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

Sure! Here's a clean solution:

Let the marked price be M and the initial discount rate be d. The cost price is 1650. A profit of 20% means the selling price is

$$SP = 1650 \times 1.2 = 1980$$

With discount d,

$$M(1-d) = 1980$$

If the discount is doubled, the selling price becomes (M(1-2d)), and the profit is 110, so

$$M(1-2d) - 1650 = 110 \Rightarrow M(1-2d) = 1760$$

Subtracting the two equations:

$$M(1-d) - M(1-2d) = 1980 - 1760 \Rightarrow Md = 220 \Rightarrow M = \frac{220}{d}$$

Plug into the first equation:

$$ \frac{220}{d}(1-d) = 1980 \Rightarrow \frac{1-d}{d} = 9 \Rightarrow 1-d = 9d \Rightarrow d = 0.1 $$

So the initial discount rate is 10% and the marked price is

$$M = \frac{220}{0.1} = 2200 $$

Now, let the discount rate be (r) such that the profit percentage equals the discount percentage. Then

$$\frac{2200(1-r) - 1650}{1650} = r$$

Simplify:

$$2200(1-r) - 1650 = 1650 r \Rightarrow 2200 - 2200r - 1650 = 1650r \Rightarrow 550 = 3850 r \Rightarrow r = \frac{550}{3850} \approx 0.142857 $$

Thus, the required discount rate is approximately 14%