ML Aggarwal Class 8 Maths Solutions Chapter 07 Percentage

Exercise 7.1

Question 1. Express the following percentages as fractions:
(i) 356%
(ii) 2 ½%
(iii) 16 2/3 %
Answer:
(i) To convert 356% to a fraction, divide it by 100. This gives 356/100. When we reduce this by cancelling common factors, we get 89/25, which equals 3 14/25 as a mixed number.
(ii) First, express 2 ½ as an improper fraction: 5/2. Now write the percentage as a fraction by dividing by 100: (5/2)/100, which simplifies to 5/(2 × 100) = 5/200 = 1/40.
(iii) Convert 16 2/3 to an improper fraction: 50/3. Write this as a percentage fraction: (50/3)/100 = 50/(3 × 100) = 50/300 = 1/6.
In simple words: To turn a percentage into a fraction, divide the number by 100 and reduce it as much as you can.

Exam Tip: Always simplify the fraction to its lowest terms by finding the greatest common divisor of the numerator and denominator.

Question 2. Express the following fractions as percentages:
(i) 3/2
(ii) 9/20
(iii) 1 ¼
Answer:
(i) To change 3/2 to a percentage, multiply it by 100: (3/2) × 100% = (3 × 100)/2 % = 300/2 % = 150%.
(ii) For 9/20, multiply by 100: (9/20) × 100% = (9 × 100)/20 % = 900/20 % = 45%.
(iii) First convert 1 ¼ to an improper fraction: 5/4. Then multiply by 100: (5/4) × 100% = (5 × 100)/4 % = 500/4 % = 125%.
In simple words: To turn a fraction into a percentage, multiply it by 100.

Exam Tip: Remember that multiplying by 100 and adding the % symbol converts any fraction into its percentage form instantly.

Question 3. Express the following fractions as decimals. Then express the decimals as percentages:
(i) ¾
(ii) 5/8
(iii) 3/16
Answer:
(i) Divide 3 by 4 to get the decimal: 3 ÷ 4 = 0.75. Convert to percentage by multiplying by 100: 0.75 × 100% = 75%.
(ii) Divide 5 by 8: 5 ÷ 8 = 0.625. Multiply by 100 to get the percentage: 0.625 × 100% = 62.5%.
(iii) Divide 3 by 16: 3 ÷ 16 = 0.1875. Convert to percentage: 0.1875 × 100% = 18.75%.
In simple words: First divide the top number by the bottom number to get a decimal. Then multiply that decimal by 100 to turn it into a percentage.

Exam Tip: Use long division to convert fractions to decimals, and remember that multiplying by 100 shifts the decimal point two places to the right.

Question 4. Express the following fractions as decimals correct to four decimal places. Then express the decimals as percentages:
(i) 2/3
(ii) 5/6
(iii) 4/7
Answer:
(i) When we divide 2 by 3, we get 0.6667 (rounded to four decimal places). Multiply by 100 to find the percentage: 0.6667 × 100% = 66.67%.
(ii) Dividing 5 by 6 gives 0.8333 (to four decimal places). Converting to percentage: 0.8333 × 100% = 83.33%.
(iii) Dividing 4 by 7 gives 0.5714 (to four decimal places). The percentage is: 0.5714 × 100% = 57.14%.
In simple words: Divide the numbers and round to four decimals. Then multiply by 100 to get the percentage, rounded to two decimal places.

Exam Tip: When rounding, check the fifth decimal place to decide whether to round up or down. This ensures accuracy to the required number of places.

Question 5. Express the following ratios as percentages:
(i) 17:20
(ii) 13:18
(iii) 93:80
Answer:
(i) Write the ratio 17:20 as a fraction: 17/20. Multiply by 100 to convert to percentage: (17/20) × 100% = (17 × 5)% = 85%.
(ii) Express 13:18 as a fraction: 13/18. Multiply by 100: (13/18) × 100% = (1300/18)% = (650/9)% = 72 2/9%.
(iii) Write 93:80 as a fraction: 93/80. Multiply by 100: (93/80) × 100% = (9300/80)% = (465/4)% = 116.25%.
In simple words: Change a ratio to a fraction by putting the first number on top and the second number on the bottom. Then multiply by 100 to make it a percentage.

Exam Tip: Always reduce fractions to simplest form before multiplying by 100 to make calculation easier and reduce the chance of errors.

Question 6. Express the following percentages as decimals:
(i) 20%
(ii) 2%
(iii) 3 ¼%
Answer:
(i) To change 20% to a decimal, divide by 100: 20/100 = 0.2.
(ii) For 2%, divide by 100: 2/100 = 0.02.
(iii) First change 3 ¼ to an improper fraction: 13/4. Now express as a decimal percentage: (13/4)/100 = 13/400 = 0.0325.
In simple words: To turn a percentage into a decimal, divide the number by 100. You can think of it as moving the decimal point two places to the left.

Exam Tip: Remember that dividing by 100 moves the decimal point exactly two places to the left, making percentage-to-decimal conversion quick and straightforward.

Question 7. Find the value of:
(i) 27 % of Rs.50
(ii) 6 ¼ % of 25 kg
Answer:
(i) To find 27% of Rs.50, write it as a fraction: (27/100) × 50 = (27 × 50)/100 = 1350/100 = Rs.13.50.
(ii) First change 6 ¼% to a fraction: 25/4%. Now calculate: (25/4)/100 × 25 = (25 × 25)/(4 × 100) = 625/400 = 25/16 = 1 9/16 kg.
In simple words: To find a percentage of something, change the percentage to a fraction or decimal and multiply it by the amount.

