The ratio z to y is equivalent to the ratio 9 to 5. If the value of z is 162,...

1. TRANSLATE the problem information

• Given information:
  • The ratio of z to y equals the ratio of 9 to 5
  • z = 162
  • Need to find: y

• What this tells us: We can set up a proportion equation

2. TRANSLATE the ratio into mathematical form

• 'The ratio z to y is equivalent to the ratio 9 to 5'
• This becomes: \(\frac{\mathrm{z}}{\mathrm{y}} = \frac{9}{5}\)

3. SIMPLIFY by substituting the known value

• Substitute z = 162 into the equation:
• \(\frac{162}{\mathrm{y}} = \frac{9}{5}\)

4. SIMPLIFY using cross multiplication

• Cross multiply to eliminate fractions:
• \(162 \times 5 = 9 \times \mathrm{y}\)
• \(810 = 9\mathrm{y}\)

5. SIMPLIFY to solve for y

• Divide both sides by 9:
• \(\mathrm{y} = \frac{810}{9} = 90\)

Answer: 90

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may set up the proportion backwards, writing \(\frac{\mathrm{y}}{\mathrm{z}} = \frac{9}{5}\) instead of \(\frac{\mathrm{z}}{\mathrm{y}} = \frac{9}{5}\). This happens because they don't carefully map which variable corresponds to which number in the given ratio statement.

Using the incorrect setup \(\frac{\mathrm{y}}{\mathrm{z}} = \frac{9}{5}\) with z = 162 leads to:

\(\frac{162}{\mathrm{y}} = \frac{5}{9}\), giving \(\mathrm{y} = \frac{162 \times 9}{5} = 291.6\)

This leads to an incorrect numerical answer.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\frac{\mathrm{z}}{\mathrm{y}} = \frac{9}{5}\) but make arithmetic errors during cross multiplication or division. Common mistakes include calculating \(162 \times 5 = 800\) instead of 810, or making division errors with \(810 \div 9\).

These arithmetic errors lead to incorrect final answers that are close to but not equal to 90.

The Bottom Line:

This problem tests whether students can accurately translate ratio language into mathematical notation and then execute the algebraic steps without arithmetic errors. The key insight is recognizing that 'ratio A to B' means A/B, not B/A.