If 8/5 = 32/k, what is the value of k?

1. TRANSLATE the problem information

• Given information:
  • We have the proportion: \(\frac{8}{5} = \frac{32}{\mathrm{k}}\)
  • We need to find the value of k

2. INFER the solution approach

• This is a proportion (two equal ratios)
• The most efficient method is cross multiplication
• Cross multiplication means: if \(\frac{\mathrm{a}}{\mathrm{b}} = \frac{\mathrm{c}}{\mathrm{d}}\), then \(\mathrm{ad} = \mathrm{bc}\)

3. SIMPLIFY by applying cross multiplication

• Cross multiply: \(\frac{8}{5} = \frac{32}{\mathrm{k}}\) becomes \(8\mathrm{k} = 5 \times 32\)
• Calculate the right side: \(8\mathrm{k} = 160\)
• Solve for k: \(\mathrm{k} = 160 \div 8 = 20\)

4. Verify the answer

• Check: \(\frac{8}{5} = 1.6\) and \(\frac{32}{20} = 1.6\) ✓

Answer: D) 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't recognize this as a standard proportion problem requiring cross multiplication. Instead, they try to manipulate the fractions directly, perhaps attempting to make denominators equal or getting confused about which numbers to multiply or divide.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up cross multiplication but make arithmetic errors. For example, calculating \(5 \times 32\) incorrectly as 150 instead of 160, leading to \(\mathrm{k} = 150 \div 8 = 18.75\).

Since 18.75 isn't among the choices, this causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem requires students to immediately recognize the proportion structure and confidently apply cross multiplication. Students who don't see this pattern quickly often waste time on inefficient approaches or make calculation errors under pressure.