Exam Tip: Convert percentages to fractions first, then multiply by the given quantity to avoid calculation errors, especially with mixed number percentages.

Question 8. What percent is:
(i) 300 g of 2 kg
(ii) Rs.7.50 of Rs.6
Answer:
(i) First, express both quantities in the same units. Convert 2 kg to grams: 2 kg = 2000 g. Now find what percent 300 g is of 2000 g: (300/2000) × 100% = (300/20) ÷ 10 × 100% = (30/2)% = 15%.
(ii) To find what percent Rs.7.50 is of Rs.6: (7.50/6) × 100% = (7.50 × 100)/6 % = 750/6 % = 125%.
In simple words: Divide the first amount by the second amount, then multiply by 100 to get the percentage.

Exam Tip: Always ensure both quantities are in the same unit of measurement before calculating the percentage to avoid incorrect answers.

Question 9. What percent of:
(i) 50 kg is 65 kg
(ii) Rs.9 is Rs.4
Answer:
(i) Let x% of 50 kg equal 65 kg. Set up the equation: (x/100) × 50 = 65. Simplify: x/2 = 65. Solve for x: x = 130. Therefore, 130% of 50 kg is 65 kg.
(ii) Let x% of Rs.9 equal Rs.4. Set up: (x/100) × 9 = 4. Solve: x = (4 × 100)/9 = 400/9 = 44 4/9. Therefore, 44 4/9 % of Rs.9 is Rs.4.
In simple words: Set up an equation using x for the unknown percentage, then solve to find what percent one number is of another.

Exam Tip: Use the format (x/100) × base = result and cross-multiply to solve for the unknown percentage quickly.

Question 10. (i) If 16 2/3 % of a number is 25, find the number.
(ii) If 13.25% of a number is 159, find the number.
Answer:
(i) Let the number be x. We know 16 2/3 % of x equals 25. Change 16 2/3 to an improper fraction: 50/3. Write the equation: (50/3)/100 × x = 25, which simplifies to (50/300) × x = 25. Solve: x = (25 × 300)/50 = 25 × 6 = 150. The number is 150.
(ii) Let the number be x. We have 13.25% of x equals 159. Write: (13.25/100) × x = 159. Solve: x = (159 × 100)/13.25. To simplify, multiply numerator and denominator by 100: x = (159 × 10000)/1325 = (159 × 400)/53 = 3 × 400 = 1200. The number is 1200.
In simple words: When a percentage of an unknown number is given, divide that percentage amount by the percentage itself (written as a decimal) to find the number.

Exam Tip: Convert mixed number percentages to improper fractions before setting up equations to minimize calculation errors.

Question 11. (i) Increase the number 60 by 30 %
(ii) Decrease the number 750 by 10%
Answer:
(i) When increasing 60 by 30%, the new number is: 60 × (1 + 30/100) = 60 × (1 + 3/10) = 60 × 13/10 = (60 × 13)/10 = 780/10 = 78.
(ii) When decreasing 750 by 10%, the new number is: 750 × (1 - 10/100) = 750 × (1 - 1/10) = 750 × 9/10 = (750 × 9)/10 = 6750/10 = 675.
In simple words: To increase a number, multiply it by (1 + percent/100). To decrease a number, multiply it by (1 - percent/100).

Exam Tip: The key formula is: New value = Original value × (1 ± percent/100), where you add for increase and subtract for decrease.

Question 12. (i) What number when increased by 15% becomes 299?
(ii) On decreasing the number by 18%, it becomes 697. Find the number.
Answer:
(i) Let the original number be x. When increased by 15%, it becomes: x × (1 + 15/100) = 299. Simplify: x × (1 + 3/20) = 299, so x × 23/20 = 299. Solve: x = (299 × 20)/23 = (13 × 20) = 260. The original number is 260.
(ii) Let the original number be x. When decreased by 18%, it becomes: x × (1 - 18/100) = 697. Simplify: x × (1 - 18/100) = 697, which gives x × 82/100 = 697. Solve: x = (697 × 100)/82 = (697 × 50)/41 = 17 × 50 = 850. The original number is 850.
In simple words: If you know the final value after a percentage increase or decrease, divide the final value by the decimal form of (1 ± percent) to find the original number.

Exam Tip: Always set up the equation carefully: if a value increased becomes 299, then original × 1.15 = 299; if a value decreased becomes 697, then original × 0.82 = 697.

Question 13. Mr. Khanna spent 83% of his salary and saved Rs.1870. Calculate his monthly salary.
Answer: If Mr. Khanna spent 83% of his salary, then the amount he saved is the remaining percentage: 100% - 83% = 17%. Since 17% of his salary equals Rs.1870, we can find his total salary. Divide the saved amount by the saved percentage: Salary = (1870 × 100)/17 = 110 × 100 = Rs.11000.
In simple words: If someone spends 83%, they save 17%. If the 17% saved is Rs.1870, then the full salary is found by dividing 1870 by 0.17.

Exam Tip: Always check that the percentages for spent and saved add up to 100% - this is a good way to verify your understanding of the problem.

Question 14. In school, 38% of the students are girls. If the number of boys is 1023, find the total strength of the school.
Answer: If 38% are girls, then the percentage of boys is 100% - 38% = 62%. Let the total strength be x. We know that 62% of x equals 1023. Set up: (62/100) × x = 1023. Solve: x = (1023 × 100)/62 = 1023 × 50/31 = 33 × 50 = 1650. The total strength of the school is 1650.
In simple words: If boys are 62% of all students and there are 1023 boys, divide 1023 by 0.62 to find the total number of students.

Exam Tip: Remember that the two groups (girls and boys) must add up to 100%, so if you know one percentage, you can find the other immediately.

