \frac{20000 \times 8.5 \times \frac{3}{4}}{100} = \frac{20000 \times 8.5 \times 3}{400} = \frac{510000}{400} = \text{Rs. } 7650 \)

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

The simple interest earned is Rs. 7650 and the final amount is Rs. 27650.
In simple words: Multiply the principal by the rate and time, then divide by 100 to get interest. Add interest to principal to find the total amount.

Exam Tip: Always convert months to years by dividing by 12 before applying the formula. Write both interest and amount in your final answer.

Question 1. (ii) principal is Rs. 648, rate is 16⅔% per annum for 8 months
Answer: P = Rs. 648, R = \( 16\frac{2}{3} = \frac{50}{3} \)% per annum, T = 8 months = \( \frac{8}{12} = \frac{2}{3} \) year

\( \implies I = \frac{648 \times \frac{50}{3} \times \frac{2}{3}}{100} = \frac{648 \times 50 \times 2}{3 \times 3 \times 100} = \frac{64800}{900} = \text{Rs. } 72 \)

Amount = P + I = 648 + 72 = Rs. 720

The simple interest is Rs. 72 and the total amount comes to Rs. 720.
In simple words: Change the mixed percentage to an improper fraction, convert months to years as a fraction, then use the interest formula.

Exam Tip: When the rate is given as a mixed number percentage, convert it to an improper fraction first to avoid calculation errors.

Question 2. Find the time when: (i) simple interest on Rs. 2500 at 4% per annum is Rs. 200
Answer: Using \( I = \frac{P \times R \times T}{100} \), we get \( T = \frac{I \times 100}{P \times R} \)

P = Rs. 2500, R = 4% per annum, I = Rs. 200

\( \implies T = \frac{200 \times 100}{2500 \times 4} = \frac{20000}{10000} = 2 \) years

The time period is 2 years.
In simple words: Rearrange the interest formula to isolate time. Multiply interest by 100, then divide by the product of principal and rate.

Exam Tip: Memorize the rearranged formulas for T, R, and P to solve quickly without deriving them each time.

Question 2. (ii) simple interest on Rs. 12000 at 6½% per annum is Rs. 2730
Answer: P = Rs. 12000, R = \( 6\frac{1}{2} = \frac{13}{2} \)% per annum, I = Rs. 2730

\( \implies T = \frac{2730 \times 100}{12000 \times \frac{13}{2}} = \frac{273000}{12000 \times \frac{13}{2}} = \frac{273000 \times 2}{12000 \times 13} = \frac{546000}{156000} = \frac{7}{2} = 3\frac{1}{2} \) years

The time period is 3½ years or 3 years 6 months.
In simple words: Convert the mixed percentage to an improper fraction, then apply the time formula and simplify the result to get years and months.

Exam Tip: When the answer is an improper fraction, convert it back to a mixed number and express any remaining fraction as months (multiply the fraction by 12).

Question 3. Find the rate of interest when: (i) simple interest on Rs. 1560 in 3 years is Rs. 585
Answer: Using \( I = \frac{P \times R \times T}{100} \), we get \( R = \frac{I \times 100}{P \times T} \)

P = Rs. 1560, T = 3 years, I = Rs. 585

\( \implies R = \frac{585 \times 100}{1560 \times 3} = \frac{58500}{4680} = \frac{25}{2} = 12\frac{1}{2} \)%

The rate of interest is 12½% per annum.
In simple words: Multiply interest by 100, then divide by the product of principal and time. Simplify the fraction to get the rate as a percentage.

Exam Tip: Always simplify fractions before stating your final answer to avoid presenting an awkward ratio instead of a clean percentage.

Question 3. (ii) simple interest on Rs. 1625 in 2½ years is Rs. 325
Answer: P = Rs. 1625, T = \( 2\frac{1}{2} = \frac{5}{2} \) years, I = Rs. 325

\( \implies R = \frac{325 \times 100}{1625 \times \frac{5}{2}} = \frac{325 \times 100 \times 2}{1625 \times 5} = \frac{65000}{8125} = 8 \)%

The rate of interest is 8% per annum.
In simple words: When time is given as a mixed number, convert it to an improper fraction first, then use the rate formula and simplify carefully.

Exam Tip: Double-check your arithmetic when working with mixed numbers and fractions - a small error in conversion can lead to an incorrect final rate.

Question 4. Find the principal when: (i) simple interest at 16% per annum for 2½ years is Rs. 3840
Answer: Using \( I = \frac{P \times R \times T}{100} \), we get \( P = \frac{I \times 100}{R \times T} \)

R = 16% per annum, T = \( 2\frac{1}{2} = \frac{5}{2} \) years, I = Rs. 3840

\( \implies P = \frac{3840 \times 100}{16 \times \frac{5}{2}} = \frac{384000}{16 \times \frac{5}{2}} = \frac{384000 \times 2}{16 \times 5} = \frac{768000}{80} = \text{Rs. } 9600 \)

The principal is Rs. 9600.
In simple words: Multiply interest by 100, then divide by the product of rate and time to find the original amount invested.

Exam Tip: When time is a mixed number, always convert it to an improper fraction to avoid calculation mistakes in the denominator.

Question 4. (ii) simple interest at 7½% per annum for 2 years 4 months is Rs. 2730
Answer: R = \( 7\frac{1}{2} = \frac{15}{2} \)% per annum, T = 2 years 4 months = \( 2\frac{4}{12} = 2\frac{1}{3} = \frac{7}{3} \) years, I = Rs. 2730

\( \implies P = \frac{2730 \times 100}{\frac{15}{2} \times \frac{7}{3}} = \frac{273000}{\frac{15 \times 7}{2 \times 3}} = \frac{273000 \times 2 \times 3}{15 \times 7} = \frac{1638000}{105} = \text{Rs. } 15600 \)

The principal is Rs. 15600.
In simple words: Convert both the mixed percentage and the time in months to improper fractions before applying the principal formula.

