RS Aggarwal Class 7 Mathematics Solutions Chapter 11 Profit and Loss

Exercise 11A

Important Facts

Cost Price: The price at which an article is purchased is known as its cost price, written as C.P.

Selling Price: The price at which an article is sold is known as its selling price, written as S.P.

Profit or Gain: If S.P is greater than C.P, the seller is said to have a profit or gain.

Loss: If S.P is less than C.P, the seller is said to have suffered a loss.

Important Formulae

• Gain = (S.P) - (C.P)
• Loss = (C.P) - (S.P)
• Loss or gain is always reckoned on C.P
• Gain Percentage: (Gain %) = \( \left( \frac{\text{Gain} \times 100}{\text{C.P}} \right) \)
• Loss Percentage: (Loss %) = \( \left( \frac{\text{Loss} \times 100}{\text{C.P}} \right) \)
• Selling Price (S.P) = \( \left[ \frac{(100 + \text{Gain }\%)}{{100}} \times \text{C.P} \right] \)
• Selling Price (S.P) = \( \left[ \frac{(100 - \text{Loss }\%)}{{100}} \times \text{C.P} \right] \)
• Cost Price (C.P) = \( \left[ \frac{100}{(100 + \text{Gain }\%)} \times \text{S.P} \right] \)
• Cost Price (C.P) = \( \left[ \frac{100}{(100 - \text{Loss }\%)} \times \text{S.P} \right] \)
• If an article is sold at a gain of say 35%, then S.P = 135% of C.P
• If an article is sold at a loss of say 35%, then S.P = 65% of C.P
• When a person sells two similar items, one at a gain of say x%, and the other at a loss of x%, then the seller always incurs a loss given by: Loss % = \( \left( \frac{\text{Common Loss and Gain }\%}{10} \right)^2 = \left( \frac{x}{10} \right)^2 \)
• If a trader professes to sell his goods at cost price, but uses false weights, then: Gain % = \( \left[ \frac{\text{Error}}{(\text{True Value}) - (\text{Error})} \right] \times 100 \) %

Question 1. Find the selling price when: (i) C.P = Rs. 950 and Gain = 6% (ii) C.P = Rs. 9600 and Gain = 16 \( \frac{2}{3} \) % (iii) C.P = Rs. 1540 and Loss = 4% (iv) C.P = Rs. 8640 and Loss = 12 \( \frac{1}{2} \) %
Answer: (i) C.P = Rs. 950, Gain = 6%

S.P = \( \left[ \frac{(100 + \text{Gain }\%)}{{100}} \times \text{C.P} \right] \)

= \( \left[ \frac{(100 + 6)}{{100}} \times 950 \right] \)

= \( \frac{106}{100} \times 950 \)

= \( \frac{100700}{100} \)

= Rs. 1007

(ii) C.P = Rs. 9600, Gain = \( 16\frac{2}{3} \) % = \( \frac{50}{3} \) %

S.P = \( \left[ \frac{(100 + \text{Gain }\%)}{{100}} \times \text{C.P} \right] \)

= \( \left[ \frac{(100 + \frac{50}{3})}{{100}} \times 9600 \right] \)

= \( \frac{350}{300} \times 9600 \)

= \( \frac{3360000}{300} \)

= Rs. 11200

(iii) C.P = Rs. 1540, Loss = 4%

S.P = \( \left[ \frac{(100 - \text{Loss }\%)}{{100}} \times \text{C.P} \right] \)

= \( \left[ \frac{(100 - 4)}{{100}} \times 1540 \right] \)

= \( \frac{96}{100} \times 1540 \)

= \( \frac{147840}{100} \)

= Rs. 1478.40

(iv) C.P = Rs. 8640, Loss = \( 12\frac{1}{2} \) % = \( \frac{25}{2} \) %

S.P = \( \left[ \frac{(100 - \text{Loss }\%)}{{100}} \times \text{C.P} \right] \)

= \( \left[ \frac{(100 - \frac{25}{2})}{{100}} \times 8640 \right] \)

= \( \frac{175}{200} \times 8640 \)

= \( \frac{1512000}{200} \)

= Rs. 7560
In simple words: Use the formula to find selling price: multiply the cost price by the gain or loss percentage added to or subtracted from 100, then divide by 100.

