To find the marked price at which a pharmacist must price an item that costs [tex]$2.60 such that even after a 25% discount the sale would yield a gross profit of 35%, follow the steps below:
1. Determine the desired profit: Start by calculating the desired selling price that includes a 35% profit margin on the cost price.
Given:
- Cost price (C) = $[/tex]2.60
- Desired profit margin = 35% (or 0.35 in decimal form)
To find the selling price with the desired profit (SP):
[tex]\[
SP = C \times (1 + \text{Desired Profit Margin})
\][/tex]
Substituting the values:
[tex]\[
SP = 2.60 \times (1 + 0.35) = 2.60 \times 1.35
\][/tex]
This yields:
[tex]\[
SP = 3.51
\][/tex]
2. Adjust for the discount: Next, we need to determine the marked price before the discount is applied. Given that the selling price after applying the 25% discount should be [tex]$3.51, we calculate the marked price (MP).
Given:
- Discount = 25% (or 0.25 in decimal form)
- Selling price after discount (SP) = $[/tex]3.51
The marked price can be found using the formula:
[tex]\[
SP = MP \times (1 - \text{Discount})
\][/tex]
Rearrange to solve for MP:
[tex]\[
MP = \frac{SP}{1 - \text{Discount}}
\][/tex]
Substituting the values:
[tex]\[
MP = \frac{3.51}{1 - 0.25} = \frac{3.51}{0.75}
\][/tex]
This gives us:
[tex]\[
MP = 4.68
\][/tex]
Therefore, the pharmacist must mark the item at [tex]$4.68 to ensure that even after a 25% discount, the sale still yields a gross profit of 35% on the cost price of $[/tex]2.60.