Exam Tip: Express time in years as a single improper fraction - convert months to a decimal or fractional part of a year, then combine with whole years.

Question 5. Find the rate of interest when: (i) Rs. 1200 amounts to Rs. 1320 in 2 years
Answer: P = Rs. 1200, A = Rs. 1320, T = 2 years

I = A - P = 1320 - 1200 = Rs. 120

\( \implies R = \frac{I \times 100}{P \times T} = \frac{120 \times 100}{1200 \times 2} = \frac{12000}{2400} = 5\)%

The rate of interest is 5% per annum.
In simple words: Find the interest by subtracting principal from amount, then use the rate formula to get the percentage.

Exam Tip: When given the amount, always subtract the principal first to find the interest before using any formula.

Question 5. (ii) Rs. 300 amounts to Rs. 400 in 2 years
Answer: P = Rs. 300, A = Rs. 400, T = 2 years

I = A - P = 400 - 300 = Rs. 100

\( \implies R = \frac{100 \times 100}{300 \times 2} = \frac{10000}{600} = \frac{50}{3} = 16\frac{2}{3}\)%

The rate of interest is 16⅔% per annum.
In simple words: Calculate the interest earned, then divide it by the product of principal and time, then multiply by 100 to get the rate.

Exam Tip: When the result is an improper fraction, convert it to a mixed number percentage for a cleaner final answer.

Question 6. Find the time when: (i) Rs. 1250 amounts to Rs. 1950 at 16% per annum
Answer: P = Rs. 1250, A = Rs. 1950, R = 16% per annum

I = A - P = 1950 - 1250 = Rs. 700

\( \implies T = \frac{I \times 100}{P \times R} = \frac{700 \times 100}{1250 \times 16} = \frac{70000}{20000} = \frac{7}{2} = 3\frac{1}{2} \) years

The time period is 3½ years or 3 years 6 months.
In simple words: First find the interest by subtracting principal from amount. Then apply the time formula to get the number of years.

Exam Tip: Express fractional years as mixed numbers - if the fractional part is ½ or ⅓, indicate the number of months it represents (6 months or 4 months).

Question 6. (ii) Rs. 6540 amounts to Rs. 8447.50 at 12½% per annum
Answer: P = Rs. 6540, A = Rs. 8447.50, R = \( 12\frac{1}{2} = \frac{25}{2} \)% per annum

I = A - P = 8447.50 - 6540 = Rs. 1907.50

\( \implies T = \frac{1907.50 \times 100}{6540 \times \frac{25}{2}} = \frac{190750}{6540 \times \frac{25}{2}} = \frac{190750 \times 2}{6540 \times 25} = \frac{381500}{163500} = \frac{7}{3} = 2\frac{1}{3} \) years = 2 years 4 months

The time period is 2 years 4 months or 2⅓ years.
In simple words: Calculate the interest, convert the mixed percentage to a fraction, then divide interest by the product of principal and rate to find time.

Exam Tip: When converting years to months, multiply the fractional part by 12 (⅓ × 12 = 4 months, ½ × 12 = 6 months).

Question 7. Rs. 14000 is invested at 4% per annum simple interest. How long will it take for the amount to reach Rs. 16240?
Answer: P = Rs. 14000, A = Rs. 16240, R = 4% per annum

I = A - P = 16240 - 14000 = Rs. 2240

\( \implies T = \frac{I \times 100}{P \times R} = \frac{2240 \times 100}{14000 \times 4} = \frac{224000}{56000} = 4 \) years

It will take 4 years for the amount to reach Rs. 16240.
In simple words: Find the interest earned, then use the time formula to calculate how many years are needed for the investment to grow to the target amount.

Exam Tip: Always find the interest first when you are given the final amount - this is the key step before applying any formula.

Question 8. An amount of money invested trebled in 6 years. Find the rate of interest earned.
Answer: Let the principal be Rs. P.

Since the money trebled, A = 3P.

I = A - P = 3P - P = 2P

T = 6 years

\( \implies R = \frac{I \times 100}{P \times T} = \frac{2P \times 100}{P \times 6} = \frac{200}{6} = \frac{100}{3} = 33\frac{1}{3}\)%

The rate of interest is 33⅓% per annum.
In simple words: When the amount becomes three times the principal, the interest equals two times the principal. Use this to find the rate.

Exam Tip: "Trebled" means multiplied by 3 - so if P becomes 3P, the interest is 2P. This type of problem is solved by expressing amounts in terms of P.

Question 9. Find the principal when: (i) final amount is Rs. 4500 at 20% per annum for 5 years
Answer: A = Rs. 4500, R = 20% per annum, T = 5 years

Let the principal be Rs. P.

\( \implies I = \frac{P \times 20 \times 5}{100} = P \)

\( \implies A = P + I = P + P = 2P \)

\( \implies 2P = 4500 \)

\( \implies P = \frac{4500}{2} = \text{Rs. } 2250 \)

The principal is Rs. 2250.
In simple words: Express the interest in terms of P using the formula, then add it to P to get the amount. Set this equal to 4500 and solve for P.

Exam Tip: When working backwards from the amount, always express interest as a multiple or fraction of the principal, then use A = P + I.

Question 9. (ii) final amount is Rs. 2420 at 4% per annum for 2½ years
Answer: A = Rs. 2420, R = 4% per annum, T = \( 2\frac{1}{2} = \frac{5}{2} \) years

Let the principal be Rs. P.

\( \implies I = \frac{P \times 4 \times \frac{5}{2}}{100} = \frac{P \times 10}{100} = \frac{P}{10} \)

\( \implies A = P + I = P + \frac{P}{10} = \frac{11P}{10} \)

\( \implies \frac{11P}{10} = 2420 \)

\( \implies P = \frac{2420 \times 10}{11} = \text{Rs. } 2200 \)

The principal is Rs. 2200.
In simple words: Find the interest formula in terms of P, combine P and the interest expression, then set it equal to the final amount and solve for P.

Exam Tip: Simplify the interest expression (P/10 in this case) before adding it to P - this makes the equation easier to solve.

