The ratio of the length of line segment XY to the length of line segment ZV is 6:1. If the...

1. TRANSLATE the problem information

• Given information:
  • The ratio of XY to ZV is 6 to 1
  • \(\mathrm{XY} = 102\) inches
  • Need to find ZV

• What this tells us: We can write this ratio as \(\frac{\mathrm{XY}}{\mathrm{ZV}} = \frac{6}{1} = 6\)

2. INFER the solution approach

• Since we know the ratio and one length, we can set up an equation
• Strategy: Substitute the known value into the ratio equation and solve

3. SIMPLIFY by setting up and solving the equation

• Start with: \(\frac{\mathrm{XY}}{\mathrm{ZV}} = 6\)

• Substitute: \(\frac{102}{\mathrm{ZV}} = 6\)

• Multiply both sides by ZV: \(102 = 6 \times \mathrm{ZV}\)

• Divide both sides by 6: \(\mathrm{ZV} = \frac{102}{6} = 17\)

Answer: A. 17

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the ratio direction and set up \(\frac{\mathrm{ZV}}{\mathrm{XY}} = \frac{6}{1}\) instead of \(\frac{\mathrm{XY}}{\mathrm{ZV}} = \frac{6}{1}\).

When they substitute, they get \(\frac{\mathrm{ZV}}{102} = 6\), leading to \(\mathrm{ZV} = 6 \times 102 = 612\). This may lead them to select Choice D (612).

Second Most Common Error:

Poor INFER reasoning: Students recognize they need to use ratios but get confused about what the ratio means, thinking that if XY to ZV is 6 to 1, then ZV should be the same length as XY.

This leads them to think \(\mathrm{ZV} = 102\) inches, causing them to select Choice C (102).

The Bottom Line:

This problem tests whether students can accurately translate ratio language into mathematical relationships and maintain the correct order when setting up proportional equations.