8p + 3r = 42A fitness center charges members monthly fees based on their membership type. The equation above represents...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{8p + 3r = 42}\) (where p = premium memberships, r = regular memberships)
• Known information: The fitness center has 3 premium memberships
• This means: \(\mathrm{p = 3}\)

2. INFER the solution approach

• Since we know \(\mathrm{p = 3}\), we can substitute this value into the equation
• This will give us a simple equation with only r as the unknown
• Then we can solve directly for r

3. SIMPLIFY through substitution and algebra

• Substitute \(\mathrm{p = 3}\): \(\mathrm{8(3) + 3r = 42}\)
• Calculate: \(\mathrm{24 + 3r = 42}\)
• Subtract 24 from both sides: \(\mathrm{3r = 18}\)
• Divide by 3: \(\mathrm{r = 6}\)

Answer: D. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the algebraic steps

Many students correctly set up \(\mathrm{8(3) + 3r = 42}\) and get \(\mathrm{24 + 3r = 42}\), but then calculate \(\mathrm{42 - 24 = 16}\) instead of 18, leading to \(\mathrm{r = 16/3 ≈ 5.33}\). Since this doesn't match an exact answer choice, they might round to the nearest option.

This may lead them to select Choice C (5) or causes confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret which variable represents which quantity

Some students might confuse premium and regular memberships, thinking that \(\mathrm{r = 3}\) instead of \(\mathrm{p = 3}\). This leads them to solve \(\mathrm{8p + 3(3) = 42}\), getting \(\mathrm{8p + 9 = 42}\), so \(\mathrm{8p = 33}\), and \(\mathrm{p = 33/8 ≈ 4.1}\).

This may lead them to select Choice B (4) as the closest integer value.

The Bottom Line:

This problem tests whether students can accurately translate word problems into mathematical notation and then execute algebraic manipulations without arithmetic errors. The key insight is that substitution problems become straightforward once the correct value is identified and carefully substituted.