Selina Concise Solutions for ICSE Class 7 Mathematics Chapter 9 Profit Loss and Discount

Points to Remember

1. The price at which we buy any item is its Cost Price (C.P.).
2. The price at which we sell any item is its Selling Price (S.P.).
3. When the selling price is higher than the cost price, we make a profit or gain.
   \( \text{Profit} = \text{Selling Price} - \text{Cost Price} \)
   This means: \( \text{Profit (gain)} = \text{S.P.} - \text{C.P.} \) and \( \text{S.P.} = \text{C.P.} + \text{Gain} \)
4. When the selling price is lower than the cost price, we suffer a loss.
   \( \text{Loss} = \text{Cost Price} - \text{Selling Price} \)
   This means: \( \text{Loss} = \text{C.P.} - \text{S.P.} \) and \( \text{S.P.} = \text{C.P.} - \text{Loss} \)
5. We always figure out profit percentage and loss percentage using the cost price (C.P.).
   (i) \( \text{Profit \%} = \frac{\text{Profit}}{\text{C.P.}} \times 100\% \)
   (ii) \( \text{Loss \%} = \frac{\text{Loss}}{\text{C.P.}} \times 100\% \)
6. The selling price is the marked price minus any discount.
   \( \text{S.P.} = \text{M.P.} - \text{Discount} \)
   Note: (i) Discounts are calculated on the marked price (M.P.).
   (ii) Marked price is also called the List price.

Exercise 9(A)

Question 1. Find the gain or loss percent, if
(i) C.P. = Rs. 200 and S.P. = Rs. 224
(ii) C.P. = Rs. 450 and S.P. = Rs. 400
(iii) C.P. = Rs. 550 and gain = Rs. 22
(iv) C.P. = Rs. 216 and loss = Rs. 72
(v) S.P. = Rs. 500 and loss = Rs. 100
(vi) S.P. = Rs. 12 and profit = Rs. 4
(vii) C.P. = Rs. 5 and gain = 60 P
Answer:
(i) We have C.P. = Rs. 200 and S.P. = Rs. 224.
Since S.P. is larger, there is a profit.
\( \text{Profit} = \text{S.P.} - \text{C.P.} \)
\( \implies \text{Profit} = \text{Rs. } 224 - \text{Rs. } 200 = \text{Rs. } 24 \)
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)
\( \implies \text{Profit \%} = \frac{24 \times 100}{200} = 12\% \)

(ii) We have C.P. = Rs. 450 and S.P. = Rs. 400.
Since S.P. is smaller, there is a loss.
\( \text{Loss} = \text{C.P.} - \text{S.P.} \)
\( \implies \text{Loss} = \text{Rs. } 450 - \text{Rs. } 400 = \text{Rs. } 50 \)
\( \text{Loss \%} = \frac{\text{Loss} \times 100}{\text{C.P.}} \)
\( \implies \text{Loss \%} = \frac{50 \times 100}{450} = \frac{100}{9}\% = 11\frac{1}{9}\% \)

(iii) We have C.P. = Rs. 550 and profit = Rs. 22.
First, let's find the selling price.
\( \text{S.P.} = \text{C.P.} + \text{Profit} \)
\( \implies \text{S.P.} = \text{Rs. } 550 + \text{Rs. } 22 = \text{Rs. } 572 \)
Now, we calculate the profit percentage.
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)
\( \implies \text{Profit \%} = \frac{22 \times 100}{550} = 4\% \)

(iv) We have C.P. = Rs. 216 and loss = Rs. 72.
First, let's find the selling price.
\( \text{S.P.} = \text{C.P.} - \text{Loss} \)
\( \implies \text{S.P.} = \text{Rs. } 216 - \text{Rs. } 72 = \text{Rs. } 144 \)
Now, we find the loss percentage.
\( \text{Loss \%} = \frac{\text{Loss} \times 100}{\text{C.P.}} \)
\( \implies \text{Loss \%} = \frac{72 \times 100}{216} = \frac{100}{3}\% = 33\frac{1}{3}\% \)

(v) We have S.P. = Rs. 500 and loss = Rs. 100.
Let's find the cost price.
\( \text{C.P.} = \text{S.P.} + \text{Loss} \)
\( \implies \text{C.P.} = \text{Rs. } 500 + \text{Rs. } 100 = \text{Rs. } 600 \)
Now, we find the loss percentage.
\( \text{Loss \%} = \frac{\text{Loss} \times 100}{\text{C.P.}} \)
\( \implies \text{Loss \%} = \frac{100 \times 100}{600} = \frac{50}{3}\% = 16\frac{2}{3}\% \)

(vi) We have S.P. = Rs. 12 and profit = Rs. 4.
Let's find the cost price first.
\( \text{C.P.} = \text{S.P.} - \text{Profit} \)
\( \implies \text{C.P.} = \text{Rs. } 12 - \text{Rs. } 4 = \text{Rs. } 8 \)
Now, we find the profit percentage.
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)
\( \implies \text{Profit \%} = \frac{4 \times 100}{8} = 50\% \)

(vii) We have C.P. = Rs. 5 and profit = 60 P.
Let's write the profit in Rupees: 60 P = Rs. 0.60.
Now, we find the selling price.
\( \text{S.P.} = \text{C.P.} + \text{Profit} \)
\( \implies \text{S.P.} = \text{Rs. } 5 + \text{Rs. } 0.60 = \text{Rs. } 5.60 \pm \)
Now, we find the profit percentage.
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)
\( \implies \text{Profit \%} = \frac{0.60 \times 100}{5} = \frac{60}{5} = 12\% \)
In simple words: To find the profit or loss percent, first figure out the total profit or loss in money. Then, divide that amount by the cost price and multiply by 100.

Exam Tip: Remember to always use the Cost Price (C.P.) in the denominator when calculating profit or loss percent. Also, ensure both the price and profit/loss are in the same units (either both in rupees or both in paise) before calculating.

Question 2. Find the selling price, if:
(i) C.P. = Rs. 500 and gain = 25%
(ii) C.P. = Rs. 60 and loss = 12 1/2%
(iii) C.P. = Rs. 150 and loss = 20%
(iv) C.P. = Rs. 80 and gain = 2.5%
Answer:
(i) Given: C.P. = Rs. 500 and gain = 25%
We can find S.P. using the formula:
\( \text{S.P.} = \frac{\text{C.P.} \times (100 + \text{Gain \%})}{100} \)
\( \implies \text{S.P.} = \frac{500 \times (100 + 25)}{100} \)
\( \implies \text{S.P.} = \frac{500 \times 125}{100} = 5 \times 125 = \text{Rs. } 625 \)

(ii) Given: C.P. = Rs. 60 and loss = \( 12\frac{1}{2}\% = \frac{25}{2}\% \)
We use the formula:
\( \text{S.P.} = \frac{\text{C.P.} \times (100 - \text{Loss \%})}{100} \)
\( \implies \text{S.P.} = \frac{60 \times (100 - \frac{25}{2})}{100} \)
\( \implies \text{S.P.} = \frac{60 \times (\frac{200 - 25}{2})}{100} \)
\( \implies \text{S.P.} = \frac{60 \times \frac{175}{2}}{100} \)
\( \implies \text{S.P.} = \frac{30 \times 175}{100} = \frac{5250}{100} = \text{Rs. } 52.50 \)

(iii) Given: C.P. = Rs. 150 and loss = 20%
Using the loss formula:
\( \text{S.P.} = \frac{\text{C.P.} \times (100 - \text{Loss \%})}{100} \)
\( \implies \text{S.P.} = \frac{150 \times (100 - 20)}{100} \)
\( \implies \text{S.P.} = \frac{150 \times 80}{100} = 15 \times 8 = \text{Rs. } 120 \)

(iv) Given: C.P. = Rs. 80 and gain = 2.5%
Using the gain formula:
\( \text{S.P.} = \frac{\text{C.P.} \times (100 + \text{Gain \%})}{100} \)
\( \implies \text{S.P.} = \frac{80 \times (100 + 2.5)}{100} \)
\( \implies \text{S.P.} = \frac{80 \times 102.5}{100} \)
\( \implies \text{S.P.} = \frac{8200}{100} = \text{Rs. } 82 \)
In simple words: To find the selling price, either add the gain percent to 100 or subtract the loss percent from 100. Then multiply this result by the cost price and divide by 100.