Question 15. The price of an article increases from Rs.960 to Rs.1080. Find the percentage increase in the price.
Answer: First, find the actual increase in price: 1080 - 960 = Rs.120. The percentage increase is calculated on the original price: (120/960) × 100% = (1/8) × 100% = 100/8 % = 12.5%.
In simple words: Find the difference in price, then divide it by the original price and multiply by 100 to get the percentage increase.

Exam Tip: Always calculate percentage change on the original value (the starting amount), not the new value - this is a common mistake to avoid.

Question 16. In a straight contest, the loser polled 42% votes and lost by 14400 votes. Find the total number of votes polled. If the total number of eligible voters was 1 lakh, find what percentage of voters did not vote.
Answer: If the loser received 42%, the winner received 100% - 42% = 58%. The difference between them is 58% - 42% = 16% of total votes polled. Since this difference equals 14400 votes: 16% of votes polled = 14400. Solve: votes polled = (14400 × 100)/16 = 90000. Now, total eligible voters = 100000. Voters who did not vote = 100000 - 90000 = 10000. Percentage who did not vote = (10000/100000) × 100% = 10%.
In simple words: Find the percentage difference between the winner and loser, then use that to calculate the total votes. Finally, find what portion of eligible voters actually participated.

Exam Tip: Work systematically through each part: first find total votes, then find non-voters, then find the percentage of non-voters from the eligible total.

Question 17. Out of 8000 candidates, 60% were boys. If 80% of the boys and 90% of the girls passed the exam, find the number of candidates who failed.
Answer: First, find the numbers of boys and girls: Boys = 60% of 8000 = (60/100) × 8000 = 4800. Girls = 8000 - 4800 = 3200. Next, find how many passed: Passed boys = 80% of 4800 = 3840. Passed girls = 90% of 3200 = 2880. Total passed = 3840 + 2880 = 6720. Finally, find those who failed: Failed = 8000 - 6720 = 1280.
In simple words: Find how many boys and girls there are. Then find how many from each group passed. Add the passed numbers and subtract from the total to get those who failed.

Exam Tip: Always break down compound percentage problems into steps: (1) find the subgroups, (2) find the outcome for each subgroup, (3) combine the results.

Question 18. (i) Find the percentage of students who failed in any of the subjects.
(ii) Find the percentage of students who passed in both subjects.
(iii) If the number of students who failed only in English was 25, find the total number of students.
In an exam, ¼ of the students failed both in English and Maths, 35% of the students failed in Maths and 30% failed in English.
Answer:
(i) Let the total number of students be x. Failed both = x/4. Failed in Maths = 7x/20. Failed in English = 3x/10. Using the principle of inclusion-exclusion, students who failed in at least one subject = (7x/20 + 3x/10) - x/4. Finding LCM and simplifying: = (7x + 6x)/20 - x/4 = 13x/20 - x/4 = (13x - 5x)/20 = 8x/20. Percentage = (8x/20)/x × 100% = 40%.
(ii) Percentage who passed both = 100% - 40% = 60%.
(iii) Students who failed only in English = (3x/10) - (x/4) = 25. Solving: (6x - 5x)/20 = 25, so x/20 = 25, giving x = 500. Total students = 500.
In simple words: When finding how many failed in any subject, add those who failed in each subject, then subtract those who failed in both (to avoid counting them twice). Those who passed both are the remainder.

Exam Tip: Use a Venn diagram mentally to organize which students failed in which subjects, making it easier to apply the inclusion-exclusion principle correctly.

Question 19. On increasing the price of an article by 16%, it becomes Rs.1479. What was its original price?
Answer: Let the original price be x. When increased by 16%, the new price is x × (1 + 16/100) = x × 116/100 = 1479. Solve for x: x = (1479 × 100)/116 = (1479 × 25)/29. Perform the division: 1479 ÷ 29 = 51, so x = 51 × 25 = 1275. The original price was Rs.1275.
In simple words: If a number increased by 16% becomes 1479, divide 1479 by 1.16 to find the original number.

Exam Tip: When the final value is given after a percentage increase, divide by (1 + percent/100) to reverse the operation and find the original value.

Question 20. Pratibha reduced her weight by 15%. If now she weighs 59.5 kg, what was her earlier weight?
Answer: Let her original weight be x kg. After reducing by 15%, her new weight is x × (1 - 15/100) = x × 85/100 = 59.5. Solve: x = (59.5 × 100)/85 = 5950/85 = 70 kg. Her original weight was 70 kg.
In simple words: If a person's weight decreased by 15% to become 59.5 kg, divide 59.5 by 0.85 to find the original weight.

Exam Tip: To reverse a percentage decrease, divide the final value by (1 - percent/100); to reverse a percentage increase, divide by (1 + percent/100).

Question 21. In a sale, a shop reduces all its prices by 15%. Calculate:
(i) the cost of an article which was originally priced at Rs.40.
(ii) the original price of an article which was sold for Rs.20.40.
Answer:
(i) Original price = Rs.40. Reduction = 15% of 40 = (15/100) × 40 = Rs.6. Sale price = 40 - 6 = Rs.34.
(ii) Sale price = Rs.20.40. The sale price is 85% of the original price (since 15% was reduced). So 85% of original = 20.40. Original = (20.40 × 100)/85 = Rs.24.
In simple words: To find the sale price, multiply the original by (1 - reduction%). To find the original price from the sale price, divide by (1 - reduction%).

Exam Tip: Always determine what percentage remains after the reduction (85% in this case) before calculating the original price from a given sale price.

