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 answer is Rs.625.

Let the purchase price of gadget Y be $P_{\text{Y}}$ and the purchase price of gadget X be $P_{\text{X}}$.

Given that the purchase price of gadget X is 80% of the purchase price of gadget Y:

$P_{\text{X}}=0.80\times P_{\text{Y}}=0.8P_{\text{Y}}$

Step 1: Express the marked price, discount, and profit for gadget X in terms of $P_{\text{X}}$

Gadget X is tagged 60% above its purchase price:

$\text{Marked Price (MP)}=P_{\text{X}}+0.60P_{\text{X}}=1.60P_{\text{X}}$

A 20% discount is offered on gadget X:

$\text{Discount on X}=20\%\text{of Marked Price}=0.20\times 1.60P_{\text{X}}=0.32P_{\text{X}}$

The selling price (SP) of gadget X is:

$\text{Selling Price (SP)}=1.60P_{\text{X}}-0.32P_{\text{X}}=1.28P_{\text{X}}$

The profit earned on gadget X is:

$\text{Profit on X}=\text{Selling Price}-\text{Purchase Price}=1.28P_{\text{X}}-P_{\text{X}}=0.28P_{\text{X}}$

Step 2: Form an equation using the given condition

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

$\text{Discount on X}=\text{Profit on X}+20$

Substituting the values in terms of $P_{\text{X}}$:

$0.32P_{\text{X}}=0.28P_{\text{X}}+20$

$0.32P_{\text{X}}-0.28P_{\text{X}}=20$

$0.04P_{\text{X}}=20$

$P_{\text{X}}=\frac{20}{0.04}=500$

Thus, the purchase price of gadget X is Rs.500.

Step 3: Calculate the purchase price of gadget Y

Using the relationship $P_{\text{X}}=0.8P_{\text{Y}}$:

$500=0.8P_{\text{Y}}$

$P_{\text{Y}}=\frac{500}{0.8}=625$

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