Exam Tip: When dealing with fractional percentages like \( 12\frac{1}{2}\% \), convert them to improper fractions first to keep your calculations clean and avoid mistakes.

Question 3. Rohit bought a tape-recorder for Rs. 1,500 and sold it for Rs. 1,800. Calculate his profit or loss percent.
Answer:
The buying price (C.P.) of the tape-recorder is Rs. 1,500.
The selling price (S.P.) is Rs. 1,800.
Since the selling price is higher than the buying price, Rohit made a profit.
\( \text{Profit} = \text{S.P.} - \text{C.P.} \)
\( \implies \text{Profit} = \text{Rs. } 1,800 - \text{Rs. } 1,500 = \text{Rs. } 300 \)
Now, we calculate the profit percentage:
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)
\( \implies \text{Profit \%} = \frac{300 \times 100}{1500} = 20\% \)
In simple words: Rohit sold the recorder for Rs. 300 more than he paid. To find the percentage, we see that Rs. 300 is exactly 20% of his starting cost of Rs. 1,500.

Exam Tip: Always state clearly whether the transaction resulted in a profit or a loss before doing the percentage calculation.

Question 4. An article bought for Rs. 350 is sold at a profit of 20%. Find its selling price.
Answer:
The cost price (C.P.) of the item is Rs. 350.
The profit rate is 20%.
Using the selling price formula:
\( \text{S.P.} = \frac{\text{C.P.} \times (100 + \text{Profit \%})}{100} \)
\( \implies \text{S.P.} = \frac{350 \times (100 + 20)}{100} \)
\( \implies \text{S.P.} = \frac{350 \times 120}{100} \)
\( \implies \text{S.P.} = 35 \times 12 = \text{Rs. } 420 \)
In simple words: The shopkeeper wants a 20% profit on Rs. 350, which is Rs. 70. Adding this profit to the cost price gives a total selling price of Rs. 420.

Exam Tip: You can quickly check your answer by calculating 20% of Rs. 350 directly, which is Rs. 70, and then adding it to Rs. 350 to get Rs. 420.

Question 5. An old machine is bought for Rs. 1,400 and is sold at a loss of 15%. Find its selling price.
Answer:
The machine was purchased (C.P.) for Rs. 1,400.
The loss rate is 15%.
Using the formula for selling price with a loss:
\( \text{S.P.} = \frac{\text{C.P.} \times (100 - \text{Loss \%})}{100} \)
\( \implies \text{S.P.} = \frac{1400 \times (100 - 15)}{100} \)
\( \implies \text{S.P.} = \frac{1400 \times 85}{100} \)
\( \implies \text{S.P.} = 14 \times 85 = \text{Rs. } 1,190 \)
In simple words: A 15% loss on Rs. 1,400 means losing Rs. 210. So, subtracting Rs. 210 from the cost price leaves us with a selling price of Rs. 1,190.

Exam Tip: Be careful with basic multiplication like \( 14 \times 85 \). Double-check your calculation to avoid simple errors.

Question 6. Oranges are bought at 5 for Rs. 10 and sold at 6 for Rs. 15. Find profit or loss as percent.
Answer:
To solve this easily, let us find a common number of oranges using the Least Common Multiple (L.C.M.) of 5 and 6.
L.C.M. of 5 and 6 = 30.
Assume we buy and sell 30 oranges.
Cost price (C.P.) of 30 oranges = \( \frac{30 \times \text{Rs. } 10}{5} = \text{Rs. } 60 \)
Selling price (S.P.) of 30 oranges = \( \frac{30 \times \text{Rs. } 15}{6} = \text{Rs. } 75 \)
Since the selling price is higher, there is a profit.
\( \text{Profit} = \text{S.P.} - \text{C.P.} \)
\( \implies \text{Profit} = \text{Rs. } 75 - \text{Rs. } 60 = \text{Rs. } 15 \)
Now, we find the profit percentage:
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)
\( \implies \text{Profit \%} = \frac{15 \times 100}{60} = 25\% \)
In simple words: By looking at 30 oranges, we see they cost Rs. 60 to buy and are sold for Rs. 75. This gives a profit of Rs. 15, which is exactly one-quarter (25%) of the cost.

Exam Tip: Using the L.C.M. method to equalize the quantity bought and sold is the safest way to solve rate-based profit and loss problems without dealing with messy decimals.

Question 7. A certain number of articles are bought at 3 for Rs. 150 and all of them are sold at 4 for Rs. 180. Find the loss or gain as percent.
Answer:
We can find a common quantity of articles using the L.C.M. of 3 and 4.
L.C.M. of 3 and 4 = 12.
Let us assume 12 articles are bought and sold.
Cost price (C.P.) of 12 articles = \( \frac{12 \times \text{Rs. } 150}{3} = \text{Rs. } 600 \)
Selling price (S.P.) of 12 articles = \( \frac{12 \times \text{Rs. } 180}{4} = \text{Rs. } 540 \)
Since C.P. is higher than S.P., we have a loss.
\( \text{Loss} = \text{C.P.} - \text{S.P.} \)
\( \implies \text{Loss} = \text{Rs. } 600 - \text{Rs. } 540 = \text{Rs. } 60 \)
Now, calculate the loss percentage:
\( \text{Loss \%} = \frac{\text{Loss} \times 100}{\text{C.P.}} \)
\( \implies \text{Loss \%} = \frac{60 \times 100}{600} = 10\% \)
In simple words: Buying 12 items costs Rs. 600, but selling them brings in only Rs. 540. This means a loss of Rs. 60, which is exactly 10% of the cost price.

Exam Tip: Be careful when dividing: make sure you divide the total L.C.M. count by the original rate group before multiplying by the price.

Question 8. A vendor bought 120 sweets at 20 p each. In his house, 18 were consumed and he sold the remaining at 30 p each. Find his profit or loss as percent.
Answer:
Total number of sweets bought = 120.
Cost price (C.P.) of 120 sweets = \( 120 \times 20 \text{ p} = 2400 \text{ p} = \text{Rs. } 24 \).
Sweets eaten at home = 18.
Remaining sweets to sell = \( 120 - 18 = 102 \).
Selling price (S.P.) of 102 sweets = \( 102 \times 30 \text{ p} = 3060 \text{ p} = \text{Rs. } 30.60 \).
Since S.P. is more than C.P., there is a profit.
\( \text{Profit} = \text{S.P.} - \text{C.P.} \)
\( \implies \text{Profit} = \text{Rs. } 30.60 - \text{Rs. } 24 = \text{Rs. } 6.60 \)
Now, we calculate the profit percentage:
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)
\( \implies \text{Profit \%} = \frac{6.60 \times 100}{24} = \frac{660}{24} = 27.5\% \)
In simple words: The vendor spent Rs. 24 on sweets. After eating 18 of them, he sold the rest for Rs. 30.60, making a nice profit of Rs. 6.60 (which is 27.5% profit).

