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:
A$12$
B$14$
C$16$
D$18$
âš¡ AZUCATION SHORTCUT:
Profit drop = Extra discount.
Initial Profit = $20\% \text{ of } 1650 = 330$. New Profit = $110$.
Decrease in Profit $= 330 - 110 = 220$. This is the value of the original discount $D$.
Marked Price $(MP) = SP + D = (1650 + 330) + 220 = 2200$.
$CP : MP = 1650 : 2200 = 3 : 4$.
Set Profit% = Discount% $= x$: $3(1 + x) = 4(1 - x) \implies 7x = 1 \implies x = \frac{1}{7} \approx \mathbf{14.28\%}$.
Detailed Step-by-Step Solution
Step 1: Determine CP and MP
$CP = 1650$. Initial $SP = 1650 + (20\% \text{ of } 1650) = 1980$.
Let $MP = M$ and original discount $= D$.
$M - D = 1980 \quad \dots(1)$
If discount is doubled ($2D$), $SP = 1650 + 110 = 1760$.
$M - 2D = 1760 \quad \dots(2)$
Subtract (2) from (1): $D = 220$.
Substituting $D$: $M = 1980 + 220 = 2200$.
Step 2: Equate Profit% and Discount%
Let the rate be $k\%$.
$CP \times (1 + \frac{k}{100}) = MP \times (1 - \frac{k}{100})$
$1650(1 + \frac{k}{100}) = 2200(1 - \frac{k}{100})$
$3(100 + k) = 4(100 - k) \implies 300 + 3k = 400 - 4k$
$7k = 100 \implies k = 14.28\%$.
The nearest integer is **14**.
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