To determine the price at which the merchant should mark the patio set, we need to ensure that after offering a 25% discount, the merchant still achieves a 20% profit on the initial purchase price of [tex]$260.
We'll go through a step-by-step solution:
### Step 1: Calculate the Desired Selling Price for 20% Profit
First, we need to find the final selling price that includes a 20% profit over the cost price.
\[
\text{Desired Selling Price} = \text{Purchase Price} + (\text{Purchase Price} \times \text{Profit Margin})
\]
Here,
\[
\text{Purchase Price} = \$[/tex]260
\]
[tex]\[
\text{Profit Margin} = 20\% = 0.20
\][/tex]
[tex]\[
\text{Desired Selling Price} = 260 + (260 \times 0.20)
= 260 + 52
= \$312
\][/tex]
### Step 2: Determine the Marked Price Before Discount
We know that the price after a 25% discount needs to be [tex]$312. Let \( P \) be the marked price before the discount.
Given a discount of 25%:
\[
\text{Selling Price after Discount} = \text{Marked Price} \times (1 - \text{Discount Rate})
\]
\[
312 = P \times (1 - 0.25)
\]
\[
312 = P \times 0.75
\]
### Step 3: Solve for the Marked Price
To get the marked price \( P \):
\[
P = \frac{312}{0.75}
\]
\[
P = 416
\]
### Step 4: Select the Correct Option
The marked price before the discount should be \$[/tex]416.
Upon reviewing the provided options:
a. \[tex]$325
b. \$[/tex]416
c. \[tex]$312
d. \$[/tex]377
The correct answer is:
[tex]\[
\boxed{\$416}
\][/tex]