Question 22. Increase the price of Rs.200 by 10% and then decrease the new price by 10%. Is the final price same as the original one?
Answer: Starting price = Rs.200. After increasing by 10%: New price = 200 × (1 + 10/100) = 200 × 110/100 = Rs.220. After decreasing this new price by 10%: Final price = 220 × (1 - 10/100) = 220 × 90/100 = Rs.198. The final price Rs.198 is not equal to the original price Rs.200. The difference shows that a percentage increase followed by an equal percentage decrease does not return to the original value.
In simple words: Increasing by 10% and then decreasing by 10% does not bring you back to where you started, because the 10% decrease is calculated on the larger amount.

Exam Tip: Remember that percentages are calculated on different base amounts - the decrease is on 220, not on 200, so the result is always less than the original when you increase then decrease by the same percentage.

Question 23. Chandani purchased some parrots. 20% flew away and 5% died. Of the remaining, 45% were sold. Now 33 parrots remain. How many parrots had Chandani purchased?
Answer: Let the number of parrots purchased be x. Parrots that flew away = 20% of x = x/5. Parrots that died = 5% of x = x/20. Parrots remaining after losses = x - (x/5 + x/20) = x - (4x + x)/20 = x - 5x/20 = x - x/4 = 3x/4. Of these remaining parrots, 45% were sold: Sold = 45% of 3x/4 = (45/100) × (3x/4) = 27x/80. Parrots not sold = 3x/4 - 27x/80 = (60x - 27x)/80 = 33x/80. We're told 33 parrots remain unsold: 33x/80 = 33, so x = 80. Chandani purchased 80 parrots.
In simple words: Track what happens to the parrots at each step: some fly away, some die, some are sold. The 33 remaining parrots equal 33/80 of the original amount.

Exam Tip: For multi-step percentage problems, work through each step carefully and keep track of the decreasing amounts as a fraction of the original quantity.

Question 24. A candidate who gets 36% marks in an examination fails by 24 marks but another candidate, who gets 43% marks, gets 18 more marks than the minimum passmarks. Find the maximum marks and the percentage of passmarks.
Answer: Let the maximum marks be x. First candidate's marks = 36% of x = 36x/100. This candidate fails by 24 marks, meaning pass mark = 36x/100 + 24. Second candidate's marks = 43% of x = 43x/100. This candidate exceeds the pass mark by 18, meaning pass mark = 43x/100 - 18. Since both expressions equal the pass mark: 36x/100 + 24 = 43x/100 - 18. Solving: 24 + 18 = 43x/100 - 36x/100, so 42 = 7x/100, giving x = 600. Maximum marks = 600. Pass mark = (36/100 × 600) + 24 = 216 + 24 = 240. Percentage of pass marks = (240/600) × 100% = 40%.
In simple words: Set up equations using the information about both candidates. The pass mark is the same for both, so the two equations can be set equal to solve for the total marks.

Exam Tip: In problems involving two different situations with the same pass mark, equate the two expressions for the pass mark to create a solvable equation for the unknown variable.

Exercise 7.2

Question 1. Find the profit or loss percentage, when:
(i) C.P. = Rs.400, S.P. = Rs.468
(ii) C.P. = Rs.13600, S.P. = Rs.12104
Answer:
(i) Given: C.P. = Rs.400, S.P. = Rs.468
Profit = S.P. - C.P. = 468 - 400 = Rs.68
Profit % = (Profit ÷ C.P.) × 100 = (68 ÷ 400) × 100 = 17%

(ii) Given: C.P. = Rs.13600, S.P. = Rs.12104
Loss = C.P. - S.P. = 13600 - 12104 = Rs.1496
Loss % = (Loss ÷ C.P.) × 100 = (1496 ÷ 13600) × 100 = 11%
In simple words: To find profit or loss percentage, first work out if you gained or lost money. Then divide that amount by what you originally paid and multiply by 100.

Exam Tip: Always check whether it is a profit or loss situation - if S.P. is more than C.P., it is profit; if less, it is loss. Use the correct formula based on which one applies.

Question 2. By selling an article for Rs.1636.25, a dealer gains Rs.96.25. Find his gain per cent.
Answer: Given: S.P. of an article = Rs.1636.25, Gain = Rs.96.25
C.P. = S.P. - Gain = 1636.25 - 96.25 = Rs.1540
Gain % = (Gain ÷ C.P.) × 100 = (96.25 ÷ 1540) × 100 = 6.25% or 6¼%
In simple words: To find the gain percentage, take the money you made and divide it by how much you spent to buy the item, then multiply by 100.

Exam Tip: Make sure to use C.P. (not S.P.) in the denominator when calculating gain or loss percentage - this is a common mistake students make.

Question 3. By selling an article for Rs.770, a man incurs a loss of Rs.110. Find his loss percentage.
Answer: Given: S.P. of an article = Rs.770, Loss = Rs.110
C.P. = S.P. + Loss = 770 + 110 = Rs.880
Loss % = (Loss ÷ C.P.) × 100 = (110 ÷ 880) × 100 = 12.5%
In simple words: When there is a loss, add the loss amount to the selling price to get the cost price. Then calculate the loss percentage using the same formula.

Exam Tip: Remember that when loss is given, C.P. = S.P. + Loss, whereas when profit is given, C.P. = S.P. - Profit.

