The purchase price of gadget X is 80% of the purchase price of gadget Y. Gadget X is tagged 60% above its purchase price, and a 20% discount is offered on it. If the discount given on gadget X is Rs.20 more than the profit earned on gadget X, find the purchase price of gadget Y.

The correct option is Rs.625.

Step-by-Step Explanation:

Let us solve the problem by expressing all values for gadget X in terms of the purchase price of gadget Y.

Step 1: Express the Cost Price (Purchase Price) of Gadget X

Let the purchase price of gadget Y be $P_Y$.

The purchase price of gadget X (${\mathrm{CP}}_X$) is given as 80% of the purchase price of gadget Y:

${\mathrm{CP}}_X=0.80\times P_Y=0.8P_Y$

Step 2: Calculate the Tagged (Marked) Price of Gadget X

Gadget X is marked 60% above its purchase price. Therefore, its Marked Price (${\mathrm{MP}}_X$) is:

${\mathrm{MP}}_X={\mathrm{CP}}_X+(0.60\times {\mathrm{CP}}_X)=1.60\times {\mathrm{CP}}_X$

Substituting ${\mathrm{CP}}_X=0.8P_Y$:

${\mathrm{MP}}_X=1.60\times 0.8P_Y=1.28P_Y$

Step 3: Calculate the Discount Offered on Gadget X

A 20% discount is offered on gadget X based on its marked price. The discount amount ($D_X$) is:

$D_X=0.20\times {\mathrm{MP}}_X$

Substituting ${\mathrm{MP}}_X=1.28P_Y$:

$D_X=0.20\times 1.28P_Y=0.256P_Y$

Step 4: Calculate the Profit Earned on Gadget X

The Selling Price (${\mathrm{SP}}_X$) of gadget X is the Marked Price minus the Discount:

${\mathrm{SP}}_X={\mathrm{MP}}_X-D_X=1.28P_Y-0.256P_Y=1.024P_Y$

The Profit (${\mathrm{Profit}}_X$) earned on gadget X is the Selling Price minus the Cost Price:

${\mathrm{Profit}}_X={\mathrm{SP}}_X-{\mathrm{CP}}_X=1.024P_Y-0.8P_Y=0.224P_Y$

Step 5: Formulate the Equation and Find $P_Y$

We are given that the discount on gadget X is Rs.20 more than the profit earned on gadget X:

$D_X-{\mathrm{Profit}}_X=20$

Substitute the expressions in terms of $P_Y$:

$0.256P_Y-0.224P_Y=20$

$0.032P_Y=20$

$P_Y=\frac{20}{0.032}=\frac{20000}{32}=625$

Hence, the purchase price of gadget Y is Rs.625.