In a club, the ratio of junior members to senior members is 2:3. If there are s senior members in...

1. TRANSLATE the problem information

• Given information:
  • Ratio of junior members to senior members is 2 to 3
  • There are s senior members
  • Need to find: expression for number of junior members

• What this tells us: The ratio \(2:3\) means \(\frac{\mathrm{juniors}}{\mathrm{seniors}} = \frac{2}{3}\)

2. INFER the mathematical relationship

• Since \(\frac{\mathrm{juniors}}{\mathrm{seniors}} = \frac{2}{3}\), and we have s seniors
• We can write: \(\frac{\mathrm{juniors}}{\mathrm{s}} = \frac{2}{3}\)
• This means: \(\mathrm{juniors} = \frac{2}{3} \times \mathrm{s} = \frac{2}{3}\mathrm{s}\)

3. Match to answer choices

Looking at the options, \(\frac{2}{3}\mathrm{s}\) appears as choice (B).

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up which part of the ratio corresponds to which group. They see "ratio of junior to senior is 2 to 3" and incorrectly think this means juniors = \(\frac{3}{2} \times \mathrm{seniors}\).

They reason: "If the ratio is 2 to 3, then for every 2 juniors there are 3 seniors, so juniors must be larger." This flawed reasoning leads them to flip the fraction.

This may lead them to select Choice A (\(\frac{3}{2}\mathrm{s}\)).

Second Most Common Error:

Missing conceptual knowledge about ratios: Students don't understand that ratios represent multiplicative relationships. Instead, they think ratios mean "add 2 more" or similar additive reasoning.

They might think: "The ratio involves 2 and 3, so maybe we add 2 to the number of seniors."

This may lead them to select Choice C (s + 2).

The Bottom Line:

The key challenge is correctly translating ratio language into mathematical relationships. Students must recognize that "\(\mathrm{A}\text{ to }\mathrm{B}\text{ is }2\text{ to }3\)" means \(\frac{\mathrm{A}}{\mathrm{B}} = \frac{2}{3}\), which gives \(\mathrm{A} = \frac{2}{3} \times \mathrm{B}\).