Question 4. Rashida bought 25 dozen eggs at the rate of Rs.9.60 per dozen. 30 eggs were broken in the transaction and she sold the remaining eggs at one rupee each. Find her gain or loss percentage.
Answer: Given: C.P. of one dozen eggs = Rs.9.60
C.P. of 25 dozen eggs = 25 × 9.60 = Rs.240
Total eggs = 25 × 12 = 300 eggs
Eggs broken = 30
Remaining eggs = 300 - 30 = 270
S.P. of one egg = Rs.1
S.P. of 270 eggs = 270 × 1 = Rs.270
Profit = S.P. - C.P. = 270 - 240 = Rs.30
Profit % = (Profit ÷ C.P.) × 100 = (30 ÷ 240) × 100 = 12.5%
In simple words: Calculate the total cost and the number of items you can actually sell. Then find the selling total and work out the profit percentage the normal way.

Exam Tip: When some goods are damaged or lost, always reduce the quantity before calculating the selling price. Only the remaining quantity is sold.

Question 5. The cost of an article was Rs.20000 and Rs.1400 were spent on its repairs. If it is sold for a profit of 20%, find the selling price of the article.
Answer: Given: Cost of an article = Rs.20000, Cost of repair = Rs.1400, Profit = 20%
Total C.P. = 20000 + 1400 = Rs.21400
S.P. = C.P. × (100 + Profit %) ÷ 100 = 21400 × 120 ÷ 100 = Rs.25680
In simple words: Add all the costs together. Then use the profit percentage formula to find how much to sell it for.

Exam Tip: Always include all overhead costs (repairs, transportation, etc.) in the total cost before applying profit percentage.

Question 6. A shopkeeper buys 200 bicycles at Rs.1200 per bicycle. He spends Rs.30 per bicycle on transportation. He also spends Rs.4000 on advertising. Then he sells all the bicycles at Rs.1350 per piece. Find his profit or loss. Also, calculate it as a percentage.
Answer: Given: C.P. of one bicycle = Rs.1200, Quantity = 200
C.P. of 200 bicycles = 1200 × 200 = Rs.240000
Transportation cost per bicycle = Rs.30
Total transportation cost = 30 × 200 = Rs.6000
Advertising cost = Rs.4000
Total C.P. = 240000 + 6000 + 4000 = Rs.250000
S.P. of 200 bicycles at Rs.1350 each = 1350 × 200 = Rs.270000
Profit = S.P. - C.P. = 270000 - 250000 = Rs.20000
Profit % = (Profit ÷ C.P.) × 100 = (20000 ÷ 250000) × 100 = 8%
In simple words: Add the buying price, transportation, and advertising to get total cost. Subtract this from the total selling price to get profit, then calculate the percentage.

Exam Tip: Do not forget to include all additional expenses in the cost price. Each type of cost must be multiplied by the quantity if it is per unit.

Question 7. The cost price of an article is 90% of its selling price. Find his profit percentage.
Answer: Let S.P. = Rs.x
C.P. = 90% of x = 0.9x
Profit = S.P. - C.P. = x - 0.9x = 0.1x
Profit % = (Profit ÷ C.P.) × 100 = (0.1x ÷ 0.9x) × 100 = (1 ÷ 9) × 100 = 100/9 % = 11⅑%
In simple words: If cost is 90% of selling price, then the extra 10% is the profit. Work out what fraction of the cost this is to get the percentage.

Exam Tip: When profit or loss is given in relation to S.P. or a variable, assume a value or use algebra to find the relationship, then calculate the percentage.

Question 8. Rao bought notebooks at the rate of 4 for Rs.35 and sold them at the rate of 5 for Rs.58. Calculate:
(i) his gain percentage.
(ii) the number of notebooks he should sell to earn a profit of Rs.171.
Answer:
(i) Take the L.C.M. of 4 and 5, which is 20.
C.P. of 20 notebooks = (35 ÷ 4) × 20 = 35 × 5 = Rs.175
S.P. of 20 notebooks = (58 ÷ 5) × 20 = 58 × 4 = Rs.232
Gain = S.P. - C.P. = 232 - 175 = Rs.57
Gain % = (Gain ÷ C.P.) × 100 = (57 ÷ 175) × 100 = 32⁴⁄₇%

(ii) When profit = Rs.57, notebooks sold = 20
When profit = Rs.1, notebooks sold = 20 ÷ 57
When profit = Rs.171, notebooks sold = (20 ÷ 57) × 171 = 20 × 3 = 60
In simple words: For part (i), find the cost and selling price for the same number of items using L.C.M., then calculate gain percentage. For part (ii), use the unitary method - if 20 notebooks give Rs.57 profit, use this ratio to find how many give Rs.171 profit.

Exam Tip: When items are sold at different rates (e.g. 4 for Rs.35), always find the L.C.M. of the quantities to make calculations easier and avoid fractions.

Question 9. A vendor buys bananas at 3 for a rupee and sells at 4 for a rupee. Find his profit or loss percentage.
Answer: Take the L.C.M. of 3 and 4, which is 12.
C.P. of 12 bananas = (1 ÷ 3) × 12 = Rs.4
S.P. of 12 bananas = (1 ÷ 4) × 12 = Rs.3
Loss = C.P. - S.P. = 4 - 3 = Rs.1
Loss % = (Loss ÷ C.P.) × 100 = (1 ÷ 4) × 100 = 25%
In simple words: When items are bought and sold at different rates per rupee, use L.C.M. to find the cost and selling price for the same quantity, then calculate loss or profit.

Exam Tip: Buying at 3 for Rs.1 is more expensive per item than selling at 4 for Rs.1, so there will always be a loss in this case.

