Let's assume the marked price of the product is "M".

Given that the company offers a discount of 25%, the selling price after the discount will be (100% - 25%) = 75% of the marked price. So, the selling price is 0.75M.

The company wants to maintain a profit margin of 15% after offering the discount. This means the selling price should be 115% of the cost price. Mathematically, it can be represented as:

0.75M = 1.15C

Where C is the cost price.

To find the markup percentage on the cost price, we need to calculate the difference between the selling price and the cost price as a percentage of the cost price. Let's solve the equation to find the value of C:

0.75M = 1.15C

Dividing both sides by 1.15:

0.75M / 1.15 = C

Simplifying the expression:

C = 0.65M

The difference between the selling price and the cost price is (0.75M - C).

The markup percentage on the cost price can be calculated as:

$[(0.75M-C)/C]\times 100$

Substituting the value of C:

$[(0.75M-0.65M)/0.65M]\times 100$

Simplifying the expression:

$[0.1M/0.65M]\times100\\ =(0.1/0.65)\times 100\\ \approx 15.38\%$

Therefore, the markup percentage on the cost price should be approximately 15.38% to maintain a 15% profit margin after offering a 25% discount on the marked price.

Therefore, the correct answer is _E___.

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