Exam Tip: Always apply the correct formula - add the gain percentage to 100 for profit situations, and subtract the loss percentage from 100 for loss situations.

Question 2. Find the cost price when: (i) S.P = Rs. 924 and Gain = 10% (ii) S.P = Rs. 1755 and Gain = 12 \( \frac{1}{2} \) % (iii) S.P = Rs. 8510 and Loss = 8%
Answer: (i) S.P = Rs. 924, Gain = 10%

C.P = \( \left[ \frac{100}{(100 + \text{Gain }\%)} \times \text{S.P} \right] \)

= \( \left[ \frac{100}{(100 + 10)} \times 924 \right] \)

= \( \frac{92400}{110} \)

= Rs. 840

(ii) S.P = Rs. 1755, Gain = \( 12\frac{1}{2} \) % = \( \frac{25}{2} \) %

C.P = \( \left[ \frac{100}{(100 + \text{Gain }\%)} \times \text{S.P} \right] \)

= \( \left[ \frac{100}{(100 + \frac{25}{2})} \times 1755 \right] \)

= \( \left[ \frac{200}{225} \times 1755 \right] \)

= \( \frac{351000}{225} \)

= Rs. 1560

(iii) S.P = Rs. 8510, Loss = 8%

C.P = \( \left[ \frac{100}{(100 - \text{Loss }\%)} \times \text{S.P} \right] \)

= \( \left[ \frac{100}{(100 - 8)} \times 8510 \right] \)

= \( \frac{851000}{92} \)

= Rs. 9250
In simple words: Use the reverse formula: divide 100 by (100 plus gain percent) or (100 minus loss percent), then multiply by the selling price to find the cost price.

Exam Tip: Remember to reverse the operation - when finding C.P from S.P, you divide by the percentage factor instead of multiplying.

Question 3. Find the gain or loss percent: (i) C.P = Rs. 2400, S.P = Rs. 2592 (ii) C.P = Rs. 1650, S.P = Rs. 1452 (iii) C.P = Rs. 12000, S.P = Rs. 12800 (iv) C.P = Rs. 1800, S.P = Rs. 1611
Answer: (i) C.P = Rs. 2400, S.P = Rs. 2592

Gain = S.P - C.P = Rs. (2592 - 2400) = Rs. 192

Gain% = \( \left( \frac{\text{Gain}}{\text{C.P}} \times 100 \right) = \left( \frac{192}{2400} \times 100 \right) = 8 \)

(ii) C.P = Rs. 1650, S.P = Rs. 1452

Loss = C.P - S.P = (1650 - 1452) = Rs. 198

Loss% = \( \left( \frac{\text{Loss}}{\text{C.P}} \times 100 \right) = \left( \frac{198}{1650} \times 100 \right) = 12 \)

(iii) C.P = Rs. 12000 and S.P = Rs. 12800

Gain = S.P - C.P = (12800 - 12000) = Rs. 800

Gain% = \( \left( \frac{\text{Gain}}{\text{C.P}} \times 100 \right) = \left( \frac{800}{12800} \times 100 \right) = 6.66 \)

(iv) C.P = Rs. 1800, S.P = Rs. 1611

Loss = C.P - S.P = (1800 - 1611) = Rs. 189

Loss% = \( \left( \frac{\text{Loss}}{\text{C.P}} \times 100 \right) = \left( \frac{189}{1800} \times 100 \right) = 10.5 \)
In simple words: Calculate the gain or loss amount first by finding the difference between selling and cost prices. Then divide this difference by the cost price and multiply by 100 to get the percentage.

Exam Tip: Always calculate the absolute gain or loss first, then use the percentage formula - remember the denominator is always the cost price, not the selling price.

Question 4. A man purchases an amirah for Rs. 13600 and spends Rs. 400 on its transportation. He then sells it for Rs. 16800. Find his gain percent.
Answer: Cost price of the amirah = Rs. 13600

Transportation cost = Rs. 400

Total cost price = Rs. (13600 + 400) = Rs. 14000

Selling price = Rs. 16800

Since S.P > C.P

Gain = S.P - C.P = (16800 - 14000) = Rs. 2800

Gain% = \( \left( \frac{\text{Gain}}{\text{C.P}} \times 100 \right) \% \)

= \( \left( \frac{2800}{14000} \times 100 \right) \% \)

= \( \frac{28000}{140} \) %

= 20%
In simple words: When extra expenses are involved, add them to the purchase price to get the total cost. Then calculate the gain percent as usual.