Exam Tip: Remember to subtract the consumed items from the total count before calculating the total selling price. Do not calculate the selling price on all 120 sweets.

Question 9. The cost price of an article is Rs. 1,200 and selling price is 5/4 times of its cost price. Find:
(i) selling price of the article
(ii) profit or loss as percent.
Answer:
(i) Given: Cost price (C.P.) = Rs. 1,200.
Selling price (S.P.) is \( \frac{5}{4} \) times the C.P.
\( \text{S.P.} = \frac{5}{4} \times \text{Rs. } 1,200 = 5 \times 300 = \text{Rs. } 1,500 \).

(ii) Since the selling price is higher than the cost price, there is a profit.
\( \text{Profit} = \text{S.P.} - \text{C.P.} \)
\( \implies \text{Profit} = \text{Rs. } 1,500 - \text{Rs. } 1,200 = \text{Rs. } 300 \)
Now, we calculate the profit percentage:
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)
\( \implies \text{Profit \%} = \frac{300 \times 100}{1200} = 25\% \)
In simple words: The article was sold for Rs. 1,500, which is Rs. 300 more than it cost. This extra Rs. 300 is exactly a 25% profit on the original cost.

Exam Tip: If the selling price is represented as a fraction of C.P. (like \( \frac{5}{4} \)), and the numerator is larger than the denominator, it automatically indicates a profit.

Question 10. The selling price of an article is Rs. 1,200 and cost price is 5/4 times of its selling price, find :
(i) cost price of the article ;
(ii) profit or loss as percent.
Answer:
(i) Given: Selling price (S.P.) = Rs. 1,200.
Cost price (C.P.) is \( \frac{5}{4} \) times the S.P.
\( \text{C.P.} = \frac{5}{4} \times \text{Rs. } 1,200 = 5 \times 300 = \text{Rs. } 1,500 \).

(ii) Since the cost price is higher than the selling price, there is a loss.
\( \text{Loss} = \text{C.P.} - \text{S.P.} \)
\( \implies \text{Loss} = \text{Rs. } 1,500 - \text{Rs. } 1,200 = \text{Rs. } 300 \)
Now, we calculate the loss percentage on the Cost Price:
\( \text{Loss \%} = \frac{\text{Loss} \times 100}{\text{C.P.}} \)
\( \implies \text{Loss \%} = \frac{300 \times 100}{1500} = 20\% \)
In simple words: The item cost Rs. 1,500 to buy, but was sold for only Rs. 1,200. This resulted in a loss of Rs. 300, which is 20% of what it originally cost.

Exam Tip: Be very careful! Even though the loss is Rs. 300 (the same amount as the profit in Question 9), the percentage is 20% instead of 25% because the Cost Price is now higher (Rs. 1,500 instead of Rs. 1,200).

Exercise 9(B)

Question 1. Find the cost price, if:
(i) S.P. = Rs. 21 and gain = 5%
(ii) S.P. = Rs. 22 and loss = 12%
(iii) S.P. = Rs. 340 and gain = Rs. 20
(iv) S.P. = Rs. 200 and loss = Rs. 50
(v) S.P. = Re. 1 and loss = 5 p.
Answer:
(i) Given: S.P. = Rs. 21 and gain = 5%.
We can find C.P. using the formula:
\( \text{C.P.} = \frac{\text{S.P.} \times 100}{100 + \text{Gain \%}} \)
\( \implies \text{C.P.} = \frac{21 \times 100}{100 + 5} = \frac{2100}{105} = \text{Rs. } 20 \)

(ii) Given: S.P. = Rs. 22 and loss = 12%.
We use the formula:
\( \text{C.P.} = \frac{\text{S.P.} \times 100}{100 - \text{Loss \%}} \)
\( \implies \text{C.P.} = \frac{22 \times 100}{100 - 12} = \frac{2200}{88} = \text{Rs. } 25 \)

(iii) Given: S.P. = Rs. 340 and profit = Rs. 20.
Since we have the profit amount directly, we subtract it from S.P. to get C.P.:
\( \text{C.P.} = \text{S.P.} - \text{Profit} \)
\( \implies \text{C.P.} = \text{Rs. } 340 - \text{Rs. } 20 = \text{Rs. } 320 \)

(iv) Given: S.P. = Rs. 200 and loss = Rs. 50.
Since we have the loss amount directly, we add it to S.P. to get C.P.:
\( \text{C.P.} = \text{S.P.} + \text{Loss} \)
\( \implies \text{C.P.} = \text{Rs. } 200 + \text{Rs. } 50 = \text{Rs. } 250 \)

(v) Given: S.P. = Re. 1 (which is 100 p) and loss = 5 p.
Let's write both in the same unit. S.P. = Rs. 1.00 and loss = Rs. 0.05.
\( \text{C.P.} = \text{S.P.} + \text{Loss} \)
\( \implies \text{C.P.} = \text{Rs. } 1.00 + \text{Rs. } 0.05 = \text{Rs. } 1.05 \)
In simple words: When you know the profit or loss percent, use the standard formulas to find the cost price. If you already have the actual money amount of profit or loss, just add or subtract it from the selling price directly.

Exam Tip: Be careful not to confuse when to add or subtract. Add the loss to the selling price to find a higher cost price, and subtract the profit from the selling price to find a lower cost price.

Question 2. By selling an article for Rs. 810, a loss of 10 percent is suffered. Find its cost price.
Answer:
The selling price (S.P.) is Rs. 810.
The loss rate is 10%.
Using the cost price formula for a loss:
\( \text{C.P.} = \frac{\text{S.P.} \times 100}{100 - \text{Loss \%}} \)
\( \implies \text{C.P.} = \frac{810 \times 100}{100 - 10} \)
\( \implies \text{C.P.} = \frac{810 \times 100}{90} = 9 \times 100 = \text{Rs. } 900 \)
In simple words: Selling the article at a 10% loss means selling it for 90% of what it cost. If 90% is Rs. 810, then the full 100% cost price is Rs. 900.

Exam Tip: You can quickly check your answer: a 10% loss on Rs. 900 is Rs. 90. Subtracting Rs. 90 from Rs. 900 gives Rs. 810, which matches the problem description.

Question 3. By selling a scooter for Rs. 9,200, a man gains 15%. Find the cost price of the scooter.
Answer:
The selling price (S.P.) of the scooter is Rs. 9,200.
The profit rate is 15%.
Using the cost price formula for a gain:
\( \text{C.P.} = \frac{\text{S.P.} \times 100}{100 + \text{Gain \%}} \)
\( \implies \text{C.P.} = \frac{9200 \times 100}{100 + 15} \)
\( \implies \text{C.P.} = \frac{9200 \times 100}{115} \)
\( \implies \text{C.P.} = 80 \times 100 = \text{Rs. } 8,000 \pm \)
In simple words: The selling price represents 115% of the cost price. We divide Rs. 9,200 by 115 and multiply by 100 to find the original cost of Rs. 8,000.

Exam Tip: When simplifying fractions like \( \frac{9200 \times 100}{115} \), look for common factors. Here, 115 goes into 920 exactly 8 times, making the math very clean.