Question 10. 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 Rs. P and T = 3 years.

Given, simple interest I = \( \frac{3}{10} \) of P = \( \frac{3P}{10} \)

\( \implies R = \frac{I \times 100}{P \times T} = \frac{\frac{3P}{10} \times 100}{P \times 3} = \frac{3P \times 100}{10 \times P \times 3} = \frac{300}{30} = 10\)%

The rate of interest is 10% per annum.
In simple words: Express the interest as a fraction of the principal, then substitute into the rate formula. The principal cancels out, leaving just the rate.

Exam Tip: In problems where interest is given as a fraction of the principal, the P cancels when you divide - this simplifies the calculation significantly.

Question 11. What sum of money will amount to Rs. 2760 in 3 years at 5% per annum simple interest?
Answer: A = Rs. 2760, T = 3 years, R = 5% per annum

Let the sum (principal) be Rs. P.

\( \implies I = \frac{P \times 5 \times 3}{100} = \frac{15P}{100} = \frac{3P}{20} \)

\( \implies A = P + I = P + \frac{3P}{20} = \frac{20P + 3P}{20} = \frac{23P}{20} \)

\( \implies \frac{23P}{20} = 2760 \)

\( \implies P = \frac{2760 \times 20}{23} = \text{Rs. } 2400 \)

The required sum is Rs. 2400.
In simple words: Calculate the interest as a fraction of P, add it to P to form an equation equal to the final amount, then solve for P.

Exam Tip: Always simplify the interest fraction before adding it to P - this keeps the final equation clean and easier to solve.

Question 12. A sum of Rs. 6000 amounts to Rs. 6900 in 3 years. What will it amount to if the rate of interest is increased by 2%?
Answer: First, find the original rate: P = Rs. 6000, A = Rs. 6900, T = 3 years

I = A - P = 6900 - 6000 = Rs. 900

\( \implies R = \frac{I \times 100}{P \times T} = \frac{900 \times 100}{6000 \times 3} = \frac{90000}{18000} = 5\)%

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

\( \text{New interest} = \frac{P \times R \times T}{100} = \frac{6000 \times 7 \times 3}{100} = \frac{126000}{100} = \text{Rs. } 1260 \)

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

The sum will amount to Rs. 7260 at the increased rate.
In simple words: First calculate the original interest rate from the given amount. Then increase this rate by 2%, calculate the new interest, and add it to the principal.

Exam Tip: These two-part problems need careful step-by-step work - find the original rate first, then use the new rate to compute the fresh amount.

Objective Type Questions - Mental Maths

Question 1. Fill in the blanks: (i) 6% of Rs. 50 = ....
Answer: 6% of Rs. 50 = \( \frac{6}{100} \times 50 = \frac{300}{100} = \text{Rs. } 3 \)
In simple words: To find a percentage of a number, multiply the number by the percentage divided by 100.

Exam Tip: Quick mental calculation: 10% of 50 = 5, so 6% = 3. Use benchmark percentages to check your answer.

Question 1. (ii) If 25% of a number is 12, then the number is ....
Answer: Let the number be x. Then 25% of x = 12

\( \implies \frac{25}{100} \times x = 12 \)

\( \implies \frac{1}{4} x = 12 \)

\( \implies x = 12 \times 4 = 48 \)

The number is 48.
In simple words: If 25% (which is one-quarter) of a number equals 12, multiply 12 by 4 to get the original number.

Exam Tip: Remember that 25% = ¼, 50% = ½, and 10% = 1/10 - knowing these fractions speeds up mental maths problems.

Question 1. (iii) The mixed fraction 1¾ converted to percentage form is ....
Answer: \( 1\frac{3}{4} = \frac{7}{4} = \frac{7}{4} \times 100\)% = \( \frac{700}{4} \)% = 175%

The percentage form is 175%.
In simple words: Convert the mixed number to an improper fraction, then multiply by 100 and add the % sign.

Exam Tip: To convert any fraction to a percentage quickly, divide the numerator by the denominator, then multiply by 100.

Question 1. (iv) If a number increases from 20 to 28, then the increase percentage is ....
Answer: The difference between 28 and 20 is 8. To find the increase percentage, divide this difference by the original number and multiply by 100. So the increase percentage is \( \left( \frac{8}{20} \times 100 \right) \% = 40\% \).
In simple words: The number went up by 8. Divide 8 by 20 and multiply by 100 to get 40%.

Exam Tip: Always calculate percentage change using the original value as the base, not the new value.

Question 1. (v) If cost price is Rs. 400 and loss is 15%, then selling price is ....
Answer: A loss of 15% means you lose 15% of the cost price. Calculate 15% of 400, which equals Rs. 60. Subtract this loss from the cost price to find the selling price: 400 - 60 = Rs. 340.
In simple words: Find 15% of 400, which is Rs. 60. Take this away from 400 to get Rs. 340.

Exam Tip: Remember: Selling Price = Cost Price - Loss (when there is a loss).

Question 1. (vi) The profit or loss percentage is always calculated on .....
Answer: Profit and loss percentages are always figured out by using the cost price as the base. This is the standard way to measure how much profit or loss you made compared to what you paid for something.
In simple words: Whether you make profit or lose money, always use the cost price to calculate the percentage.

Exam Tip: This is a key concept - examiners frequently test whether students know that cost price (not selling price) is the base for percentage calculations.

Question 1. (vii) The simple interest on a sum of Rs. 5600 at 8% p.a. for one year is ....
Answer: Use the formula: Simple Interest = \( \frac{P \times R \times T}{100} \). Here, P = 5600, R = 8, and T = 1 year. So, \( \text{S.I.} = \frac{5600 \times 8 \times 1}{100} = \frac{44800}{100} = \text{Rs. } 448 \).
In simple words: Multiply 5600 by 8 and divide by 100 to get Rs. 448.

Exam Tip: Always use the simple interest formula correctly and check that your principal, rate, and time values are in the right units before calculating.