Exam Tip: Never forget to include all additional costs like transportation, repair, or commission in the total cost price before calculating profit percentage.

Question 5. A house costs Rs. 765000. Ravi spends Rs. 115000 on repairing it. He then sells it at a gain of 5%. How much does he get?
Answer: Cost price of the house = Rs. 765000

Cost of repairing the house = Rs. 115000

Total cost price = (765000 + 115000) = Rs. 880000

Ravi sold it at a gain of 5%.

S.P = \( \left[ \frac{(100 + \text{gain }\%)}{{100}} \times \text{C.P} \right] \)

= \( \left[ \frac{(100 + 5)}{{100}} \times 880000 \right] \)

= \( \frac{105}{100} \times 880000 \)

= Rs. 924000

He gets Rs. 924000.
In simple words: Find the total cost by adding the purchase price and repair expenses. Then apply the gain percentage formula to get the selling price.

Exam Tip: Include all expenses incurred before calculating selling price - this ensures you don't accidentally calculate profit on the purchase price alone.

Question 6. The cost price of 12 lemons is Rs. 25. Ramesh sells 5 lemons for Rs. 12. Find his gain or loss percent.
Answer: Cost price of 12 lemons (one dozen) = Rs. 25

Cost price of one lemon = Rs. \( \frac{25}{12} \)

Cost price of five lemons = 5 x \( \frac{25}{12} \) = \( \frac{125}{12} \) = Rs. 10.42

Selling price of five lemons = Rs. 12 (given)

Gain = S.P - C.P = (12 - 10.42) = Rs. 1.58

Gain % = \( \left( \frac{\text{Gain}}{\text{C.P}} \times 100 \right) \% \)

= \( \left( \frac{1.58}{10.42} \times 100 \right) \% \)

= 15.2%
In simple words: First find the cost and selling price for the same quantity. Then calculate the gain or loss percent based on those matching quantities.

Exam Tip: When prices are given for different quantities, always convert them to the same quantity before comparing - this makes calculation straightforward and prevents errors.

Question 7. The cost price of a pen is Re 1. Arun sells 12 pens for Rs. 15. Also, he sells 15 pens for Rs. 15. Find his gain percent.
Answer: Let the cost price of the pen be Re 1.

Cost price of 12 pens = Rs. 12

Selling price of 12 pens = Cost price of 15 pens = Rs. 15

Gain = S.P - C.P = Rs. (15 - 12) = Rs. 3

Gain % = \( \left( \frac{\text{Gain}}{\text{C.P}} \times 100 \right) \% \)

= \( \left( \frac{3}{12} \times 100 \right) \% \)

= 25%

Gain% = 25%
In simple words: Find what he buys and what he sells. Compare their values using the same quantity to determine the profit percentage.

Exam Tip: Carefully parse the problem statement - make sure you identify which quantity corresponds to which price (buying or selling).

Question 8. The cost price of one spoon is Re 1. A shopkeeper purchases 16 spoons for Rs. 16. He sells them such that the cost price of 15 spoons equals the selling price of 16 spoons. Find his loss percent.
Answer: Let the cost price of one spoon be Re 1.

Cost price of 16 spoons = Rs. 16

Selling price of 16 spoons = Cost price of 15 spoons = Rs. 15

Loss = C.P - S.P = (16 - 15) = Re 1

Loss % = \( \left( \frac{\text{Loss}}{\text{C.P}} \times 100 \right) \% \)

= \( \left( \frac{1}{16} \times 100 \right) \% \)

= 6.25%

Loss% = 6.25%
In simple words: Compare the buying and selling prices of the same quantity. The difference between them gives the loss amount, from which you calculate the loss percentage.

Exam Tip: Always identify the quantity being compared - sometimes the problem states the selling price in terms of a different quantity than the buying price.

