A merchant sold a desk lamp for $850, incurring a 15% loss. If the merchant sets the marked price of the same lamp at 50% above its cost price and makes a profit of $200 on the sale, find the discount percentage allowed on the marked price.

The correct answer is 20%.

Step 1: Find the Cost Price (CP) of the desk lamp
The lamp was sold for $850, incurring a 15% loss. We can calculate the Cost Price (CP) using the loss formula:

$\text{Selling Price}=\text{Cost Price}\times (1-\frac{\text{Loss \%}}{100})$

Substituting the given values into the formula:

$850=\text{Cost Price}\times (1-0.15)$

$850=\text{Cost Price}\times 0.85$

$\text{Cost Price}=\frac{850}{0.85}=1000$

So, the Cost Price of the lamp is $1000.

Step 2: Find the Marked Price (MP) of the desk lamp
The merchant sets the marked price at 50% above its cost price:

$\text{Marked Price}=\text{Cost Price}\times (1+\frac{50}{100})$

$\text{Marked Price}=1000\times 1.50=1500$

So, the Marked Price of the lamp is $1500.

Step 3: Find the actual Selling Price and Discount amount
The merchant makes a profit of $200 on the sale, so the actual selling price is:

$\text{Selling Price}=\text{Cost Price}+\text{Profit}$

$\text{Selling Price}=1000+200=1200$

The discount given on the marked price is:

$\text{Discount}=\text{Marked Price}-\text{Selling Price}$

$\text{Discount}=1500-1200=300$

So, the discount amount is $300.

Step 4: Calculate the Discount Percentage
The discount percentage allowed on the marked price is given by:

$\text{Discount Percentage}=\left(\frac{\text{Discount}}{\text{Marked Price}}\right)\times 100\%$

$\text{Discount Percentage}=\left(\frac{300}{1500}\right)\times 100\%=20\%$

Therefore, the discount percentage allowed on the marked price is 20%.