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

Correct Answer: The correct option is 14.

Step-by-Step Explanation:

Step 1: Calculate the initial selling price
Given:
Cost Price ($\mathrm{CP}$) = Rs. 1650
Initial Profit percentage = 20%

The initial profit earned in rupees is:
$\mathrm{Initial\; Profit}=20\%\×1650=\frac{20}{100}\×1650=\text{Rs. 330}$

Therefore, the initial Selling Price (${\mathrm{SP}}_1$) is:
${\mathrm{SP}}_1=\mathrm{CP}+\mathrm{Initial\; Profit}=1650+330=\text{Rs. 1980}$

Step 2: Find the Marked Price ($\mathrm{MP}$) and initial discount amount
Let the Marked Price be $\mathrm{MP}$ and the initial discount amount be $D$ rupees.
We have:
${\mathrm{SP}}_1=\mathrm{MP}-D=1980$

When the discount is doubled (i.e., new discount = $2D$), the profit becomes Rs. 110.
The new Selling Price (${\mathrm{SP}}_2$) is:
${\mathrm{SP}}_2=\mathrm{CP}+\text{New Profit}=1650+110=\text{Rs. 1760}$

We can also write:
${\mathrm{SP}}_2=\mathrm{MP}-2D=1760$

Subtracting the second equation from the first:
$(\mathrm{MP}-D)-(\mathrm{MP}-2D)=1980-1760$
$D=\text{Rs. 220}$

Substituting $D=220$ to find $\mathrm{MP}$:
$\mathrm{MP}=1980+220=\text{Rs. 2200}$

Step 3: Calculate the required discount rate
Let $x\%$ be the rate of discount where profit percentage is also $x\%$.

Expressing Selling Price in terms of discount percentage $x\%$ on $\mathrm{MP}$:
$\mathrm{SP}=2200\×(1-\frac{x}{100})$

Expressing Selling Price in terms of profit percentage $x\%$ on $\mathrm{CP}$:
$\mathrm{SP}=1650\×(1+\frac{x}{100})$

Equating both expressions:
$2200\×(1-\frac{x}{100})=1650\×(1+\frac{x}{100})$

Dividing both sides by 550:
$4\×(1-\frac{x}{100})=3\×(1+\frac{x}{100})$

$4-\frac{4x}{100}=3+\frac{3x}{100}$
$1=\frac{7x}{100}$
$x=\frac{100}{7}\≈14.2857\%$

Rounding to the nearest integer gives 14.