Question 9. The cost price of a video is Rs. 12000. Rahul sold it at a gain of 10%. Later, he sold it at a loss of 5%. Find what Rakesh pays.
Answer: Cost price of a video = Rs. 12000

Selling price of a video at a gain of 10% = \( \left[ \frac{(100 + \text{Gain }\%)}{{100}} \times \text{C.P} \right] \)

= \( \left[ \frac{(100 + 10)}{{100}} \times 12000 \right] \)

= \( \left[ \frac{110}{100} \times 12000 \right] \)

= Rs. 13200

So, Rahul purchased at a cost price of Rs. 13200.

Rahul sells it at a loss of 5%.

Selling price of a video at loss of 5% = \( \left[ \frac{(100 - \text{Loss }\%)}{{100}} \times \text{C.P} \right] \)

= \( \left[ \frac{(100 - 5)}{{100}} \times 13200 \right] \)

= \( \frac{95}{100} \times 13200 \)

= Rs. 12540

Rakesh pays = Rs. 12540
In simple words: First find what Rahul paid (selling price becomes his cost price). Then calculate what price he receives when selling at a loss.

Exam Tip: When there are multiple transactions, remember that one person's selling price becomes another person's cost price for the next transaction.

Question 10. A sofa set is sold for Rs. 21600 at a gain of 8%. What was the cost price?
Answer: Selling price of the sofa set = Rs. 21600

Gain% = 8

C.P. of the sofa set = \( \left[ \frac{100}{(100 + \text{Gain}\%)} \times \text{S.P} \right] \)

= \( \left[ \frac{100}{(100 + 8)} \times 21600 \right] \)

= \( \frac{2160000}{108} \)

= Rs. 20000

He purchased it at the cost of Rs. 20000.
In simple words: Apply the formula to work backwards from the selling price to find the cost price using the given gain percentage.

Exam Tip: When asked to find cost price from selling price and gain percent, use the formula with 100 divided by (100 plus gain percent).

Question 11. A watch is sold for Rs. 11400 at a loss of 5%. What was its cost price?
Answer: Selling price of the watch = Rs. 11400

Loss% = 5

C.P = \( \left[ \frac{100}{(100 - \text{Loss }\%)} \times \text{S.P} \right] \)

= \( \left[ \frac{100}{(100 - 5)} \times 11400 \right] \)

= \( \frac{11400}{95} \)

= Rs. 12000

He purchased it at the cost of Rs. 12000.
In simple words: To find cost price when there is a loss, divide the selling price by (100 minus loss percent), then multiply by 100.

Exam Tip: Remember to use (100 minus loss percent) in the denominator when calculating cost price from selling price at a loss.

Question 22. A shopkeeper buys pencils at Rs. 96 per dozen and sells them at Rs. 1 each. What is the gain or loss percentage?
Answer: Let the cost price of one pencil be Rs. x. Therefore, the cost price of 96 pencils will be Rs. 96x. Let the selling price of one pencil be Rs. y, so the selling price of 96 pencils is Rs. 96y. Given that the selling price of one dozen pencils is Rs. 12y. Gain = SP - CP. This gives us 12y = 96 - 96x = 96x - 96y - 12y = 96x - 84y = x - 84y/96. The gain percentage is calculated as: Gain % = (GainCP × 100)% = (12y × 96/96x + 100)% = 12y × 96/96 + 84y) × 100% = 14.28%
In simple words: When the shopkeeper buys pencils by the dozen and sells them individually at a higher per-unit price, he makes a profit. The profit works out to be about 14.28 percent.

Exam Tip: Always find the cost price and selling price for the same quantity before comparing them - this avoids confusion when prices are given in different units.

Exercise 11B

Question 1. A book is bought for Rs. 80 and sold for Rs. 100. What is the gain percentage?
Answer: The cost price of the book is Rs. 80, and the selling price is Rs. 100. The profit is calculated as Rs. (100 - 80) = Rs. 20. Using the profit percentage formula: Gain % = (Gain / CP × 100)% = (20 / 80 × 100)% = 25%
In simple words: When you buy something for Rs. 80 and sell it for Rs. 100, you gain Rs. 20. This gain is 25 percent of what you paid.

Exam Tip: Profit percentage is always calculated on the cost price, not the selling price - this is a common mistake students make.