Question 4. On selling an article for Rs. 2,640, a profit of 10 percent is made. Find
(i) cost price of the article
(ii) new selling price of it, in order to gain 15%
Answer:
(i) First, let's find the original cost price (C.P.).
Given S.P. = Rs. 2,640 and profit rate = 10%.
\( \text{C.P.} = \frac{\text{S.P.} \times 100}{100 + \text{Gain \%}} \)
\( \implies \text{C.P.} = \frac{2640 \times 100}{100 + 10} = \frac{2640 \times 100}{110} \)
\( \implies \text{C.P.} = 24 \times 100 = \text{Rs. } 2,400 \)

(ii) Now, we want to find the new selling price (S.P.) for a 15% profit.
Using the cost price of Rs. 2,400:
\( \text{New S.P.} = \frac{\text{C.P.} \times (100 + \text{New Gain \%})}{100} \)
\( \implies \text{New S.P.} = \frac{2400 \times (100 + 15)}{100} \)
\( \implies \text{New S.P.} = \frac{2400 \times 115}{100} \)
\( \implies \text{New S.P.} = 24 \times 115 = \text{Rs. } 2,760 \)
In simple words: First, we find that the article cost Rs. 2,400 to buy. To earn a higher profit of 15% on this cost, we must sell it for Rs. 2,760.

Exam Tip: Always calculate the new selling price using the original Cost Price (C.P.), never using the old Selling Price.

Question 5. A T.V. set is sold for Rs. 6800 at a loss of 15%. Find
(i) cost price of the T.V. set.
(ii) new selling price of it, in order to gain 12%
Answer:
(i) Let's find the original cost price (C.P.) of the television set.
Given: S.P. = Rs. 6,800 and loss = 15%.
\( \text{C.P.} = \frac{\text{S.P.} \times 100}{100 - \text{Loss \%}} \)
\( \implies \text{C.P.} = \frac{6800 \times 100}{100 - 15} = \frac{6800 \times 100}{85} \)
\( \implies \text{C.P.} = 80 \times 100 = \text{Rs. } 8,000 \)

(ii) Now, let's find the new selling price to make a 12% profit.
Using the calculated cost price of Rs. 8,000:
\( \text{New S.P.} = \frac{\text{C.P.} \times (100 + \text{Gain \%})}{100} \)
\( \implies \text{New S.P.} = \frac{8000 \times (100 + 12)}{100} \)
\( \implies \text{New S.P.} = \frac{8000 \times 112}{100} \)
\( \implies \text{New S.P.} = 80 \times 112 = \text{Rs. } 8,960 \)
In simple words: The television originally cost Rs. 8,000. Selling it for Rs. 6,800 meant losing money, but selling it for Rs. 8,960 will bring in a 12% profit.

Exam Tip: Double-check your division: \( 6800 \div 85 = 80 \). Breaking down 85 into factors like 5 and 17 can help simplify the calculation.

Question 6. A fruit seller bought mangoes at Rs. 90 per dozen and sold them at a loss of 8 percent. How much will a customer pay for.
(i) one mango
(ii) 40 mangoes
Answer:
Given: Cost price (C.P.) of 12 mangoes (1 dozen) = Rs. 90.
Loss percentage = 8%.
Let's calculate the total selling price (S.P.) of 12 mangoes first:
\( \text{S.P.} = \frac{\text{C.P.} \times (100 - \text{Loss \%})}{100} \)
\( \implies \text{S.P.} = \frac{90 \times (100 - 8)}{100} = \frac{90 \times 92}{100} \)
\( \implies \text{S.P.} = \frac{8280}{100} = \text{Rs. } 82.80 \)

(i) Now, let's find the selling price of a single mango:
\( \text{S.P. of 1 mango} = \frac{\text{S.P. of 12 mangoes}}{12} \)
\( \implies \text{S.P. of 1 mango} = \frac{\text{Rs. } 82.80}{12} = \text{Rs. } 6.90 \)

(ii) Next, let's find the selling price of 40 mangoes:
\( \text{S.P. of 40 mangoes} = 40 \times \text{S.P. of 1 mango} \)
\( \implies \text{S.P. of 40 mangoes} = 40 \times \text{Rs. } 6.90 = \text{Rs. } 276 \)
In simple words: The seller sold a dozen mangoes for Rs. 82.80 after an 8% loss. This means each mango costs Rs. 6.90 to buy, so 40 mangoes will cost Rs. 276.

Exam Tip: Be sure to divide by 12 first to find the rate per mango before multiplying by the target quantity of 40.

Question 7. By selling two transistors for Rs. 600 each, a shopkeeper gains 20 percent on one transistor and loses 20 percent on the other. Find :
(i) C.P. of each transistor
(ii) total C.P. and total S.P. of both the transistors
(iii) profit or loss percent on the whole.
Answer:
(i) Let's calculate the Cost Price (C.P.) for each of the two transistors.
For the first transistor (sold at 20% gain):
S.P. = Rs. 600, Gain = 20%.
\( \text{C.P.} = \frac{\text{S.P.} \times 100}{100 + \text{Gain \%}} \)
\( \implies \text{C.P.} = \frac{600 \times 100}{100 + 20} = \frac{600 \times 100}{120} = \text{Rs. } 500 \)

For the second transistor (sold at 20% loss):
S.P. = Rs. 600, Loss = 20%.
\( \text{C.P.} = \frac{\text{S.P.} \times 100}{100 - \text{Loss \%}} \)
\( \implies \text{C.P.} = \frac{600 \times 100}{100 - 20} = \frac{600 \times 100}{80} = \text{Rs. } 750 \)

(ii) Now, we find the total C.P. and total S.P.:
\( \text{Total C.P.} = \text{Rs. } 500 + \text{Rs. } 750 = \text{Rs. } 1,250 \)
\( \text{Total S.P.} = \text{Rs. } 600 + \text{Rs. } 600 = \text{Rs. } 1,200 \)

(iii) Since the total C.P. is more than the total S.P., the shopkeeper suffered a loss.
\( \text{Total Loss} = \text{Total C.P.} - \text{Total S.P.} \)
\( \implies \text{Total Loss} = \text{Rs. } 1,250 - \text{Rs. } 1,200 = \text{Rs. } 50 \)
Now, we calculate the loss percentage on the whole transaction:
\( \text{Loss \%} = \frac{\text{Total Loss} \times 100}{\text{Total C.P.}} \)
\( \implies \text{Loss \%} = \frac{50 \times 100}{1250} = 4\% \)
In simple words: The shopkeeper bought both transistors for Rs. 1,250 but sold them for Rs. 1,200 in total. This means he lost Rs. 50 overall, which is a 4% loss.

Exam Tip: Whenever the selling price is the same for two items, and the gain percent on one equals the loss percent on the other, there is always an overall loss.

Question 8. Mangoes are bought at 20 for Rs. 60. If they are sold at 33 1/3 percent profit. Find:
(i) selling price of each mango.
(ii) S.P. of 8 mangoes.
Answer:
We are given: Cost price (C.P.) of 20 mangoes = Rs. 60.
Profit percentage = \( 33\frac{1}{3}\% = \frac{100}{3}\% \).
First, let's find the selling price (S.P.) of 20 mangoes:
\( \text{S.P.} = \frac{\text{C.P.} \times (100 + \text{Gain \%})}{100} \)
\( \implies \text{S.P.} = \frac{60 \times (100 + \frac{100}{3})}{100} \)
\( \implies \text{S.P.} = \frac{60 \times \frac{400}{3}}{100} \)
\( \implies \text{S.P.} = \frac{20 \times 400}{100} = \text{Rs. } 80 \)

(i) Now, let's find the selling price of a single mango:
\( \text{S.P. of 1 mango} = \frac{\text{Rs. } 80}{20} = \text{Rs. } 4 \)

(ii) Next, let's calculate the selling price of 8 mangoes:
\( \text{S.P. of 8 mangoes} = 8 \times \text{S.P. of 1 mango} \)
\( \implies \text{S.P. of 8 mangoes} = 8 \times \text{Rs. } 4 = \text{Rs. } 32 \)
In simple words: The seller bought 20 mangoes for Rs. 60 and sold them all for Rs. 80 to make a 33 1/3% profit. This means each mango sold for Rs. 4, so 8 mangoes sold for Rs. 32.

