CAT 2025

Slot 2

QA

Question & SolutionQues & Sol

An item with a cost price of Rs. 1650 is sold at a certain discount on a fixed marked price to earn a profit of 20% on the cost price. If the discount was doubled, the profit would have been Rs. 110. The rate of discount, in percentage, at which the profit percentage would be equal to the rate of discount, is nearest to

1. Concept Used

• Topic: Profit and Loss — Marked Price, Discount, and Selling Price relationships
• Formula: $$SP = M(1 - d), \quad \text{Profit %} = \frac{SP - CP}{CP} \times 100$$

2. Calculation

Let the marked price be ( M ) and the initial discount rate be ( d ). The cost price is ( CP = 1650 ).

A 20% profit means the selling price is: $$SP = 1650 \times 1.2 = 1980$$

With discount rate ( d ), the selling price is: $$M(1 - d) = 1980 \quad \text{...(1)}$$

When the discount is doubled, the new selling price yields a profit of Rs. 110: $$M(1 - 2d) = 1650 + 110 = 1760 \quad \text{...(2)}$$

Subtracting equation (2) from equation (1): $$M(1 - d) - M(1 - 2d) = 1980 - 1760$$ $$Md = 220 \implies M = \frac{220}{d}$$

Substituting back into equation (1): $$\frac{220}{d}(1 - d) = 1980$$ $$\frac{1 - d}{d} = \frac{1980}{220} = 9$$ $$1 - d = 9d \implies 10d = 1 \implies d = 0.1$$

So the initial discount rate is 10% and the marked price is: $$M = \frac{220}{0.1} = 2200$$

Now, let the new discount rate be ( r ) such that the profit percentage equals the discount percentage (both expressed as a decimal for consistency): $$\frac{2200(1 - r) - 1650}{1650} = r$$

Expanding and simplifying: $$2200 - 2200r - 1650 = 1650r$$ $$550 = 2200r + 1650r = 3850r$$ $$r = \frac{550}{3850} = \frac{1}{7} \approx 0.14286$$

Converting to percentage: $$r \approx 14.29%$$

This is nearest to 14%.

3. Solution

Answer = Option C (14) ✅

The rate of discount at which the profit percentage equals the discount rate is approximately 14%.