Question 2. A football costs Rs. 120 and is sold for Rs. 105. What is the loss percentage?
Answer: The cost price of a football is Rs. 120, and the selling price is Rs. 105. Since the cost price is greater than the selling price, there is a loss. Loss is calculated as Rs. (120 - 105) = Rs. 15. Using the loss percentage formula: Loss % = (Loss / CP × 100)% = (15 / 120 × 100)% = 12.5%
In simple words: When you buy a football for Rs. 120 but sell it for only Rs. 105, you lose Rs. 15. This loss amounts to 12.5 percent of the original cost.

Exam Tip: Make sure to identify whether you have a gain or loss first - if SP is less than CP, it is always a loss.

Question 3. A bat is purchased for Rs. 100 and sold for Rs. 120. What is the gain percentage?
Answer: The selling price of the bat is Rs. 100 and the gain is Rs. 20. Using the gain percentage formula: Gain % = (Gain / CP × 100)% = (20 / 80 × 100)% = 25%
In simple words: A gain of Rs. 20 on a cost of Rs. 80 means you made a 25 percent profit on your investment.

Exam Tip: Double-check your arithmetic when finding CP from gain and SP - work backwards carefully using the relationship SP = CP + Gain.

Question 4. An article is sold at Rs. 198 with a gain of 10%. What is the cost price of the article?
Answer: The selling price of the racket is Rs. 198 and the gain percentage is 10. To find the cost price, use the formula: CP = (100 / (100 + Gain %)) × 100. This gives us CP = (100 / (100 + 10)) × 198 = (100 / 110) × 198 = 10/11 × 198 = Rs. 180
In simple words: If you know the selling price and the gain percentage, you can work backwards to find what was originally paid. Divide the selling price by 1 plus the gain percentage (expressed as a decimal) to find the cost.

Exam Tip: Always use the correct formula for finding CP from SP and gain - reversing the relationship or using addition instead of division is a frequent error.

Question 5. An article is sold at Rs. 144 at a loss of 5/8. If the cost price is Rs. 168, what is the new selling price if sold at Rs. 189?
Answer: Let the cost price be Rs. x. The loss is given as Rs. 5/8. Therefore, the selling price is SP = (x - 5/8) = Rs. 8/5 x. Given that SP is Rs. 144, we have 8/5 x = 144, which gives x = 144 × 5/8 = Rs. 168. The cost price is Rs. 168 and the selling price is Rs. 144. If sold at a new selling price of Rs. 189, the gain is SP - CP = Rs. (189 - 168) = Rs. 21. The gain percentage is calculated as: Gain % = (Gain / CP × 100)% = (21 / 168 × 100)% = 12.5%. The correct answer is 12.5%. All the given options are incorrect.
In simple words: When an item is bought for Rs. 168 and initially sold for Rs. 144 (a loss), but then sold again at Rs. 189, you calculate the new profit by comparing the second selling price to the cost price, not to the first selling price.

Exam Tip: When a question involves multiple transactions, always compare to the original cost price, not to an intermediate selling price.

Question 6. A pen is sold at Rs. 48 at a loss of 20%. What should be the new selling price to gain 20%?
Answer: The selling price of the pen is Rs. 48 and the loss is 20%. To find the cost price, use: CP = (100 / (100 - Loss %)) × SP = (100 / (100 - 20)) × 48 = (100 / 80) × 48 = Rs. 60. To gain 20%, the new selling price should be: SP = ((100 + Gain %) / 100) × CP = ((100 + 20) / 100) × 60 = (120 / 100) × 60 = Rs. 72
In simple words: First, work backwards from the selling price and loss to find what was originally paid. Then, work forwards from the cost price and desired gain to find the new selling price needed.

Exam Tip: These two-step problems require you to first recover the cost price accurately, then apply the new gain percentage - don't skip the first step or confuse the two selling prices.

Question 7. The cost price of 15 pencils is Rs. 15. The selling price of 12 pencils is Rs. 12, and the cost price of 15 pencils is Rs. 15. What is the loss percentage?
Answer: Assume each pencil costs Rs. 1. The cost of 15 pencils is Rs. 15. When sold, 15 pencils of cost price Rs. 12 equals Rs. 12, and the cost price of 15 pencils is Rs. 15. The difference is RS (15 - 12) = Rs. 3. Loss % = (Loss / CP × 100)% = (3 / 15 × 100)% = 20%
In simple words: If each pencil costs Rs. 1, then 15 pencils cost Rs. 15. If they sell for Rs. 12, the loss is Rs. 3, which is 20 percent of the cost.