Exam Tip: Write \( 33\frac{1}{3}\% \) as the improper fraction \( \frac{100}{3}\% \) to simplify the calculations easily.

Question 9. Find the cost price of an article, which is sold for Rs. 4050 at a loss of 10%. Also, find the new selling price of the article which must give a profit of 8%.
Answer:
Given: First selling price (S.P.) = Rs. 4,050 and loss = 10%.
First, we calculate the original cost price (C.P.):
\( \text{C.P.} = \frac{\text{S.P.} \times 100}{100 - \text{Loss \%}} \)
\( \implies \text{C.P.} = \frac{4050 \times 100}{100 - 10} = \frac{4050 \times 100}{90} \)
\( \implies \text{C.P.} = 45 \times 100 = \text{Rs. } 4,500 \)

Now, we find the new selling price to obtain an 8% profit:
\( \text{New S.P.} = \frac{\text{C.P.} \times (100 + \text{Gain \%})}{100} \)
\( \implies \text{New S.P.} = \frac{4500 \times (100 + 8)}{100} \)
\( \implies \text{New S.P.} = \frac{4500 \times 108}{100} \)
\( \implies \text{New S.P.} = 45 \times 108 = \text{Rs. } 4,860 \)
In simple words: The article originally cost Rs. 4,500. Selling it for Rs. 4,050 resulted in a loss, but to make an 8% profit instead, we need to sell it for Rs. 4,860.

Exam Tip: Finding the cost price first is a critical step because all profit or loss percentages are calculated on this base value.

Question 10. By selling an article for Rs. 825, a man loses equal to 1/3 of its selling price. Find :
(i) the cost price of the article,
(ii) the profit percent or the loss percent made, if the same article is sold for Rs. 1265.
Answer:
Given: Selling price (S.P.) = Rs. 825.
Loss is \( \frac{1}{3} \) of the selling price.
\( \text{Loss} = \frac{1}{3} \times \text{Rs. } 825 = \text{Rs. } 275 \).

(i) Now, let's find the cost price (C.P.) of the article:
\( \text{C.P.} = \text{S.P.} + \text{Loss} \)
\( \implies \text{C.P.} = \text{Rs. } 825 + \text{Rs. } 275 = \text{Rs. } 1,100 \)

(ii) In the second case, the article is sold for Rs. 1,265.
New S.P. = Rs. 1,265.
Since New S.P. is more than C.P., there is a profit.
\( \text{Profit} = \text{New S.P.} - \text{C.P.} \)
\( \implies \text{Profit} = \text{Rs. } 1,265 - \text{Rs. } 1,100 = \text{Rs. } 165 \)
Now, we find the profit percentage on C.P.:
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)
\( \implies \text{Profit \%} = \frac{165 \times 100}{1100} = 15\% \)
In simple words: The shopkeeper lost Rs. 275 (one-third of Rs. 825), meaning the item actually cost Rs. 1,100. If he sells it for Rs. 1,265, he makes Rs. 165 in profit, which is a 15% gain.

Exam Tip: Read carefully: the problem states the loss is one-third of the *selling price*, not the cost price. Calculate the loss amount first using S.P. before determining the C.P.

Question 11. Find the loss or gain as percent, if the C.P. of 10 articles, all of the same kind, is equal to S.P. of 8 articles.
Answer: Let us assume that the cost price of 10 items and the selling price of 8 items are both Rs. 80.
This means the cost price (C.P.) for one item is \( \frac{80}{10} = \text{Rs. } 8 \).
The selling price (S.P.) for one item is \( \frac{80}{8} = \text{Rs. } 10 \).
Since S.P. is more than C.P., there is a gain.
\( \text{Gain} = \text{S.P.} - \text{C.P.} = \text{Rs. } 10 - \text{Rs. } 8 = \text{Rs. } 2 \).
We can find the gain percentage using the formula:
\( \text{Gain \%} = \frac{\text{Gain} \times 100}{\text{C.P.}} \)

\( \implies \text{Gain \%} = \frac{2 \times 100}{8} = 25\% \).
In simple words: If you buy 10 things for the same price you sell 8 of them, you make a profit. That profit is equal to 25% of what you spent.

Exam Tip: When no price is given, assume a simple total like Rs. 80 or Rs. 100 to make the division easy. Always calculate profit or loss percentage on the Cost Price (C.P.).

Question 12. Find the loss or gain as percent, if the C.P. of 8 articles, all of the same kind, is equal to S.P. of 10 articles.
Answer: Let us assume that the cost price of 8 items is equal to the selling price of 10 items, which is Rs. 80.
So, the cost price (C.P.) of one item is \( \frac{80}{8} = \text{Rs. } 10 \).
The selling price (S.P.) of one item is \( \frac{80}{10} = \text{Rs. } 8 \).
Here, C.P. is more than S.P., which means there is a loss.
\( \text{Loss} = \text{C.P.} - \text{S.P.} = \text{Rs. } 10 - \text{Rs. } 8 = \text{Rs. } 2 \).
Now we find the loss percentage:
\( \text{Loss \%} = \frac{\text{Loss} \times 100}{\text{C.P.}} \)

\( \implies \text{Loss \%} = \frac{2 \times 100}{10} = 20\% \).
In simple words: If you buy 8 items for Rs. 80 and sell 10 items for the same Rs. 80, you lose money. Your loss is 20%.

Exam Tip: Remember that loss happens when C.P. is higher than S.P. Be careful to divide by the C.P. (Rs. 10) and not the S.P. when finding the percentage.

Question 13. The cost price of an article is 96% of its selling price. Find the loss or the gain as percent on the whole.
Answer: Let us assume the selling price (S.P.) is Rs. 100.
Since the cost price (C.P.) is 96% of the S.P., we have:
\( \text{C.P.} = \frac{96}{100} \times 100 = \text{Rs. } 96 \).
As the S.P. is more than the C.P., there is a gain.
\( \text{Gain} = \text{Rs. } 100 - \text{Rs. } 96 = \text{Rs. } 4 \).
Now, we calculate the gain percentage:
\( \text{Gain \%} = \frac{\text{Gain} \times 100}{\text{C.P.}} \)

\( \implies \text{Gain \%} = \frac{4}{96} \times 100\% \)

\( \implies \text{Gain \%} = \frac{25}{6}\% = 4\frac{1}{6}\% \).
In simple words: If you sell an item for Rs. 100 that cost you Rs. 96 to buy, you make a small profit. This profit is 4 and 1/6 percent.

Exam Tip: When a value is given as a percentage of another, always start by assuming the base value is Rs. 100. Write fractional percentages as mixed fractions for final marks.

Question 14. The selling price of an article is 96% of its cost price. Find the loss or the gain as percent on the whole.
Answer: Let us assume the cost price (C.P.) is Rs. 100.
The selling price (S.P.) is 96% of the C.P., so:
\( \text{S.P.} = \frac{96}{100} \times 100 = \text{Rs. } 96 \).
Since the selling price is lower than the cost price, there is a loss.
\( \text{Loss} = \text{Rs. } 100 - \text{Rs. } 96 = \text{Rs. } 4 \).
We can calculate the loss percentage as follows:
\( \text{Loss \%} = \frac{\text{Loss} \times 100}{\text{C.P.}} \)

\( \implies \text{Loss \%} = \frac{4}{100} \times 100\% = 4\% \).
In simple words: If you buy something for Rs. 100 but sell it for only Rs. 96, you lose Rs. 4. This is a 4% loss.