Question 10. A shopkeeper buys a certain number of pens. If the selling price of 5 pens is equal to the cost price of 7 pens, find his profit or loss percentage.
Answer: Let C.P. of 7 pens = Rs.x
C.P. of 1 pen = Rs.(x ÷ 7)
S.P. of 5 pens = Rs.x
S.P. of 1 pen = Rs.(x ÷ 5)
Profit on 1 pen = S.P. - C.P. = (x ÷ 5) - (x ÷ 7) = (7x - 5x) ÷ 35 = 2x ÷ 35
Profit % = (Profit ÷ C.P.) × 100 = [(2x ÷ 35) ÷ (x ÷ 7)] × 100 = (2x ÷ 35) × (7 ÷ x) × 100 = (2 ÷ 5) × 100 = 40%
In simple words: Use a variable to express C.P. and S.P. per unit. The profit on each pen is the difference, and when you divide by C.P. and multiply by 100, you get the percentage.

Exam Tip: When a ratio between C.P. and S.P. is given, use variables to represent them rather than actual numbers - this makes the algebra cleaner.

Question 11. Find the selling price, when:
(i) Cost price = Rs.2360, Profit = 8%
(ii) Cost price = Rs.380, Loss = 7.5%
Answer:
(i) Given: C.P. = Rs.2360, Profit = 8%
S.P. = C.P. × (100 + Profit %) ÷ 100 = 2360 × 108 ÷ 100 = Rs.2548.80

(ii) Given: C.P. = Rs.380, Loss = 7.5%
S.P. = C.P. × (100 - Loss %) ÷ 100 = 380 × 92.5 ÷ 100 = Rs.351.50
In simple words: To find S.P. when profit is given, add profit to 100 and multiply by C.P., then divide by 100. When loss is given, subtract loss from 100 and do the same.

Exam Tip: The formula for S.P. is straightforward - use (100 + P%) for profit situations and (100 - L%) for loss situations.

Question 12. A dealer bought a number of eggs at Rs.18 a dozen and sold them at 50% profit. Find the selling price per egg.
Answer: Given: C.P. of one dozen eggs = Rs.18, Profit = 50%
S.P. of 12 eggs = C.P. × (100 + Profit %) ÷ 100 = 18 × 150 ÷ 100 = 18 × 1.5 = Rs.27
S.P. of 1 egg = 27 ÷ 12 = Rs.2.25
In simple words: First find the selling price for one dozen eggs using the profit percentage. Then divide by 12 to get the price per single egg.

Exam Tip: When finding the price per unit from a dozen, always divide the total price by 12. Make sure to first calculate the S.P. for the whole dozen using the profit percentage.

Exercise 7.3

Question 1. Find the discount and the selling price, when:
(i) the marked price = Rs.575, discount = 12%
(ii) the printed price = Rs.12750, discount = 8⅓%
Answer:
(i) Given: M.P. = Rs.575, Discount = 12%
Amount of discount = 12% of Rs.575 = (12 ÷ 100) × 575 = Rs.69
Net selling price = M.P. - Discount = 575 - 69 = Rs.506

(ii) Given: M.P. = Rs.12750, Discount = 8⅓% = 25/3%
Amount of discount = (25 ÷ 300) × 12750 = Rs.1062.50
Net selling price = M.P. - Discount = 12750 - 1062.50 = Rs.11687.50
In simple words: Calculate the discount amount by finding the given percentage of marked price. Subtract the discount from the marked price to get the selling price customers pay.

Exam Tip: The discount is always calculated on the marked price, not the selling price. Be careful to apply the percentage to M.P. only.

Question 2. Find the discount and the discount percentage, when:
(i) marked price = Rs.780, selling price = Rs.721.50
(ii) advertised price = Rs.28500, selling price = Rs.24510
Answer:
(i) Given: M.P. = Rs.780, S.P. = Rs.721.50
Discount = M.P. - S.P. = 780 - 721.50 = Rs.58.50
Discount % = (Discount ÷ M.P.) × 100 = (58.50 ÷ 780) × 100 = 7.5%

(ii) Given: Advertised price = Rs.28500, S.P. = Rs.24510
Discount = 28500 - 24510 = Rs.3990
Discount % = (Discount ÷ Advertised price) × 100 = (3990 ÷ 28500) × 100 = 14%
In simple words: When both marked price and selling price are given, subtract S.P. from M.P. to find the discount amount. Divide this by M.P. and multiply by 100 to get the percentage.

Exam Tip: Always use the marked price (not selling price) as the denominator when calculating discount percentage.

Question 3. A notebook is marked at Rs.30. Find the price a student pays for a dozen notebooks if he gets 15% discount.
Answer: Given: M.P. of one notebook = Rs.30
M.P. of 12 notebooks = 30 × 12 = Rs.360
Discount = 15%
Amount of discount = 15% of Rs.360 = (15 ÷ 100) × 360 = Rs.54
Price a student pays = 360 - 54 = Rs.306
In simple words: Find the total marked price for 12 notebooks. Calculate the discount on this total amount. Subtract the discount from the total to get what the student actually pays.

Exam Tip: When buying multiple items with a discount, always calculate the total marked price first, then apply the discount to the total.

Question 4. A dealer gave 9% discount on an electric fan and charges Rs.728 from the customer. Find the marked price of the fan.
Answer: Let M.P. = Rs.x
Discount = 9% of x = 0.09x
S.P. = M.P. - Discount
728 = x - 0.09x
728 = 0.91x
x = 728 ÷ 0.91 = Rs.800
Therefore, the marked price is Rs.800.
In simple words: If the customer pays Rs.728 after a 9% discount, that means they paid 91% of the marked price. Divide the amount paid by 0.91 to get the original marked price.

Exam Tip: When you need to find M.P. from S.P. and discount percentage, use the relationship: S.P. = M.P. × (100 - Discount%) ÷ 100, then solve for M.P.

