The ratio 140:m is equivalent to the ratio 4:28. What is the value of m?

1. TRANSLATE the problem information

• Given information:
  • Ratio 140 to m is equivalent to ratio 4 to 28
• What this tells us: We can set up equal fractions since equivalent ratios have the same value

2. TRANSLATE to mathematical notation

• Write the equivalent ratios as a proportion:
  \(\frac{140}{\mathrm{m}} = \frac{4}{28}\)

3. SIMPLIFY using cross-multiplication

• Cross-multiply to eliminate fractions:
  \(140 \times 28 = 4 \times \mathrm{m}\)
• Calculate the left side (use calculator):
  \(3920 = 4\mathrm{m}\)

4. SIMPLIFY to solve for m

• Divide both sides by 4:
  \(\mathrm{m} = 3920 \div 4 = 980\)

Answer: 980

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Setting up the proportion incorrectly by misinterpreting which numbers correspond to which position in the ratios.

Students might write \(\frac{140}{4} = \frac{\mathrm{m}}{28}\) instead of \(\frac{140}{\mathrm{m}} = \frac{4}{28}\), confusing the order of terms in "140 to m" versus "4 to 28." This leads to \(\mathrm{m} = \frac{140 \times 28}{4} = 980\), which by coincidence gives the same answer, but shows flawed reasoning. However, if they set it up as \(\frac{140}{4} = \frac{28}{\mathrm{m}}\), they would get \(\mathrm{m} = \frac{4 \times 28}{140} = 0.8\), leading to confusion and potential guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors during the calculation steps.

Students correctly set up \(\frac{140}{\mathrm{m}} = \frac{4}{28}\) but make mistakes when computing \(140 \times 28\) (getting something like 3820 instead of 3920) or when dividing \(3920 \div 4\). These calculation errors lead to incorrect final answers and confusion about whether their method was right.

The Bottom Line:

This problem tests whether students can accurately translate ratio language into mathematical proportions and then execute multi-step algebra without computational errors. The key insight is recognizing that "A to B equivalent to C to D" means \(\frac{\mathrm{A}}{\mathrm{B}} = \frac{\mathrm{C}}{\mathrm{D}}\).