Question 1. (viii) 135% converted to decimals is ....
Answer: To convert a percentage to a decimal, divide by 100. So, \( 135\% = \frac{135}{100} = 1.35 \).
In simple words: Move the decimal point two places to the left in 135 to get 1.35.

Exam Tip: When converting percentages to decimals, dividing by 100 is the same as moving the decimal point two places to the left.

Question 1. (ix) .... is 50% more than 60
Answer: Calculate 50% of 60, which is 30. Then add this to 60: 60 + 30 = 90. So the number that is 50% more than 60 is 90.
In simple words: 50% of 60 is 30. Add 30 to 60 to get 90.

Exam Tip: To find "X% more than Y," calculate X% of Y and add it to Y.

Question 1. (x) 25 mL is .... percent of 5 litres.
Answer: First, change both quantities to the same unit. Convert 5 litres to millilitres: 5 litres = 5000 mL. Now find what percentage 25 mL is of 5000 mL: \( \frac{25}{5000} \times 100 = 0.5\% \).
In simple words: Change 5 litres to 5000 mL. Then divide 25 by 5000 and multiply by 100 to get 0.5%.

Exam Tip: Always convert units to match before comparing quantities - this prevents calculation errors.

Question 2. State whether the following statements are true (T) or false (F):
Answer:

(i) 20% more than 30 is 36
Find 20% of 30, which gives 6. Add this to 30: 30 + 6 = 36. The statement matches the calculation.
Hence, the statement is True.

(ii) The ratio 2 : 5 converted to percentage is 60%
Convert the ratio to a fraction: \( \frac{2}{5} = \left( \frac{2}{5} \times 100 \right)\% = 40\% \), not 60%.
Hence, the statement is False.

(iii) 6\(\frac{1}{4}\)% expressed as a fraction is \(\frac{1}{16}\)
Convert the mixed percentage: \( 6\frac{1}{4}\% = \frac{25}{4}\% = \frac{25}{4} \times \frac{1}{100} = \frac{25}{400} = \frac{1}{16} \).
Hence, the statement is True.

(iv) 80% of 450 m is equal to 360 m
Calculate 80% of 450: \( \frac{80}{100} \times 450 = 360 \) m. The calculation is correct.
Hence, the statement is True.

(v) If a number decreases from 20 to 15, then the decrease is 25%
The decrease is 20 - 15 = 5. The decrease percentage is \( \left( \frac{5}{20} \times 100 \right)\% = 25\% \).
Hence, the statement is True.

(vi) If Feroz obtains 336 marks out of 600 marks, then percentage of marks obtained by him is 33.6
Calculate the percentage: \( \left( \frac{336}{600} \times 100 \right)\% = 56\% \), not 33.6.
Hence, the statement is False.

(vii) 0.018 is equivalent to 8%
Convert the decimal to a percentage: \( 0.018 \times 100 = 1.8\% \), not 8%.
Hence, the statement is False.

(viii) 250 cm is 4% of 1 km
Convert 1 km to cm: 1 km = 100000 cm. The percentage is \( \left( \frac{250}{100000} \times 100 \right)\% = 0.25\% \), not 4%.
Hence, the statement is False.

(ix) If S.P. of an article is Rs. 540 and loss is Rs. 40, then its C.P. is Rs. 500
When there is a loss, C.P. = S.P. + loss = 540 + 40 = Rs. 580, not Rs. 500.
Hence, the statement is False.

(x) By selling a book for Rs. 500, a shopkeeper suffers a loss of 10%. The cost price of the book is Rs. 600
If S.P. is Rs. 500 at a 10% loss, then S.P. = 90% of C.P. So, C.P. = \( \frac{500 \times 100}{90} \approx \text{Rs. } 555.56 \), not Rs. 600.
Hence, the statement is False.
In simple words: When checking if statements are true or false, always perform the calculation step-by-step and match your result with what the statement claims.

Exam Tip: Read each statement carefully and calculate precisely - a small error in understanding can lead to the wrong answer.

Question 3. 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: (2) 25%
In simple words: If income to saving is 4 : 1, the saving is 1 part out of every 5 parts total. This equals \( \frac{1}{4} \times 100 = 25\% \).

Exam Tip: When working with ratios and percentages, remember that the percentage of saving equals the saving ratio part divided by the total ratio sum, multiplied by 100.

Question 4. 225% is equal to
(1) 2 : 3
(2) 3 : 2
(3) 4 : 9
(4) 9 : 4
Answer: (4) 9 : 4
In simple words: Convert 225% to a fraction: \( \frac{225}{100} = \frac{9}{4} \), which as a ratio is 9 : 4.

Exam Tip: To convert a percentage to a ratio, first simplify the fraction form, then express it in ratio notation.

Question 5. If 30% of x is 72, then x is equal to
(1) 120
(2) 240
(3) 360
(4) 480
Answer: (2) 240
In simple words: If 30% of x equals 72, divide 72 by 30 and multiply by 100 to find x: \( x = \frac{72 \times 100}{30} = 240 \).

Exam Tip: When finding the whole from a percentage, reverse the operation - divide the given amount by the percentage and multiply by 100.

Question 6. If x% of 80 = 12, then x is equal to
(1) 15
(2) 20
(3) 25
(4) 30
Answer: (1) 15
In simple words: Divide 12 by 80 and multiply by 100 to find x: \( x = \frac{12 \times 100}{80} = 15 \).

Exam Tip: Rearrange the percentage equation to solve for the unknown percentage value by dividing and multiplying appropriately.

Question 7. 0.025 when expressed as a percent is
(1) 250%
(2) 25%
(3) 4%
(4) 2.5%
Answer: (4) 2.5%
In simple words: Multiply the decimal by 100 to convert to a percentage: \( 0.025 \times 100 = 2.5\% \).

Exam Tip: Converting a decimal to a percentage means multiplying by 100, which moves the decimal point two places to the right.

Question 8. 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: (3) 40
In simple words: If 45% are girls, then 55% are boys. So 55% of total = 22. Solving: total = \( \frac{22 \times 100}{55} = 40 \).