Question 5. The list price of an article is Rs.800 and a dealer is selling it at a discount of 20%. Find:
(i) the selling price of the article.
(ii) the cost price of the article if he makes 25% profit on selling it.
Answer:
(i) Given: M.P. = Rs.800, Discount = 20%
S.P. = M.P. × (100 - Discount %) ÷ 100 = 800 × 80 ÷ 100 = Rs.640

(ii) Given: S.P. = Rs.640, Profit = 25%
C.P. = S.P. × 100 ÷ (100 + Profit %) = 640 × 100 ÷ 125 = Rs.512
In simple words: First find the selling price using the discount. Then work backwards from the selling price to find the cost price using the profit percentage.

Exam Tip: This is a two-step problem - first use the discount formula to find S.P., then use the profit formula to find C.P. from S.P.

Question 6. A shopkeeper marks his goods at such a price that would give him a profit of 10% after allowing a discount of 12%. If an article is marked at Rs.2250, find its:
(i) selling price
(ii) cost price.
Answer:
(i) Given: M.P. = Rs.2250, Discount = 12%
S.P. = M.P. × (100 - Discount %) ÷ 100 = 2250 × 88 ÷ 100 = Rs.1980

(ii) Given: S.P. = Rs.1980, Profit = 10%
C.P. = S.P. × 100 ÷ (100 + Profit %) = 1980 × 100 ÷ 110 = Rs.1800
In simple words: Use the discount to find what customers actually pay. Then use the profit percentage to work out how much the shopkeeper spent to buy the item.

Exam Tip: In problems with both discount and profit, calculate S.P. first from M.P. and discount, then C.P. from S.P. and profit. The order matters.

Question 7. A shopkeeper purchased a calculator for Rs.650. He sells it at a discount of 20% and still makes a profit of 20%. Find:
(i) the selling price
(ii) marked price
Answer:
(i) Given: C.P. = Rs.650, Profit = 20%
S.P. = C.P. × (100 + Profit %) ÷ 100 = 650 × 120 ÷ 100 = Rs.780

(ii) Given: S.P. = Rs.780, Discount = 20%
M.P. = S.P. × 100 ÷ (100 - Discount %) = 780 × 100 ÷ 80 = Rs.975
In simple words: First find the selling price by adding the profit to the cost. Then work backwards using the discount to find what marked price was needed to give that selling price.

Exam Tip: When working with profit first, find S.P. from C.P. Then use S.P. to find M.P. from discount. This is the reverse of the usual problem order.

Question 8. A shopkeeper buys a dinner set for Rs.1200 and marks it 80% above the cost price. If he gives 15% discount on it, find:
(i) the marked price
(ii) the selling price
(iii) his profit percentage.
Answer:
(i) Given: C.P. = Rs.1200
M.P. = C.P. + 80% of C.P. = 1200 + (80 ÷ 100) × 1200 = 1200 + 960 = Rs.2160

(ii) Given: M.P. = Rs.2160, Discount = 15%
S.P. = M.P. × (100 - Discount %) ÷ 100 = 2160 × 85 ÷ 100 = Rs.1836

(iii) Profit = S.P. - C.P. = 1836 - 1200 = Rs.636
Profit % = (Profit ÷ C.P.) × 100 = (636 ÷ 1200) × 100 = 53%
In simple words: Add the mark-up to the cost to get M.P. Subtract the discount from M.P. to get S.P. Finally, divide the profit by C.P. and multiply by 100 for the percentage.

Exam Tip: When an article is marked "above" the cost price, add that percentage to the C.P. Mark-up and discount are different concepts - one increases, the other decreases the price.

Question 9. The cost price of an article is Rs.1600, which is 20% below the marked price. If the article is sold at a discount of 16%, find:
(i) the marked price
(ii) the selling price
(iii) profit percentage.
Answer:
(i) Given: C.P. = Rs.1600, and C.P. is 20% below M.P.
This means C.P. = 80% of M.P.
1600 = 0.80 × M.P.
M.P. = 1600 ÷ 0.80 = Rs.2000

(ii) Given: M.P. = Rs.2000, Discount = 16%
S.P. = M.P. × (100 - Discount %) ÷ 100 = 2000 × 84 ÷ 100 = Rs.1680

(iii) Profit = S.P. - C.P. = 1680 - 1600 = Rs.80
Profit % = (Profit ÷ C.P.) × 100 = (80 ÷ 1600) × 100 = 5%
In simple words: If C.P. is 20% below M.P., then C.P. equals 80% of M.P. Divide C.P. by 0.80 to find M.P. Then calculate S.P. using the discount, and finally the profit percentage.

Exam Tip: "X is 20% below Y" means X = 80% of Y, not X = Y - 20%. Be careful with this phrasing - it is different from "20% discount on Y".

Question 9. A shopkeeper allows 20% discount on his goods and still earns a profit of 20%. If an article is sold for Rs 360, find: (i) the marked price (ii) the cost price.
Answer: The shopkeeper gives a 20% discount on the marked price. The selling price is Rs 360. Using the discount formula, we can write: S.P. = M.P. × (100 - discount%) / 100. Substituting 360 = M.P. × 80/100, we get M.P. = 360 × 100/80 = Rs 450. For the cost price, since the shopkeeper earns 20% profit on cost, we use: S.P. = C.P. × (100 + profit%) / 100. So 360 = C.P. × 120/100, which gives C.P. = 360 × 100/120 = Rs 300.
In simple words: When there is a 20% discount, you pay 80% of the marked price. The marked price is Rs 450. The shopkeeper bought the article for Rs 300 and made a profit of Rs 60.

Exam Tip: Remember to apply the discount formula first to find M.P., then use the profit formula separately to find C.P. — do not mix the two calculations together.

