Directions: Solve the following problem and select the correct option.

A tech retailer purchased two types of displays: 4K Monitors and HD Monitors. He sold the 4K Monitors at a 25% profit and the HD Monitors at a 10% loss. The cost price of the 4K Monitors was 40% higher than that of the HD Monitors. If his overall profit from selling both items was $100, what was the purchase price of each type of display?

The correct option is $560 and $400.

Step-by-Step Explanation:

1. Define the cost prices:
Let the purchase price of HD Monitors be:
${\mathrm{CP}}_{\text{HD}}$
Since the cost price of 4K Monitors is 40% higher than that of HD Monitors:
${\mathrm{CP}}_{\text{4K}}={\mathrm{CP}}_{\text{HD}}+0.40\times {\mathrm{CP}}_{\text{HD}}=1.40\times {\mathrm{CP}}_{\text{HD}}$

2. Calculate profit and loss:
Profit from selling 4K Monitors at a 25% profit:
${\text{Profit}}_{\text{4K}}=25\%\times {\mathrm{CP}}_{\text{4K}}=0.25\times (1.40\times {\mathrm{CP}}_{\text{HD}})=0.35\times {\mathrm{CP}}_{\text{HD}}$
Loss from selling HD Monitors at a 10% loss:
${\text{Loss}}_{\text{HD}}=10\%\times {\mathrm{CP}}_{\text{HD}}=0.10\times {\mathrm{CP}}_{\text{HD}}$

3. Formulate the overall profit equation:
Given that the overall profit from selling both items is $100:
$\text{Overall Profit}={\text{Profit}}_{\text{4K}}-{\text{Loss}}_{\text{HD}}$
Substitute the expressions into the equation:
$0.35\times {\mathrm{CP}}_{\text{HD}}-0.10\times {\mathrm{CP}}_{\text{HD}}=100$
$0.25\times {\mathrm{CP}}_{\text{HD}}=100$

4. Solve for the purchase price of HD Monitors:
${\mathrm{CP}}_{\text{HD}}=\frac{100}{0.25}=400$
The purchase price of HD Monitors is $400.

5. Find the purchase price of 4K Monitors:
${\mathrm{CP}}_{\text{4K}}=1.40\times 400=560$
The purchase price of 4K Monitors is $560.

Therefore, the purchase prices of 4K Monitors and HD Monitors are $560 and $400, respectively.