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CAT 2025Slot 2QAQues & Sol

ArithmeticMedium

Question

A retailer purchases an item whose cost price is Rs. 1650. He marks the item at a certain price and offers a fixed rate of discount on the marked price, thereby earning a profit of 20% on the cost price. It is observed that if the discount rate were doubled, the absolute profit earned would reduce to Rs. 110. Now, the retailer wishes to offer a new discount rate on the same marked price such that the profit percentage (on cost price) equals the discount rate (in percentage). The new discount rate, in percentage, is nearest to

Options

Solution

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%.