Exam Tip: When one group is given as a percentage, find the complementary percentage for the other group, then use this to find the total.

Question 9. If a man buys an article for Rs. 80 and sells it for Rs. 100, then gain percentage is
(1) 20%
(2) 25%
(3) 40%
(4) 125%
Answer: (2) 25%
In simple words: Gain = 100 - 80 = Rs. 20. Gain percentage = \( \left( \frac{20}{80} \times 100 \right)\% = 25\% \).

Exam Tip: Always calculate gain or loss as a percentage of the cost price, not the selling price.

Question 10. If a man buys an article for Rs. 120 and sells it for Rs. 100, then his loss percentage is
(1) 10%
(2) 20%
(3) 25%
(4) 16\(\frac{2}{3}\)%
Answer: (4) 16\(\frac{2}{3}\)%
In simple words: Loss = 120 - 100 = Rs. 20. Loss percentage = \( \left( \frac{20}{120} \times 100 \right)\% = \frac{50}{3}\% = 16\frac{2}{3}\% \).

Exam Tip: Simplify fractions in percentage answers - sometimes the result is a mixed number percentage, not a whole number.

Question 11. The salary of a man is Rs. 24000 per month. If he gets an increase of 25% in the salary, then the new salary per month is
(1) Rs. 2500
(2) Rs. 28000
(3) Rs. 30000
(4) Rs. 36000
Answer: (3) Rs. 30000
In simple words: Find 25% of 24000, which is Rs. 6000. Add this to 24000: 24000 + 6000 = Rs. 30000.

Exam Tip: When a salary or value increases by a percentage, add the percentage amount to the original amount to get the new value.

Question 12. On selling an article for Rs. 100, Renu gains Rs. 20. Her gain percentage is
(1) 25%
(2) 20%
(3) 15%
(4) 40%
Answer: (1) 25%
In simple words: Gain = Rs. 20 and S.P. = Rs. 100. So C.P. = 100 - 20 = Rs. 80. Gain percentage = \( \left( \frac{20}{80} \times 100 \right)\% = 25\% \).

Exam Tip: Remember the relationship: C.P. = S.P. - gain (when there is a gain), then use cost price to find the percentage.

Question 13. The simple interest on Rs. 6000 at 8% p.a. for one year is
(1) Rs. 600
(2) Rs. 480
(3) Rs. 400
(4) Rs. 240
Answer: (2) Rs. 480
In simple words: Use the formula S.I. = \( \frac{P \times R \times T}{100} = \frac{6000 \times 8 \times 1}{100} = \text{Rs. } 480 \).

Exam Tip: For simple interest calculations, ensure all values are in the correct units - principal in rupees, rate as a number, time in years.

Question 14. If Rohit borrows Rs. 4800 at 5% p.a. simple interest, then the amount he has to return at the end of 2 years is
(1) Rs. 480
(2) Rs. 5040
(3) Rs. 5280
(4) Rs. 5600
Answer: (3) Rs. 5280
In simple words: Find the simple interest: S.I. = \( \frac{4800 \times 5 \times 2}{100} = \text{Rs. } 480 \). Amount to return = Principal + S.I. = 4800 + 480 = Rs. 5280.

Exam Tip: The amount to return (or total amount) is always the principal plus the simple interest earned over the period.

Question 15. Statement I: 125% is equal to 10 : 8 | Statement II: 10 : 8 and 4 : 5 are equal ratios.
(1) Statement I is true but statement II is false
(2) Statement I is false but statement II is true
(3) Both Statement I and statement II are true
(4) Both Statement I and statement II are false
Answer: (1) Statement I is true but statement II is false
In simple words: Statement I: \( 125\% = \frac{125}{100} = \frac{5}{4} = 10 : 8 \) (True). Statement II: \( 10 : 8 = 5 : 4 \) while \( 4 : 5 \) is different, so they are not equal (False).

Exam Tip: Always simplify ratios to their lowest terms before comparing them for equality.

Question 16. Statement I: If Aman buys a table for Rs. 4000 and sells it for Rs. 4400, he has a gain percentage of 10% | Statement II: Profit and loss percentage is calculated on the cost price.
(1) Statement I is true but statement II is false
(2) Statement I is false but statement II is true
(3) Both Statement I and statement II are true
(4) Both Statement I and statement II are false
Answer: (3) Both Statement I and statement II are true
In simple words: Statement I: Gain = 4400 - 4000 = Rs. 400. Gain percentage = \( \left( \frac{400}{4000} \times 100 \right)\% = 10\% \) (True). Statement II: Profit and loss percentages use cost price as the base (True).

Exam Tip: Both statements address key concepts in profit-loss calculations - verify each independently before selecting your answer.

Question 17. Statement I: Akash borrowed Rs. 10000 for 3 years at 6% p.a. simple interest. At the end of 3 years, he has to return Rs. 13000 | Statement II: Simple interest = \(\frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100}\)
(1) Statement I is true but statement II is false
(2) Statement I is false but statement II is true
(3) Both Statement I and statement II are true
(4) Both Statement I and statement II are false
Answer: (3) Both Statement I and statement II are true
In simple words: Statement I: S.I. = \( \frac{10000 \times 6 \times 3}{100} = \text{Rs. } 1800 \). Amount = 10000 + 1800 = Rs. 11800 (Wait - this does not match Rs. 13000, so Statement I is False). Let me recalculate: Actually, Statement I claims Rs. 13000, but the correct amount is Rs. 11800, so Statement I is False. Statement II gives the correct formula (True). The correct answer should be (2).

Exam Tip: When checking statement-based questions, calculate each value carefully and verify the claim made in the statement before confirming if it is true or false.

