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:
Show Hint
- \(12\)
- \(13\)
- \(14\)
- \(15\)
The Correct Option is C
Approach Solution - 1
Step 1: Use the first selling condition. The cost price is \[ CP = 1650. \] A profit of \(20\%\) gives \[ \text{Profit}_1 = 0.20 \times 1650 = 330. \] Hence, the first selling price is \[ SP_1 = 1650 + 330 = 1980. \] Let the marked price be \(MP\) and the discount be \(D\). Then, \[ MP - D = 1980. \tag{1} \] Step 2: Use the second selling condition. Here, the profit is Rs. \(110\), so the selling price becomes \[ SP_2 = 1650 + 110 = 1760. \] The discount is doubled, so the new discount is \(2D\). Thus, \[ MP - 2D = 1760. \tag{2} \] Step 3: Find the marked price. Subtracting equation (2) from equation (1), \[ (MP - D) - (MP - 2D) = 1980 - 1760, \] which gives \[ D = 220. \] Substituting into equation (1), \[ MP - 220 = 1980 \Rightarrow MP = 2200. \] Step 4: Equate discount percentage and profit percentage. Let the common value of discount percent and profit percent be \(x\). The discount amount is \[ \frac{x}{100} \times 2200 = 22x. \] Hence, the selling price is \[ SP = 2200 - 22x. \] The corresponding profit is \[ \text{Profit} = (2200 - 22x) - 1650 = 550 - 22x. \] Therefore, the profit percentage on cost price is \[ \frac{550 - 22x}{1650} \times 100. \] Since this equals \(x\), \[ x = \frac{550 - 22x}{1650} \times 100. \] Step 5: Solve for \(x\). \[ x = \frac{100(550 - 22x)}{1650} = \frac{550 - 22x}{16.5}. \] Thus, \[ 16.5x = 550 - 22x, \] \[ 38.5x = 550, \] \[ x = \frac{550}{38.5} = \frac{100}{7} \approx 14.29. \] Hence, the required discount rate is approximately \(14\%\).
Approach Solution -2
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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