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

The correct option is 14.

First, let's determine the initial Selling Price (${\mathrm{SP}}_1$) and the relation between Marked Price ($\mathrm{MP}$) and the initial discount ($D$).
We are given that the Cost Price ($\mathrm{CP}$) is Rs. 1650 and the initial profit is 20%.

$\text{Initial Profit}=0.20\times 1650=330$

${\mathrm{SP}}_1=\mathrm{CP}+\text{Profit}=1650+330=1980$

Since the selling price is the marked price minus the discount, we can write our first equation:

$\mathrm{MP}-D=1980$

Next, we use the information for the second scenario. If the discount is doubled (i.e., new discount is $2D$), the profit is Rs. 110.

${\mathrm{SP}}_2=\mathrm{CP}+\text{New Profit}=1650+110=1760$

This gives us our second equation:

$\mathrm{MP}-2D=1760$

Now, we can subtract the second equation from the first to solve for the discount $D$:

$(\mathrm{MP}-D)-(\mathrm{MP}-2D)=1980-1760$

$D=220$

Substituting the value of $D$ back into the first equation to find the $\mathrm{MP}$:

$\mathrm{MP}-220=1980$
$\mathrm{MP}=1980+220=2200$

The problem asks for the percentage rate of discount, let's call it $x$, at which the profit percentage is also equal to $x$. We can express the actual discount amount and profit amount in terms of $x$:

$\text{Discount}=\frac{x}{100}\times \mathrm{MP}=\frac{x}{100}\times 2200=22x$

$\text{Profit}=\frac{x}{100}\times \mathrm{CP}=\frac{x}{100}\times 1650=16.5x$

The Selling Price ($\mathrm{SP}$) can be written in two equivalent ways: either by subtracting the discount from the Marked Price or by adding the profit to the Cost Price:

$\mathrm{SP}=\mathrm{MP}-\text{Discount}=\mathrm{CP}+\text{Profit}$

Equating both expressions for $\mathrm{SP}$:

$2200-22x=1650+16.5x$

Now, let's group the terms to solve for $x$:

$2200-1650=22x+16.5x$
$550=38.5x$

$x=\frac{550}{38.5}=\frac{5500}{385}=\frac{100}{7}$

$x\approx 14.28$

The rate of discount nearest to an integer value is 14%. Thus, the given correct answer is 14.