The ratio x to y is equivalent to the ratio 12 to t. When x = 156, what is the...

1. TRANSLATE the problem information

• Given information:
  • The ratio x to y equals the ratio 12 to t
  • \(\mathrm{x} = 156\)
  • We need y in terms of t

• In mathematical notation: \(\frac{\mathrm{x}}{\mathrm{y}} = \frac{12}{\mathrm{t}}\)

2. INFER the solution approach

• Since we know \(\mathrm{x} = 156\) and have the proportion \(\frac{\mathrm{x}}{\mathrm{y}} = \frac{12}{\mathrm{t}}\), we can substitute and solve for y
• This will give us y expressed in terms of t

3. TRANSLATE the substitution

• Substitute \(\mathrm{x} = 156\) into the proportion:
  \(\frac{156}{\mathrm{y}} = \frac{12}{\mathrm{t}}\)

4. SIMPLIFY through cross multiplication

• Cross multiply to get: \(156\mathrm{t} = 12\mathrm{y}\)
• Divide both sides by 12: \(\mathrm{y} = \frac{156\mathrm{t}}{12}\)
• Reduce the fraction: \(\mathrm{y} = 13\mathrm{t}\)

Answer: A. 13t

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may set up the proportion incorrectly, perhaps writing \(\frac{\mathrm{x}}{\mathrm{y}} = \frac{\mathrm{t}}{12}\) instead of \(\frac{\mathrm{x}}{\mathrm{y}} = \frac{12}{\mathrm{t}}\), misinterpreting which quantities correspond to which positions in the ratio.

This leads to \(\frac{156}{\mathrm{y}} = \frac{\mathrm{t}}{12}\), which gives \(\mathrm{y} = \frac{156(12)}{\mathrm{t}} = \frac{1872}{\mathrm{t}}\), which doesn't match any answer choice and causes confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equation but make arithmetic errors when calculating 156 ÷ 12, perhaps getting 12 instead of 13.

This may lead them to select Choice B (12t).

The Bottom Line:

This problem tests whether students can accurately translate ratio language into mathematical notation and then execute straightforward algebraic steps. The key insight is recognizing that ratios create proportional relationships that can be solved using cross multiplication.