Exam Tip: Make sure to read carefully whether the percentage is of the Cost Price or the Selling Price, as this changes the base value.

Question 15. Hundred oranges are bought for Rs. 350 and all of them are sold at the rate of Rs. 48 per dozen. Find the profit percent or loss percent made.
Answer: First, let us find the cost price of a single orange.
\( \text{C.P. of one orange} = \frac{\text{Rs. } 350}{100} = \text{Rs. } 3.50 \).
Next, we find the selling price of one orange. Since a dozen contains 12 oranges:
\( \text{S.P. of one orange} = \frac{\text{Rs. } 48}{12} = \text{Rs. } 4 \).
Since the selling price is more than the cost price, there is a gain.
\( \text{Gain} = \text{Rs. } 4 - \text{Rs. } 3.50 = \text{Rs. } 0.50 \).
Now we calculate the gain percentage:
\( \text{Gain \%} = \frac{\text{Gain} \times 100}{\text{C.P.}} \)

\( \implies \text{Gain \%} = \frac{0.50}{3.50} \times 100\% = \frac{50}{3.50}\% = 14\frac{2}{7}\% \).
In simple words: One orange costs Rs. 3.50 to buy and sells for Rs. 4. This gives a profit of 50 paise per orange, which is 14 and 2/7 percent.

Exam Tip: When dealing with rates per dozen, divide the rate by 12 to find the cost of a single item so that you can compare it easily.

Question 16. Oranges are bought at 100 for Rs. 80 and all of them are sold at Rs. 80 for Rs. 100. Find the loss or gain as percent in this transaction.
Answer: Let us work out the cost price (C.P.) of a single orange:
\( \text{C.P. of one orange} = \frac{\text{Rs. } 80}{100} = \text{Rs. } 0.80 \).
Now, let us find the selling price (S.P.) of a single orange:
\( \text{S.P. of one orange} = \frac{\text{Rs. } 100}{80} = \text{Rs. } 1.25 \).
Since the selling price is higher than the cost price, there is a profit.
\( \text{Profit} = \text{Rs. } 1.25 - \text{Rs. } 0.80 = \text{Rs. } 0.45 \).
Let us calculate the profit percentage:
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)

\( \implies \text{Profit \%} = \frac{0.45}{0.80} \times 100\% = \frac{45}{80} \times 100\% = \frac{450}{8}\% = 56.25\% \).
In simple words: An orange costs Rs. 0.80 and is sold for Rs. 1.25. The profit is Rs. 0.45 per orange, which is 56.25%.

Exam Tip: Always calculate the unit cost price and unit selling price first when dealing with bulk items of different quantities.

Question 17. An article is bought for Rs. 5,700 and Rs. 1,300 is spent on its repairing, transportion, etc. For how much should this article be sold in order to gain 20% on the whole.
Answer: The original price paid for the item is Rs. 5,700.
Extra money spent on repairs and transport is Rs. 1,300.
So, the total cost price (C.P.) is:
\( \text{Total C.P.} = \text{Rs. } 5,700 + \text{Rs. } 1,300 = \text{Rs. } 7,000 \).
The target gain is 20%.
We can find the required selling price (S.P.) using this formula:
\( \text{S.P.} = \frac{\text{Total C.P.} \times (100 + \text{Gain \%})}{100} \)

\( \implies \text{S.P.} = \frac{7,000 \times (100 + 20)}{100} \)

\( \implies \text{S.P.} = \frac{7,000 \times 120}{100} = \text{Rs. } 8,400 \).
Thus, the article should be sold for Rs. 8,400.
In simple words: The total cost of the item, including repairs, is Rs. 7,000. To make a 20% profit, you must sell it for Rs. 8,400.

Exam Tip: Remember that overhead expenses like repair and transport are always added to the buying price to find the total cost price.

Exercise 9 (C)

Question 1. A machine is marked at Rs. 5000 and is sold at a discount of 10%. Find the selling price of the machine.
Answer: The marked price (M.P.) of the machine is Rs. 5,000.
The discount rate offered is 10%.
Let us find the discount value:
\( \text{Discount} = \text{Rs. } 5,000 \times \frac{10}{100} = \text{Rs. } 500 \).
Now we subtract the discount from the marked price to get the selling price:
\( \text{Selling Price (S.P.)} = \text{M.P.} - \text{Discount} \)

\( \implies \text{S.P.} = \text{Rs. } 5,000 - \text{Rs. } 500 = \text{Rs. } 4,500 \).
In simple words: The machine has a tag of Rs. 5,000. A 10% discount means you save Rs. 500, so you pay Rs. 4,500.

Exam Tip: Always calculate the discount amount on the marked price (M.P.) first, and then subtract it to find the selling price.

Question 2. A shopkeeper marked a dinner set for Rs. 1000. He sold it at Rs. 900, what percent discount did he give ?
Answer: The marked price (M.P.) of the dinner set is Rs. 1,000, and it is sold for Rs. 900.
The discount amount is:
\( \text{Discount} = \text{M.P.} - \text{S.P.} = \text{Rs. } 1,000 - \text{Rs. } 900 = \text{Rs. } 100 \).
Now we calculate the discount percentage:
\( \text{Discount \%} = \frac{\text{Discount} \times 100}{\text{M.P.}} \)

\( \implies \text{Discount \%} = \frac{100 \times 100}{1,000} = 10\% \).
In simple words: The dinner set was marked Rs. 1,000 but sold for Rs. 900. This means the shopkeeper gave a Rs. 100 discount, which is 10%.

Exam Tip: Make sure to use the marked price as the denominator when calculating the discount percentage.

Question 3. A pair of shoes marked at Rs. 320, are sold at a discount of 15 percent. Find : (i) discount (ii) selling price of the shoes.
Answer: The marked price (M.P.) of the shoes is Rs. 320, and the discount rate is 15%.
(i) Let us find the discount value:
\( \text{Discount} = \text{Rs. } 320 \times \frac{15}{100} = \text{Rs. } 48 \).
(ii) Now we calculate the selling price of the shoes:
\( \text{Selling Price} = \text{M.P.} - \text{Discount} \)

\( \implies \text{Selling Price} = \text{Rs. } 320 - \text{Rs. } 48 = \text{Rs. } 272 \).
In simple words: The shoes cost Rs. 320 on the label. A 15% discount saves you Rs. 48, so the final price is Rs. 272.

Exam Tip: Clearly label parts (i) and (ii) in your answer as requested in the question to ensure you get full step-marks.

Question 4. The list price of an article is Rs. 450 and it is sold for Rs. 360. Find : (i) discount (ii) discount percent
Answer: The marked price or list price (M.P.) is Rs. 450, and the selling price (S.P.) is Rs. 360.
(i) Let us find the discount given on the article:
\( \text{Discount} = \text{M.P.} - \text{S.P.} = \text{Rs. } 450 - \text{Rs. } 360 = \text{Rs. } 90 \).
(ii) Next, we find the discount percentage:
\( \text{Discount \%} = \frac{\text{Discount} \times 100}{\text{M.P.}} \)

\( \implies \text{Discount \%} = \frac{90 \times 100}{450} = 20\% \).
In simple words: You save Rs. 90 on a Rs. 450 item, which is a 20% discount.

