Let's solve the given problem step-by-step to find the required profit percentage.
1. Define the Terms:
- Let the cost price (CP) of the article be 100 units (for simplicity).
- Let [tex]\( X \% \)[/tex] be the given discount in the first scenario.
- Let [tex]\( SP1 \)[/tex] be the selling price after giving [tex]\( X \% \)[/tex] discount and earning a 40% profit.
- Let [tex]\( SP2 \)[/tex] be the selling price after giving [tex]\( 2X \% \)[/tex] discount and earning a 5% profit.
2. Formulate the Equations:
- For the first condition ([tex]\( X \% \)[/tex] discount and 40% profit):
[tex]\[
SP1 = CP \times (1 - \frac{X}{100}) \times (1 + \frac{40}{100}) = 100 \times (1 - \frac{X}{100}) \times 1.40
\][/tex]
- For the second condition ([tex]\( 2X \% \)[/tex] discount and 5% profit):
[tex]\[
SP2 = CP \times (1 - \frac{2X}{100}) \times (1 + \frac{5}{100}) = 100 \times (1 - \frac{2X}{100}) \times 1.05
\][/tex]
3. Equate the Selling Prices:
Since the original selling prices [tex]\( SP1 \)[/tex] and [tex]\( SP2 \)[/tex] must be equal:
[tex]\[
100 \times (1 - \frac{X}{100}) \times 1.40 = 100 \times (1 - \frac{2X}{100}) \times 1.05
\][/tex]
4. Simplify the Equation:
[tex]\[
(1 - \frac{X}{100}) \times 1.40 = (1 - \frac{2X}{100}) \times 1.05
\][/tex]
[tex]\[
1.40 - \frac{1.40X}{100} = 1.05 - \frac{2.10X}{100}
\][/tex]
[tex]\[
1.40 - 1.05 = \frac{2.10X}{100} - \frac{1.40X}{100}
\][/tex]
[tex]\[
0.35 = \frac{0.70X}{100}
\][/tex]
[tex]\[
X = \frac{0.35 \times 100}{0.70} = 50
\][/tex]
5. Determine the Profit After Giving a 25% Discount:
- Let the discount be 25%. We need to find the profit percentage here.
- The new selling price [tex]\( SP \)[/tex]:
[tex]\[
SP = CP \times (1 - \frac{25}{100}) \times (1 + \frac{40}{100}) = 100 \times 0.75 \times 1.40 = 105
\][/tex]
- Therefore, the profit percentage when the Cost Price (CP) is 100:
[tex]\[
\text{Profit Percent} = \left( \frac{SP}{CP \times (1 - \frac{25}{100})} - 1 \right) \times 100
\][/tex]
[tex]\[
\text{Profit Percent} = \left( \frac{105}{75} - 1 \right) \times 100 = \left(1.4 - 1 \right) \times 100 = 0.4 \times 100 = 40\%
\][/tex]
Thus, if Ram sells the article after giving a discount of 25%, he would earn an approximate profit percentage of 40%.