Exam Tip: When dealing with quantities, find the cost and selling price per unit first - this simplifies the entire calculation.

Question 8. The cost price of three toffees is Rs. 3, and the selling price of four toffees is Rs. 4. What is the gain percentage?
Answer: Let the cost price of each toffee be Rs. 1. The cost of three toffees is Rs. 3. The selling price of four toffees is Rs. 4, so each toffee sells for Rs. 1. When three toffees are sold at Rs. 1 each, the selling price is Rs. 3, making the cost price Rs. 3. The gain is SP - CP = Rs. (4 - 3) = Rs. 1. The gain percentage is: Gain % = (Gain / CP × 100)% = (1 / 3 × 100)% = 33.33%, or 33 1/3 %
In simple words: When you buy three toffees for Rs. 3 total and sell four for Rs. 4 total, you are selling each at a slightly higher rate than you bought it, creating a gain of about 33 percent.

Exam Tip: With different quantities for buying and selling, convert everything to a cost price and selling price for the same quantity before calculating percentage.

Question 9. An article is sold at Rs. 144 at a loss of 10%. What should be the selling price to gain 10%?
Answer: The selling price of an article is Rs. 144 and the loss is 10%. To find the cost price: CP = (100 / (100 - Loss %)) × SP = (100 / (100 - 10)) × 144 = (100 / 90) × 144 = Rs. 160. To gain 10%, the new selling price should be: SP = ((100 + Gain %) / 100) × CP = ((100 + 10) / 100) × 160 = (110 / 100) × 160 = Rs. 176
In simple words: First, find the original cost by reversing the 10 percent loss calculation. Then, apply a 10 percent gain to that cost to find the new selling price.

Exam Tip: Use the complementary percentages correctly - loss means subtract from 100, gain means add to 100 - mixing these up will give you the wrong cost price.

Question 10. Six lemons cost Rs. 1, and four lemons cost Rs. 1. What is the gain or loss percentage?
Answer: The cost price of six lemons is Rs. 1, so the cost price of one lemon is Rs. 1/6. The cost price of four lemons is Rs. 1/8. The cost price of four lemons is Rs. 4/8. The selling price of four lemons is Rs. 1. The gain is 1 - 4/8 = 2/8 = Rs. 1/3. The gain percentage is: Gain % = (Gain / CP × 100)% = (3/2 ÷ 1 × 100)% = 100/50 = 50%
In simple words: When you buy four lemons for Rs. 1/6 each but sell them for Rs. 1 per four, you are getting more money per lemon, which means you have a 50 percent gain.

Exam Tip: Problems with fractional quantities require careful unit conversion - always express both cost price and selling price for the same item (e.g., per lemon or per dozen).

Question 11. A chair is sold at Rs. 720 and the gain is 20%. What is the cost price of the chair?
Answer: The selling price of the chair is Rs. 720 and the gain percentage is 20. To find the cost price, use the formula: CP = (100 / (100 + Profit percentage)) × SP = (100 / 120) × 720 = 7200 / 120 = Rs. 600
In simple words: If an article is sold at Rs. 720 with a 20 percent gain, the original cost must have been lower. Divide the selling price by 1.2 (which represents the 120 percent) to recover the cost price.

Exam Tip: When SP and gain % are given, use the direct formula for CP rather than trying to work backwards step-by-step - it is faster and less error-prone.

Question 12. A stool is sold at Rs. 630 at a loss of 10%. What is the cost price of the stool?
Answer: The selling price of a stool is Rs. 630 and the loss is 10. To find the cost price, use: CP = (100 / (100 - Loss %)) × SP = (100 / (100 - 10)) × 630 = (100 / 90) × 630 = 10/9 × 630

Exam Tip: When loss percentage is given with selling price, remember to subtract the loss percentage from 100 in the denominator - subtracting from 100 gives the percentage the selling price represents of the cost price.