Exam Tip: Remember that "list price" is another term for "marked price". Always use this value to calculate both the discount amount and its percentage.

Question 5. A shopkeeper buys an article for Rs. 300. He increases its price by 20% and then gives 10% discount on the new price. Find: (i) the new price (marked price) of the article. (ii) the discount given by the shopkeeper. (iii) the selling price. (iv) profit percent made by the shopkeeper.
Answer: The cost price (C.P.) is Rs. 300.
(i) The price is increased by 20% to set the marked price (M.P.):
\( \text{M.P.} = \text{Rs. } 300 \times \frac{100 + 20}{100} = \text{Rs. } 300 \times \frac{120}{100} = \text{Rs. } 360 \).
(ii) A 10% discount is offered on this new price of Rs. 360:
\( \text{Discount} = \text{Rs. } 360 \times \frac{10}{100} = \text{Rs. } 36 \).
(iii) The final selling price (S.P.) is:
\( \text{S.P.} = \text{M.P.} - \text{Discount} = \text{Rs. } 360 - \text{Rs. } 36 = \text{Rs. } 324 \).
(iv) The profit earned by the shopkeeper is:
\( \text{Profit} = \text{S.P.} - \text{C.P.} = \text{Rs. } 324 - \text{Rs. } 300 = \text{Rs. } 24 \).
Now, we calculate the profit percentage:
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)

\( \implies \text{Profit \%} = \frac{24 \times 100}{300} = 8\% \).
In simple words: The shopkeeper bought an item for Rs. 300, raised the price to Rs. 360, and then sold it for Rs. 324 after a 10% discount. This gave a final profit of 8%.

Exam Tip: Make sure to find the discount on the increased marked price (Rs. 360), not the original cost price.

Question 6. A car is marked at Rs. 50,000. The dealer gives 5% discount on first Rs. 20,000 and 2% discount on the remaining Rs. 30,000. Find : (i) the total discount. (ii) the price charged by the dealer.
Answer: The total marked price of the car is Rs. 50,000.
Let us find the discount for each part:
First discount on the initial Rs. 20,000 at 5%:
\( \text{First Discount} = \text{Rs. } 20,000 \times \frac{5}{100} = \text{Rs. } 1,000 \).
Second discount on the remaining Rs. 30,000 at 2%:
\( \text{Second Discount} = \text{Rs. } 30,000 \times \frac{2}{100} = \text{Rs. } 600 \).
(i) The total discount is the sum of these two amounts:
\( \text{Total Discount} = \text{Rs. } 1,000 + \text{Rs. } 600 = \text{Rs. } 1,600 \).
(ii) The final price charged by the dealer is:
\( \text{Final Price} = \text{Rs. } 50,000 - \text{Rs. } 1,600 = \text{Rs. } 48,400 \).
In simple words: The car is listed at Rs. 50,000. After getting a Rs. 1,000 discount on the first part and Rs. 600 on the rest, the total savings are Rs. 1,600, so you pay Rs. 48,400.

Exam Tip: When a discount is split into parts, calculate each discount separately and add them up to find the total.

Question 7. A dealer buys a T.V. set for Rs. 2500. He marks it at Rs. 3,200 and then gives a discount of 10% on it. Find : (i) the selling price of the T.V. set (ii) the profit percent made by the dealer.
Answer: The cost price (C.P.) of the TV is Rs. 2,500, and the marked price (M.P.) is Rs. 3,200.
The discount rate is 10% on the marked price:
\( \text{Discount} = \text{Rs. } 3,200 \times \frac{10}{100} = \text{Rs. } 320 \).
(i) The selling price (S.P.) is:
\( \text{S.P.} = \text{Rs. } 3,200 - \text{Rs. } 320 = \text{Rs. } 2,880 \).
(ii) The profit earned by the dealer is:
\( \text{Profit} = \text{S.P.} - \text{C.P.} = \text{Rs. } 2,880 - \text{Rs. } 2,500 = \text{Rs. } 380 \).
Now we calculate the profit percentage:
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)

\( \implies \text{Profit \%} = \frac{380 \times 100}{2,500} = \frac{380}{25}\% = 15.2\% \text{ or } 15\frac{1}{5}\% \).
In simple words: The TV was bought for Rs. 2,500 and sold for Rs. 2,880 after a 10% discount on its Rs. 3,200 price tag. This gives a profit of 15.2%.

Exam Tip: Do not calculate the discount on the cost price of Rs. 2,500. Discount is always applied to the marked price (Rs. 3,200).

Question 8. A sells his goods at 15% discount. Find the price of an article which is sold for Rs. 680.
Answer: The selling price (S.P.) of the article is Rs. 680, and the discount given is 15%.
Let us assume the marked price (M.P.) is Rs. 100.
In this case, the selling price would be:
\( \text{S.P.} = \text{Rs. } 100 - \text{Rs. } 15 = \text{Rs. } 85 \).
By using the unitary method:
When the S.P. is Rs. 85, the M.P. is Rs. 100.
When the S.P. is Rs. 680, the M.P. is:
\( \text{M.P.} = \frac{100 \times 680}{85} = \text{Rs. } 800 \).
Therefore, the marked price of the article is Rs. 800.
In simple words: An item is sold for Rs. 680 after a 15% discount. This means the original price on the tag was Rs. 800.

Exam Tip: When finding the marked price from the selling price and discount, using the ratio method or assuming a base of Rs. 100 helps avoid calculation mistakes.

Question 9. A shopkeeper allows 20% discount on the marked price of his articles. Find the marked price of an article for which he charges Rs. 560.
Answer: Let us assume the marked price (M.P.) of the item is Rs. 100.
With a 20% discount, the selling price (S.P.) is:
\( \text{S.P.} = \text{Rs. } 100 - \text{Rs. } 20 = \text{Rs. } 80 \).
Using the unitary method:
If the selling price is Rs. 80, the marked price is Rs. 100.
If the selling price is Rs. 560, the marked price is:
\( \text{M.P.} = \frac{100 \times 560}{80} = \text{Rs. } 700 \).
So, the marked price of the article is Rs. 700.
In simple words: The customer paid Rs. 560, which includes a 20% discount. The original price marked on the item was Rs. 700.

Exam Tip: Always establish the relation between the assumed marked price (Rs. 100) and selling price (Rs. 80) first to set up the unitary ratio correctly.

Question 10. An article is bought for Rs. 1,200 and Rs. 100 is spent on its transportation, etc. Find : (i) the total C.P. of the article. (ii) the selling price of it in order to gain 20% on the whole.
Answer: The initial cost of the article is Rs. 1,200, and Rs. 100 is spent on transport.
(i) Let us find the total cost price (C.P.):
\( \text{Total C.P.} = \text{Rs. } 1,200 + \text{Rs. } 100 = \text{Rs. } 1,300 \).
(ii) To make a profit of 20% on this total cost price:
\( \text{S.P.} = \frac{\text{Total C.P.} \times (100 + \text{Gain \%})}{100} \)

\( \implies \text{S.P.} = \frac{1,300 \times (100 + 20)}{100} \)

\( \implies \text{S.P.} = \frac{1,300 \times 120}{100} = \text{Rs. } 1,560 \).
In simple words: Buying and moving the item cost Rs. 1,300 in total. To get a 20% profit, you must sell it for Rs. 1,560.

Exam Tip: For any question with overhead expenses, calculate the total C.P. first. Never find the selling price using only the initial buying price.