Question 1. Convert the following percentages into fractions in the simplest form:
(i) \( 12\frac{1}{2}\% \)
(ii) \( 66\frac{2}{3}\% \)
(iii) \( 8\frac{1}{3}\% \)
Answer: To change a percentage into a fraction, replace the % symbol with \( \frac{1}{100} \) and simplify to its lowest terms.
(i) \( 12\frac{1}{2}\% = \frac{25}{2}\% = \frac{25}{2} \times \frac{1}{100} = \frac{25}{200} = \frac{1}{8} \)

Therefore, \( 12\frac{1}{2}\% = \frac{1}{8} \)

(ii) \( 66\frac{2}{3}\% = \frac{200}{3}\% = \frac{200}{3} \times \frac{1}{100} = \frac{200}{300} = \frac{2}{3} \)

Therefore, \( 66\frac{2}{3}\% = \frac{2}{3} \)

(iii) \( 8\frac{1}{3}\% = \frac{25}{3}\% = \frac{25}{3} \times \frac{1}{100} = \frac{25}{300} = \frac{1}{12} \)

Therefore, \( 8\frac{1}{3}\% = \frac{1}{12} \)
In simple words: Write the mixed number as an improper fraction, multiply by \( \frac{1}{100} \), and reduce the result to lowest terms.

Exam Tip: Always convert mixed percentages to improper fractions first, then multiply by \( \frac{1}{100} \) and cancel common factors carefully to reach the simplest form.

Question 2. Express each of the following fractions as a percentage:
(i) \( \frac{5}{8} \)
(ii) \( \frac{13}{40} \)
(iii) \( \frac{7}{6} \)
Answer: To change a fraction into a percentage, multiply the fraction by 100 and write the % symbol.
(i) \( \frac{5}{8} = \left( \frac{5}{8} \times 100 \right)\% = \frac{500}{8}\% = 62.5\% = 62\frac{1}{2}\% \)

Therefore, \( \frac{5}{8} = 62\frac{1}{2}\% \)

(ii) \( \frac{13}{40} = \left( \frac{13}{40} \times 100 \right)\% = \frac{1300}{40}\% = 32.5\% = 32\frac{1}{2}\% \)

Therefore, \( \frac{13}{40} = 32\frac{1}{2}\% \)

(iii) \( \frac{7}{6} = \left( \frac{7}{6} \times 100 \right)\% = \frac{700}{6}\% = \frac{350}{3}\% = 116\frac{2}{3}\% \)

Therefore, \( \frac{7}{6} = 116\frac{2}{3}\% \)
In simple words: Multiply the fraction by 100 to get a decimal or whole number, then add the % sign.

Exam Tip: Divide the numerator by the denominator after multiplying by 100, simplify any remainders as fractions, and express mixed numbers where required.

Question 3. Express each of the following percentages as a decimal:
(i) 122%
(ii) 2.2%
(iii) \( 3\frac{1}{8}\% \)
Answer: To turn a percentage into a decimal, replace the % sign with \( \frac{1}{100} \) and write the result as a decimal.
(i) 122% \( = \frac{122}{100} = 1.22 \)

Therefore, 122% = 1.22

(ii) 2.2% \( = \frac{2.2}{100} = 0.022 \)

Therefore, 2.2% = 0.022

(iii) \( 3\frac{1}{8}\% = \frac{25}{8}\% = \frac{25}{8} \times \frac{1}{100} = \frac{25}{800} = 0.03125 \)

Therefore, \( 3\frac{1}{8}\% = 0.03125 \)
In simple words: Remove the % sign and divide the number by 100. Move the decimal point two places to the left.

Exam Tip: When the percentage is a mixed number, convert it to an improper fraction first, then multiply by \( \frac{1}{100} \) before writing as a decimal.

Question 4. Express 0.0345 as a percentage.
Answer: To change a decimal into a percentage, multiply by 100 and add the % sign.
\( (0.0345 \times 100)\% = 3.45\% \)

Therefore, 0.0345 = 3.45%
In simple words: Multiply the decimal number by 100 and put the % symbol at the end.

Exam Tip: Multiplying by 100 is the same as moving the decimal point two places to the right.

Question 5. Convert each part of the ratio 5 : 6 : 9 to a percentage.
Answer: The given ratio is 5 : 6 : 9.

Total number of parts = 5 + 6 + 9 = 20.

Percentage of first part \( = \left( \frac{5}{20} \times 100 \right)\% = 25\% \)

Percentage of second part \( = \left( \frac{6}{20} \times 100 \right)\% = 30\% \)

Percentage of third part \( = \left( \frac{9}{20} \times 100 \right)\% = 45\% \)

Therefore, the three parts are 25%, 30% and 45%.
In simple words: Add all the parts of the ratio to get the total. For each part, divide it by the total and multiply by 100 to get its percentage.

Exam Tip: Always verify that all three percentages add up to 100% as a final check.

Question 6. (i) What percent of a day is half an hour?
Answer: We know that 1 day = 24 hours = 24 × 60 = 1440 minutes.

Half an hour = 30 minutes.

Required percentage \( = \left( \frac{30}{1440} \times 100 \right)\% = \frac{3000}{1440}\% = \frac{25}{12}\% = 2\frac{1}{12}\% \)

Therefore, half an hour is \( 2\frac{1}{12}\% \) of a day.
In simple words: Express both quantities in the same unit (minutes), then divide the smaller by the larger and multiply by 100.

Exam Tip: Convert all measurements to the same unit before calculating the percentage; this prevents errors in comparison.

Question 6. (ii) What percent is \( \frac{3}{4} \) metres of \( 4\frac{1}{2} \) metres?
Answer: Here, \( 4\frac{1}{2} \) m = \( \frac{9}{2} \) m.

Required percentage \( = \left( \frac{3}{4} \div \frac{9}{2} \times 100 \right)\% = \left( \frac{3}{4} \times \frac{2}{9} \times 100 \right)\% = \left( \frac{6}{36} \times 100 \right)\% = \left( \frac{1}{6} \times 100 \right)\% = \frac{100}{6}\% = 16\frac{2}{3}\% \)

Therefore, \( \frac{3}{4} \) metres is \( 16\frac{2}{3}\% \) of \( 4\frac{1}{2} \) metres.
In simple words: Convert mixed numbers to improper fractions, divide the first by the second, and multiply the result by 100.

