To determine the profit percentage, let's analyze the given information in a step-by-step manner.
1. Define Variables:
- Let [tex]\( \text{CP} \)[/tex] be the cost price of one hat.
- Let [tex]\( \text{SP} \)[/tex] be the selling price of one hat.
2. Given Information:
- The selling price of 20 hats is equal to the cost price of 25 hats.
- This can be mathematically expressed as:
[tex]\[
\text{SP}_{20} = \text{CP}_{25}
\][/tex]
- Where [tex]\( \text{SP}_{20} \)[/tex] is the total selling price of 20 hats and [tex]\( \text{CP}_{25} \)[/tex] is the total cost price of 25 hats.
3. Translate to Expressions:
- Total selling price of 20 hats is:
[tex]\[
\text{SP}_{20} = 20 \times \text{SP}
\][/tex]
- Total cost price of 25 hats is:
[tex]\[
\text{CP}_{25} = 25 \times \text{CP}
\][/tex]
4. Set Up Equation:
- From the given information, we have:
[tex]\[
20 \times \text{SP} = 25 \times \text{CP}
\][/tex]
5. Solve for [tex]\(\text{SP}\)[/tex]:
- Divide both sides of the equation by 20 to find the selling price of one hat in terms of cost price ([tex]\(\text{CP}\)[/tex]):
[tex]\[
\text{SP} = \frac{25 \times \text{CP}}{20}
\][/tex]
- Simplify the fraction:
[tex]\[
\text{SP} = \frac{5 \times \text{CP}}{4} = 1.25 \times \text{CP}
\][/tex]
6. Calculate Profit:
- Profit per hat is the difference between the selling price and the cost price:
[tex]\[
\text{Profit per hat} = \text{SP} - \text{CP} = 1.25 \times \text{CP} - \text{CP} = 0.25 \times \text{CP}
\][/tex]
7. Calculate Profit Percentage:
- Profit percentage is calculated as:
[tex]\[
\text{Profit Percentage} = \left( \frac{\text{Profit per hat}}{\text{CP}} \right) \times 100\%
\][/tex]
- Substitute the profit per hat:
[tex]\[
\text{Profit Percentage} = \left( \frac{0.25 \times \text{CP}}{\text{CP}} \right) \times 100\% = 0.25 \times 100\% = 25\%
\][/tex]
Therefore, the profit percentage is [tex]\( 25\% \)[/tex]. The correct answer is:
c. [tex]\( 25\% \)[/tex]
