A retailer marked up the prices of two products, Product M and Product N, by 30% and 40% respectively above their cost prices, and then offered a discount of 20% on Product M and 15% on Product N. If the difference between the selling prices of the two products is Rs. 820, find the cost price (in Rs.) of Product N, given that the ratio of the cost prices of Product M and Product N is 3:4, respectively.

The correct option is 2000.

Step 1: Define the cost prices of both products using a common variable.
Given that the ratio of the cost prices of Product M and Product N is 3 : 4, let:

Cost price of Product M, $C{P}_M=3x$

Cost price of Product N, $C{P}_N=4x$

Step 2: Calculate the selling price (SP) of Product M.
Product M is marked up by 30% above its cost price:

Marked Price of M, $M{P}_M=3x\times (1+\frac{30}{100})=3x\times 1.30=3.9x$

A discount of 20% is then offered on Product M:

Selling Price of M, $S{P}_M=M{P}_M\times (1-\frac{20}{100})=3.9x\times 0.80=3.12x$

Step 3: Calculate the selling price (SP) of Product N.
Product N is marked up by 40% above its cost price:

Marked Price of N, $M{P}_N=4x\times (1+\frac{40}{100})=4x\times 1.40=5.6x$

A discount of 15% is then offered on Product N:

Selling Price of N, $S{P}_N=M{P}_N\times (1-\frac{15}{100})=5.6x\times 0.85=4.76x$

Step 4: Find the value of $x$ using the difference between selling prices.
The difference between the selling prices of Product N and Product M is given as Rs. 820:

$S{P}_N-S{P}_M=820$

$4.76x-3.12x=820$

$1.64x=820$

$x=\frac{820}{1.64}=500$

Step 5: Calculate the cost price of Product N.
Substitute $x=500$ back into the cost price expression for Product N:

Cost price of Product N = $4x=4\times 500=2000$

Therefore, the cost price of Product N is Rs. 2000.