Exercise 7.4

Question 1. Find the buying price of each of the following when 5% S.T. is added on the purchase of (i) a towel of Rs 50 (ii) 5 kg of flour at Rs 15 per kg.
Answer: For the towel, the sales tax at 5% on Rs 50 is calculated as: (50 × 5) / 100 = Rs 2.50. The total buying price becomes 50 + 2.50 = Rs 52.50. For the flour, the cost of 5 kg at Rs 15 per kg is: 15 × 5 = Rs 75. The sales tax at 5% on Rs 75 is: (75 × 5) / 100 = Rs 3.75. The total buying price is 75 + 3.75 = Rs 78.75.
In simple words: The tax is added to the original price to get the final amount you pay. For the towel, you pay an extra Rs 2.50. For the flour, you pay an extra Rs 3.75.

Exam Tip: Always calculate the tax percentage on the original price, then add it to find the total buying price. Keep your calculations clear in steps.

Question 2. If 8% of VAT is included in the prices, find the original price of (i) a TV bought for Rs 13500 (ii) a shampoo bottle bought for Rs 180.
Answer: When VAT is already included in the given price, the original price is found by the formula: Original price = (Price with VAT × 100) / (100 + VAT%). For the TV: Original price = (13500 × 100) / 108 = Rs 12500. For the shampoo bottle: Original price = (180 × 100) / 108 = 500/3 = Rs 166.67.
In simple words: The price you are given includes the tax already. To find what the item cost before tax was added, divide by (100 plus the tax percentage).

Exam Tip: Do not subtract VAT as a simple percentage from the given price — always use the formula with (100 + VAT%) in the denominator to get the correct original price.

Question 3. Utkarsh bought an AC for Rs 34992 including a VAT of 8%. Find the price of AC before VAT was added.
Answer: The price paid by Utkarsh is Rs 34992, and this includes 8% VAT. To find the original price before VAT, we apply: Original price = (34992 × 100) / (100 + 8) = (34992 × 100) / 108 = Rs 32400.
In simple words: The Air Conditioner cost Rs 32400 before tax. The remaining Rs 2592 was the 8% VAT that was added on top.

Exam Tip: When working backwards from a price that includes VAT, always divide by (100 + VAT%), not subtract the VAT percentage directly from the amount.

Question 4. Gaurav bought a shirt for Rs 1296 including VAT. If the original price of the shirt is Rs 1200, find the rate of VAT.
Answer: Gaurav paid Rs 1296 in total, which includes VAT. The original price before VAT was Rs 1200. The VAT amount = 1296 - 1200 = Rs 96. To find the rate: VAT% = (VAT amount / Original price) × 100 = (96 / 1200) × 100 = 8%.
In simple words: The tax added was Rs 96 on a shirt that originally cost Rs 1200. This works out to be an 8% tax.

Exam Tip: Always subtract the original price from the final price to find the actual tax amount, then calculate the percentage by dividing the tax by the original price and multiplying by 100.

Question 5. Anjana buys a purse for Rs 523.80 including 8% VAT. Find the new selling price of the purse if VAT increases to 10%.
Answer: Anjana paid Rs 523.80, which includes 8% VAT. First, we find the original price: Original price = (523.80 × 100) / (100 + 8) = (523.80 × 100) / 108 = Rs 485. Now, if VAT increases to 10%, the new VAT amount = (485 × 10) / 100 = Rs 48.50. The new selling price = 485 + 48.50 = Rs 535.50.
In simple words: The purse itself costs Rs 485. When the tax was 8%, the total was Rs 523.80. When the tax becomes 10%, an extra Rs 48.50 is added, making the new price Rs 535.50.

Exam Tip: Always extract the original price first by reversing the old VAT, then apply the new VAT percentage to get the updated selling price.

Question 6. A wall hanging is marked for Rs 4800. The shopkeeper offers 10% discount on it. If VAT is received 8% from the customer, find the amount paid by the customer to purchase the wall hanging.
Answer: The marked price is Rs 4800. With a 10% discount, the net sale price = 4800 × (100 - 10) / 100 = 4800 × 90 / 100 = Rs 4320. On this discounted price, 8% VAT is charged: VAT amount = 4320 × 8 / 100 = Rs 345.60. The final amount paid = 4320 + 345.60 = Rs 4665.60.
In simple words: First, the discount brings the price down to Rs 4320. Then, 8% tax is added on this reduced price, raising it by Rs 345.60 to a final total of Rs 4665.60.

Exam Tip: Apply discount first to reduce the marked price, then calculate VAT on the discounted amount — VAT is always charged on the price after discount, not before.

Question 7. Amit goes to a shop to buy a washing machine. The marked price of the washing machine is Rs 10900 excluding 9% VAT. Amit bargains with the shopkeeper and convinces him for Rs 10900 including VAT as the final cost of the washing machine. Find the amount reduced by the shopkeeper.
Answer: Initially, the marked price is Rs 10900, and 9% VAT would be added separately. Let the reduced price be Rs x. When 9% VAT is added to x, the total becomes x + (9x/100) = 109x/100. According to Amit's bargain, this total equals Rs 10900. So 109x/100 = 10900, which gives x = (10900 × 100) / 109 = Rs 10000. The amount reduced by the shopkeeper = 10900 - 10000 = Rs 900.
In simple words: Amit negotiated so that Rs 10900 (including tax) is the final price instead of Rs 10900 plus additional tax. The shopkeeper reduced the marked price from Rs 10900 to Rs 10000, a cut of Rs 900.

Exam Tip: Set up an equation where the reduced price plus its VAT equals the bargained amount, then solve for the original marked price to find the reduction.