To determine how much above the cost price a shopkeeper should mark his goods to achieve a 20% gain after allowing a 25% discount on the marked price, we need to follow these steps:
1. Define the variables:
- Let [tex]\( \text{Cost Price} = CP \)[/tex].
- Let [tex]\( \text{Marked Price} = MP \)[/tex].
- Let [tex]\( \text{Selling Price} = SP \)[/tex].
2. Understand the relationships:
- The selling price [tex]\( SP \)[/tex] is obtained by giving a 25% discount on the marked price [tex]\( MP \)[/tex].
[tex]\[
SP = MP \times (1 - 0.25) = MP \times 0.75
\][/tex]
- A 20% gain on the cost price means the selling price is 120% of the cost price, or:
[tex]\[
SP = CP \times (1 + 0.20) = CP \times 1.20
\][/tex]
3. Equate the selling prices:
[tex]\[
MP \times 0.75 = CP \times 1.20
\][/tex]
4. Solve for the Marked Price ([tex]\( MP \)[/tex]):
[tex]\[
MP = \frac{CP \times 1.20}{0.75}
\][/tex]
5. Simplify the expression:
[tex]\[
MP = \frac{1.20}{0.75} \times CP = 1.60 \times CP
\][/tex]
6. Calculate the percentage above the cost price:
The marked price [tex]\( MP \)[/tex] is 1.60 times the cost price [tex]\( CP \)[/tex], which means it is 60% higher than the cost price.
[tex]\[
\text{Percentage above the cost price} = (1.60 - 1) \times 100\% = 0.60 \times 100\% = 60\%
\][/tex]
Therefore, the shopkeeper should mark his goods 60% above the cost price to achieve a 20% gain after allowing a 25% discount on the marked price.
