Question:

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:

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When the same marked price is used under different discount and profit conditions, set up equations using \(SP = MP - \text{Discount}\) and \(SP = CP + \text{Profit}\) for each scenario. Once the marked price is known, you can introduce a variable discount rate and equate the profit percentage to that rate to solve such “rate equals rate” problems.
Updated On: Dec 4, 2025
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The Correct Option is C

Solution and Explanation

Step 1: Use the first scenario (20% profit). Cost price: \[ CP = 1650. \] Profit = \(20%\) of \(1650\): \[ \text{Profit}_1 = 0.20 \times 1650 = 330. \] So the selling price in the first scenario: \[ SP_1 = CP + \text{Profit}_1 = 1650 + 330 = 1980. \] Let the marked price be \(MP\) and the initial discount be \(D\). Then: \[ MP - D = 1980. \tag{1} \]
Step 2: Use the second scenario (doubled discount and profit Rs.110). New profit: \[ \text{Profit}_2 = 110. \] So the second selling price: \[ SP_2 = CP + \text{Profit}_2 = 1650 + 110 = 1760. \] Discount is doubled \(\Rightarrow\) new discount is \(2D\), so: \[ MP - 2D = 1760. \tag{2} \]
Step 3: Solve for \(MP\) and \(D\). Subtract (2) from (1): \[ (MP - D) - (MP - 2D) = 1980 - 1760 \] \[ D = 220. \] Substitute back in (1): \[ MP - 220 = 1980 \Rightarrow MP = 2200. \] So the marked price is Rs.2200.
Step 4: Let the discount rate be \(x%\) such that \[ \text{Discount%} = \text{Profit%} = x. \] Discount amount: \[ \text{Discount} = \frac{x}{100} \times 2200 = 22x. \] New selling price: \[ SP = MP - \text{Discount} = 2200 - 22x. \] Profit amount: \[ \text{Profit} = SP - CP = (2200 - 22x) - 1650 = 550 - 22x. \] Profit percentage (on cost price) is: \[ \text{Profit%} = \frac{\text{Profit}}{CP} \times 100 = \frac{550 - 22x}{1650} \times 100. \] We are told this percentage equals \(x\): \[ x = \frac{550 - 22x}{1650} \times 100. \]
Step 5: Solve for \(x\). \[ x = \frac{100(550 - 22x)}{1650} = \frac{550 - 22x}{16.5}. \] So, \[ 16.5x = 550 - 22x \Rightarrow 16.5x + 22x = 550 \Rightarrow 38.5x = 550 \Rightarrow x = \frac{550}{38.5}. \] Simplify: \[ x = \frac{5500}{385} = \frac{1100}{77} = \frac{100}{7} \approx 14.2857. \] Thus, the required discount rate is approximately \(14.3%\), which is nearest to \(14%\).
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