Step 1: [tex]$\frac{32}{40} = \frac{20}{PR}$[/tex]

Step 2: [tex]$20PR = (32)(48)$[/tex]

Step 3: [tex]$2 \times PR = 1,536$[/tex]

Step 4: [tex]$PR = 76.8 \text{ km}$[/tex]

What is Valek's first error?

A. Valek did not correctly divide 1,536 by 20 going from step 3 to step 4.

B. Valek did not find the correct product of 32 and 48 going from step 2 to step 3.

C. Valek should have written the proportion in step 1 as [tex]$\frac{32}{20} = \frac{PR}{48}$[/tex].

D. Valek should have written the cross-product in step 2 as [tex]$32PR = (20)(48)$[/tex].

Let's walk through the steps to find Valek's first error:

1. Step 1: Understanding the Equation
The proportion given is:
[tex]\[
\frac{32}{40} = \frac{20}{PR}
\][/tex]
The goal is to solve for [tex]$ PR $[/tex].

2. Step 2: Setting Up the Cross-Product
According to the rules of cross-multiplication, the setup would be:
[tex]\[
32 \times PR = 20 \times 40
\][/tex]
This equation simplifies the comparison in terms of [tex]$ PR $[/tex].

3. Step 3: Simplification
The equation simplifies further to:
[tex]\[
PR = \frac{20 \times 40}{32}
\][/tex]

4. Step 4: Solve [tex]$ PR $[/tex] Calculation Issue
Valek attempts to solve for [tex]$ PR $[/tex] using:
[tex]\[
20 \times PR = 32 \times 48
\][/tex]
Hence,
[tex]\[
PR = \frac{1536}{20}
\][/tex]

5. Step 5: Valek's Error Identification
Valek states in Step 4 that:
[tex]\[
PR = 76.8 \text{ km}
\][/tex]
However, the calculation for:
[tex]\[
PR = \frac{1536}{20} = 76.8
\][/tex]
is correct, indicating a calculation mistake in their earlier stages.

Based on the analysis:
- Valek did not correctly implement the cross-product initially which should have been:
[tex]\[
32 \times PR = 20 \times 40
\][/tex]
Instead, they put it as:
[tex]\[
20 \times PR = 32 \times 48
\][/tex]

So, Valek's first error is in setting up the cross-product improperly. The correct answer is: Valek should have written the cross-product in step 2 as [tex]$ 32 \times PR = (20 \times 40) $[/tex].