Question 11. 40 pens are bought at 4 for Rs. 50 and all of them are sold at 5 for Rs. 80 Find : (i) C.P. of one pen. (ii) S/P. of one pen. (iii) Profit made by selling one pen. (iv) Profit percent made by selling one pen. (v) C.P. of 40 pens (vi) S.P. of 40 pens. (vii) Profit made by selling 40 pens. (viii) Profit percent made by selling 40 pens. Are the results of parts (iv) and (viii) same? What conclusion do you draw from the above result ?
Answer: Let us solve each part step by step:
(i) Since 4 pens cost Rs. 50, the cost price of 1 pen is:
\( \text{C.P. of 1 pen} = \frac{\text{Rs. } 50}{4} = \text{Rs. } 12.50 \).
(ii) Since 5 pens are sold for Rs. 80, the selling price of 1 pen is:
\( \text{S.P. of 1 pen} = \frac{\text{Rs. } 80}{5} = \text{Rs. } 16 \).
(iii) The profit made on a single pen is:
\( \text{Profit on 1 pen} = \text{S.P.} - \text{C.P.} = \text{Rs. } 16 - \text{Rs. } 12.50 = \text{Rs. } 3.50 \).
(iv) The profit percentage on one pen is:
\( \text{Profit \% on 1 pen} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)

\( \implies \text{Profit \% on 1 pen} = \frac{3.50 \times 100}{12.50} = 28\% \).
(v) The total cost price of 40 pens is:
\( \text{C.P. of 40 pens} = 40 \times \text{Rs. } 12.50 = \text{Rs. } 500 \).
(vi) The total selling price of 40 pens is:
\( \text{S.P. of 40 pens} = 40 \times \text{Rs. } 16 = \text{Rs. } 640 \).
(vii) The total profit made on 40 pens is:
\( \text{Total Profit} = \text{Total S.P.} - \text{Total C.P.} = \text{Rs. } 640 - \text{Rs. } 500 = \text{Rs. } 140 \).
(viii) The profit percentage on 40 pens is:
\( \text{Profit \% on 40 pens} = \frac{\text{Total Profit} \times 100}{\text{Total C.P.}} \)

\( \implies \text{Profit \% on 40 pens} = \frac{140 \times 100}{500} = 28\% \).
Yes, the profit percentages calculated in parts (iv) and (viii) are the same.
**Conclusion**: The profit percentage remains identical whether we calculate it for a single item or for any larger quantity of the same items.
In simple words: Buying and selling 1 pen or 40 pens gives the exact same profit rate of 28%. The percentage rate of profit does not change with the quantity.

Exam Tip: Be careful with calculations in part (i) and (ii). Working out the unit prices first makes solving all the remaining parts much simpler.

Question 12. The C.P. of 5 identical articles is equal to S.P. of 4 articles. Calculate the profit percent or loss percent made if all the articles bought are sold.
Answer: Let us assume the cost price of 5 articles is equal to the selling price of 4 articles, which is Rs. 100.
This gives:
\( \text{C.P. of 1 article} = \frac{\text{Rs. } 100}{5} = \text{Rs. } 20 \).
\( \text{S.P. of 1 article} = \frac{\text{Rs. } 100}{4} = \text{Rs. } 25 \).
Since the selling price is higher than the cost price, there is a profit:
\( \text{Profit} = \text{Rs. } 25 - \text{Rs. } 20 = \text{Rs. } 5 \).
Now we calculate the profit percentage:
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} \)

\( \implies \text{Profit \%} = \frac{5 \times 100}{20} = 25\% \).
In simple words: If 5 things cost the same as 4 things sell for, you get a 25% profit when you sell them.

Exam Tip: Assuming a simple total value like Rs. 100 makes calculating individual cost and selling prices straightforward and clean.

Question 13. The C.P. of 8 pens is same as S.P. of 10 pens. Calculate the profit or loss percent made, if all the pens bought are considered to be sold
Answer: Let us assume that the cost price of 8 pens is equal to the selling price of 10 pens, which is Rs. 100.
The cost price of a single pen is:
\( \text{C.P. of 1 pen} = \frac{\text{Rs. } 100}{8} = \text{Rs. } 12.50 \).
The selling price of a single pen is:
\( \text{S.P. of 1 pen} = \frac{\text{Rs. } 100}{10} = \text{Rs. } 10 \).
Since the cost price is higher than the selling price, there is a loss:
\( \text{Loss} = \text{Rs. } 12.50 - \text{Rs. } 10 = \text{Rs. } 2.50 \).
Now we find the loss percentage:
\( \text{Loss \%} = \frac{\text{Loss} \times 100}{\text{C.P.}} \)

\( \implies \text{Loss \%} = \frac{2.50 \times 100}{12.50} = \frac{250}{12.5} = 20\% \).
In simple words: Buying 8 pens costs the same as selling 10 pens. This results in a loss of 20%.

Exam Tip: Make sure to use the unit cost price (Rs. 12.50) as the base (denominator) to calculate the loss percentage.

Question 14. A certain number of articles are bought at Rs. 450 per dozen and all of them are sold at a profit of 20%. Find the S.P. of: (i) one article (ii)seven articles.
Answer: The cost price (C.P.) for a dozen (12 articles) is Rs. 450.
The profit rate is 20%.
First, let us find the selling price (S.P.) for a dozen articles:
\( \text{S.P. of 12 articles} = \frac{\text{C.P.} \times (100 + \text{Profit \%})}{100} \)

\( \implies \text{S.P.} = \frac{450 \times (100 + 20)}{100} \)

\( \implies \text{S.P.} = \frac{450 \times 120}{100} = \text{Rs. } 540 \).
(i) Now we find the selling price of a single article:
\( \text{S.P. of 1 article} = \frac{\text{Rs. } 540}{12} = \text{Rs. } 45 \).
(ii) Next, we calculate the selling price of 7 articles:
\( \text{S.P. of 7 articles} = \text{Rs. } 45 \times 7 = \text{Rs. } 315 \).
In simple words: A dozen items are bought for Rs. 450 and sold with a 20% profit for Rs. 540. This means each item sells for Rs. 45, and 7 items cost Rs. 315.

Exam Tip: Always remember that 1 dozen is equal to 12 items. Find the S.P. of the entire dozen first to keep the arithmetic simple.

Question 15. An article is marked 60% above the cost price and sold at 20% discount. Find the profit percent made.
Answer: Let us assume the cost price (C.P.) of the article is Rs. 100.
Since the price is marked 60% above the C.P., the marked price (M.P.) is:
\( \text{M.P.} = \text{Rs. } 100 + \text{Rs. } 60 = \text{Rs. } 160 \).
The discount rate given is 20% on this marked price:
\( \text{S.P.} = \frac{\text{M.P.} \times (100 - \text{Discount \%})}{100} \)

\( \implies \text{S.P.} = \frac{160 \times (100 - 20)}{100} \)

\( \implies \text{S.P.} = \frac{160 \times 80}{100} = \text{Rs. } 128 \).
The profit is the difference between the selling price and the cost price:
\( \text{Profit} = \text{S.P.} - \text{C.P.} = \text{Rs. } 128 - \text{Rs. } 100 = \text{Rs. } 28 \).
Now we find the profit percentage:
\( \text{Profit \%} = \frac{\text{Profit} \times 100}{\text{C.P.}} = \frac{28 \times 100}{100} = 28\% \).
In simple words: An item bought for Rs. 100 is marked up to Rs. 160. After a 20% discount, it is sold for Rs. 128, leaving a net profit of 28%.

Exam Tip: When both markup and discount are percentages, assuming the initial C.P. as Rs. 100 simplifies calculations because the final profit amount equals the profit percentage directly.