Question:If t/5 = 3, what is the value of 10/t?1/31/22/33/22

1. INFER the solution strategy

• Given: \(\frac{\mathrm{t}}{5} = 3\)
• Need to find: \(\frac{10}{\mathrm{t}}\)
• Key insight: To evaluate \(\frac{10}{\mathrm{t}}\), we first need to find the value of t

2. SIMPLIFY to solve for t

• From \(\frac{\mathrm{t}}{5} = 3\), multiply both sides by 5:
  \(\mathrm{t} = 3 \times 5 = 15\)

3. SIMPLIFY by substituting and reducing

• Substitute \(\mathrm{t} = 15\) into the expression \(\frac{10}{\mathrm{t}}\):
  \(\frac{10}{\mathrm{t}} = \frac{10}{15}\)
• Reduce the fraction: \(\frac{10}{15} = \frac{2}{3}\)

Answer: C. \(\frac{2}{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when solving for t, calculating \(3 \times 5\) incorrectly (perhaps getting 8 instead of 15), or they solve correctly but fail to reduce \(\frac{10}{15}\) to its simplest form.

If they get \(\mathrm{t} = 8\), they would calculate \(\frac{10}{8} = \frac{5}{4} = 1.25\), which doesn't match any answer choice, leading to confusion and guessing. If they leave their answer as \(\frac{10}{15}\) without reducing, they won't recognize it equals \(\frac{2}{3}\).

Second Most Common Error:

Poor INFER reasoning: Students attempt to work directly with ratios or proportions without first finding t, leading to incorrect setup of relationships between the given equation and the target expression.

This causes them to get stuck early in the problem and resort to guessing among the answer choices.

The Bottom Line:

This problem tests whether students can recognize that substitution requires finding the variable's value first, then can execute the arithmetic accurately while remembering to simplify their final answer.