Exam Tip: When dividing fractions, multiply by the reciprocal of the divisor, then simplify before multiplying by 100.

Question 7. The population of a town decreased from 25000 to 24500. Find the percentage decrease.
Answer: Original population = 25000 and new population = 24500.

Decrease in population = 25000 - 24500 = 500.

Percentage decrease \( = \left( \frac{500}{25000} \times 100 \right)\% = \frac{50000}{25000}\% = 2\% \)

Therefore, the percentage decrease = 2%.
In simple words: Find the drop in population, divide it by the original population, and multiply by 100 to get the percentage.

Exam Tip: Always use the original (starting) value as the denominator when calculating percentage decrease, not the new value.

Question 8. Arun bought a car for Rs 350000. The next year, the price went upto Rs 370000. What was the percentage increase in the price?
Answer: Original price = Rs 350000 and new price = Rs 370000.

Increase in price = 370000 - 350000 = Rs 20000.

Percentage increase \( = \left( \frac{20000}{350000} \times 100 \right)\% = \frac{2000000}{350000}\% = \frac{40}{7}\% = 5\frac{5}{7}\% \)

Therefore, the percentage increase in price = \( 5\frac{5}{7}\% \).
In simple words: Subtract the original price from the new price, divide the difference by the original price, and multiply by 100.

Exam Tip: For percentage increase, the denominator must be the original (starting) price, not the final price.

Question 9. The population of a village has decreased by 6%. If the original population was 3650, find the population after decrease.
Answer: Original population = 3650.

Decrease = 6% of 3650 = \( \frac{6}{100} \times 3650 = 219 \)

Population after decrease = 3650 - 219 = 3431.

Therefore, the population after decrease = 3431.
In simple words: Calculate 6% of the original population, then subtract this amount from the original to get the new population.

Exam Tip: When finding the decrease, always multiply the percentage by the original value, then subtract the result from that original value.

Question 10. 43% of the students in a school are girls. If the number of boys is 1482, find:
(i) the total strength of the school
(ii) number of girls in the school.
Answer: Percentage of girls = 43%, so percentage of boys = (100 - 43)% = 57%.

(i) Let the total strength of the school be x.

57% of x = 1482

\( \Rightarrow \frac{57}{100} \times x = 1482 \)

\( \Rightarrow x = \frac{1482 \times 100}{57} = 2600 \)

Therefore, the total strength of the school = 2600.

(ii) Number of girls = total strength - number of boys = 2600 - 1482 = 1118.

Therefore, the number of girls in the school = 1118.
In simple words: Use the percentage of boys to find the total by forming an equation. Then subtract the boys from the total to find the girls.

Exam Tip: Set up the percentage equation carefully: if 57% represents 1482 boys, divide 1482 by the decimal form of 57% (0.57) to find the total.

Question 11. On selling an article for Rs 1027, Meena suffered a loss of Rs 273. Find her loss percentage.
Answer: Selling Price (S.P.) of the article = Rs 1027 and loss = Rs 273.

Cost Price (C.P.) = S.P. + loss = 1027 + 273 = Rs 1300.

Loss percentage \( = \left( \frac{273}{1300} \times 100 \right)\% = \frac{27300}{1300}\% = 21\% \)

Therefore, the loss percentage = 21%.
In simple words: Add the loss amount to the selling price to get the cost price. Then divide the loss by the cost price and multiply by 100.

Exam Tip: Remember that loss% is always calculated based on the cost price, not the selling price. Use the formula: Loss% = \( \frac{\text{Loss}}{C.P.} \times 100 \).

Question 12. By selling a lamp for Rs 710, a trader suffers a loss of Rs 40. Find the cost price of the lamp. At what price this lamp should be sold in order to gain 10%?
Answer: Selling Price (S.P.) of the lamp = Rs 710 and loss = Rs 40.

Cost Price (C.P.) = S.P. + loss = 710 + 40 = Rs 750.

To gain 10%:

Gain = 10% of 750 = \( \frac{10}{100} \times 750 = \text{Rs } 75 \)

New S.P. = C.P. + gain = 750 + 75 = Rs 825.

Therefore, the cost price of the lamp = Rs 750 and to gain 10% it should be sold for Rs 825.
In simple words: Find the cost price by adding loss to the selling price. Then calculate the desired gain as a percentage of cost price and add it to the cost price to find the new selling price.

Exam Tip: When a gain is required, the formula is New S.P. = C.P. + (Gain% of C.P.). Always calculate the gain percentage on the cost price, not the selling price.

Question 13. If Rs 6000 is borrowed at 6.5% per annum simple interest, find the interest and the amount to be paid at the end of 3 years.
Answer: Here, P = Rs 6000, R = 6.5% per annum and T = 3 years.

\( \Rightarrow I = \frac{P \times R \times T}{100} \)

\( \Rightarrow I = \frac{6000 \times 6.5 \times 3}{100} = \frac{117000}{100} = \text{Rs } 1170 \)

Amount = P + I = Rs 6000 + Rs 1170 = Rs 7170.

Therefore, the interest = Rs 1170 and the amount = Rs 7170.
In simple words: Use the simple interest formula to multiply principal, rate, and time, then divide by 100. Add this interest to the principal to get the total amount.

Exam Tip: Always verify that the amount equals principal plus interest. Remember the formula: Simple Interest = \( \frac{P \times R \times T}{100} \).

Question 14. How long will it take for Rs 1860 invested at the rate of 9.5% per annum simple interest to amount to Rs 2449?
Answer: Here, P = Rs 1860, A = Rs 2449 and R = 9.5% per annum.

I = A - P = 2449 - 1860 = Rs 589.

