The ratio of t to u is 1 to 2, and t = 10. What is the value of u?

1. TRANSLATE the problem information

• Given information:
  • The ratio of t to u is 1 to 2
  • t = 10
  • Need to find: value of u
• What this tells us: We can write this as the proportion \(\mathrm{t/u = 1/2}\)

2. INFER the solving approach

• Since we know t = 10, we can substitute this value into our proportion
• This creates a simple equation with one unknown that we can solve

3. SIMPLIFY by substitution and solving

• Substitute t = 10 into the proportion:
  \(\mathrm{10/u = 1/2}\)
• Cross multiply to solve:
  \(\mathrm{10 × 2 = u × 1}\)
  \(\mathrm{20 = u}\)

Answer: D. 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students reverse the ratio relationship and write \(\mathrm{u/t = 1/2}\) instead of \(\mathrm{t/u = 1/2}\).

When they substitute t = 10, they get \(\mathrm{u/10 = 1/2}\), leading to \(\mathrm{u = 5}\).

This may lead them to select Choice B (5).

Second Most Common Error:

Poor attention to what the question asks: Students correctly solve for \(\mathrm{u = 20}\) but then select the given value instead of their calculated answer.

They see t = 10 in the problem and mistakenly think this is what they're looking for.

This may lead them to select Choice C (10).

The Bottom Line:

Ratio problems require careful attention to the order of terms and what quantity you're solving for. The key is translating "ratio of A to B" correctly as A/B, not B/A.