\( \Rightarrow T = \frac{I \times 100}{P \times R} \)

\( \Rightarrow T = \frac{589 \times 100}{1860 \times 9.5} = \frac{58900}{17670} = \frac{10}{3} = 3\frac{1}{3} \text{ years} = 3 \text{ years } 4 \text{ months} \)

Therefore, it will take 3 years 4 months.
In simple words: Subtract the principal from the final amount to get the interest. Then use the time formula: T = \( \frac{\text{Interest} \times 100}{\text{Principal} \times \text{Rate}} \).

Exam Tip: When the result is a fraction like \( \frac{10}{3} \), convert the fractional part to months by multiplying by 12. Here, \( \frac{1}{3} \times 12 = 4 \) months.

Question 15. At what rate will Rs 7200 fetch a simple interest of Rs 3024 in 4 years?
Answer: Here, P = Rs 7200, I = Rs 3024 and T = 4 years.

\( \Rightarrow R = \frac{I \times 100}{P \times T}\% \)

\( \Rightarrow R = \frac{3024 \times 100}{7200 \times 4}\% = \frac{302400}{28800}\% = 10.5\% \)

Therefore, the rate of interest = 10.5% p.a.
In simple words: Rearrange the simple interest formula to solve for rate: R = \( \frac{\text{Interest} \times 100}{\text{Principal} \times \text{Time}} \).

Exam Tip: Check your answer by working backwards: calculate the simple interest using the found rate and confirm it equals the given interest amount.

Question 16. What sum of money will yield a simple interest of Rs 1155 in 3 years 6 months at 11% p.a.?
Answer: Here, I = Rs 1155, R = 11% per annum and T = 3 years 6 months = \( 3\frac{1}{2} = \frac{7}{2} \) years.

\( \Rightarrow P = \frac{I \times 100}{R \times T} \)

\( \Rightarrow P = \frac{1155 \times 100}{11 \times \frac{7}{2}} = \frac{115500}{11 \times 3.5} = \frac{115500}{38.5} = \text{Rs } 3000 \)

Therefore, the principal = Rs 3000.
In simple words: Convert the mixed time period to an improper fraction, then rearrange the simple interest formula to solve for principal: P = \( \frac{\text{Interest} \times 100}{\text{Rate} \times \text{Time}} \).

Exam Tip: Always convert years and months to a single fraction before substituting into the formula (e.g., 3 years 6 months = \( \frac{7}{2} \) years).

Question 17. Medha deposited 20% of her money in a bank. After spending 20% of the remainder, she has ₹ 48000 left with her. How much did she originally have?
Answer: Suppose Medha started with ₹ x. She put 20% of x into the bank, which equals x/5. This leaves her with 4x/5. From this remaining amount, she spent 20%, which is (20/100) × (4x/5) = 4x/25. So the money she has now is (4x/5) - (4x/25). When we simplify this, we get (20x - 4x)/25 = 16x/25. We know this equals ₹ 48000, so 16x/25 = 48000. Solving for x: x = (48000 × 25)/16 = 75000.
In simple words: Work backwards from what she has left. If 16 parts equal 48000, then 25 parts (her original money) equal 75000.

Exam Tip: Always track what fraction remains after each deduction - deposit first, then spending from the remainder. Use a variable for the unknown total and set up the equation carefully.

Question 18. If Mohan's income is 25% more than Raman's income, then by what percent is Raman's income less than Mohan's income?
Answer: Assume Raman earns ₹ 100. Since Mohan earns 25% more, Mohan gets 100 + 25 = ₹ 125. The difference between their incomes is 125 - 100 = ₹ 25. To find what percent this difference is of Mohan's income: (25/125) × 100 = 2500/125 = 20%. Therefore, Raman's income is 20% less than Mohan's.
In simple words: When comparing how much less Raman has, you must divide the difference by Mohan's amount, not Raman's. The base for the percentage calculation changes depending on who you are comparing to.

Exam Tip: Notice that a 25% increase is not the same as a 25% decrease when reversing the comparison. Always identify which amount is the base for your percentage calculation.

Question 19. A person preparing medicine wants to convert 15% alcohol solution into 32% alcohol solution. Find how much pure alcohol should he mix with 400 mL of 15% alcohol solution to obtain it.
Answer: In 400 mL of 15% solution, the pure alcohol content is (15/100) × 400 = 60 mL. Let x mL be the quantity of pure alcohol added. After mixing, the total volume becomes (400 + x) mL, and the total pure alcohol becomes (60 + x) mL. For the final mixture to be 32% alcohol: (60 + x)/(400 + x) = 32/100. Cross-multiplying: 100(60 + x) = 32(400 + x), which gives 6000 + 100x = 12800 + 32x. Rearranging: 68x = 6800, so x = 100 mL. Therefore, 100 mL of pure alcohol must be added.
In simple words: The pure alcohol already in the mixture plus the pure alcohol you add, divided by the new total volume, must equal 32%.

Exam Tip: Set up the percentage equation using final amounts over final total volume. Always account for both the starting pure alcohol and the added pure alcohol in your numerator.

Question 20. A manufacturer sells an item to an agency at a profit of 25%. The agency sells the item to a shopkeeper at 10% profit and shopkeeper sells the item at a profit of 20%. If the selling price of the item is ₹ 594, find the manufacturing price.
Answer: Work backwards through the chain. The shopkeeper sold for ₹ 594 with a 20% profit, so the shopkeeper's cost price is 594/(1.20) = (594 × 100)/120 = ₹ 495. This ₹ 495 was the agency's selling price. The agency bought at 10% profit, so the agency's cost price is 495/(1.10) = (495 × 100)/110 = ₹ 450. This ₹ 450 was the manufacturer's selling price. The manufacturer sold at 25% profit, so the manufacturing price is 450/(1.25) = (450 × 100)/125 = ₹ 360.
In simple words: Each buyer's selling price becomes the next person's cost price. To find cost from selling price when you know the profit percent, divide by (1 + profit rate as a decimal).

Exam Tip: In multi-step profit problems, always work backwards from the final selling price, converting each profit amount step by step. Check your answer by multiplying forward: ₹ 360 × 1.25 × 1.10 × 1.20 should equal ₹ 594.