The table above gives the typical amounts of energy per gram, expressed in both food calories and kilojoules, of the...

1. TRANSLATE the problem information

• Given information:
  • Table showing food calories (x) and corresponding kilojoules (k) for different macronutrients
  • Need to find the relationship between x and k

• What this tells us: We have pairs of (x, k) values that should follow the same relationship

2. INFER the mathematical approach

• Since we're looking for "the relationship" between two quantities with multiple data points, this suggests a proportional relationship
• For proportional relationships, we expect \(\mathrm{k = mx}\) where m is the constant of proportionality
• Strategy: Calculate the ratio \(\mathrm{k/x}\) for different data points to find the constant

3. SIMPLIFY to find the constant

• Calculate \(\mathrm{k/x}\) for each macronutrient:
  • Protein: \(\mathrm{16.7 \div 4.0 = 4.175}\)
  • Fat: \(\mathrm{37.7 \div 9.0 \approx 4.189}\) (use calculator)
  • Carbohydrate: \(\mathrm{16.7 \div 4.0 = 4.175}\)

• All ratios are approximately 4.2

4. TRANSLATE back to equation form

• Since \(\mathrm{k/x = 4.2}\), we get \(\mathrm{k = 4.2x}\)

Answer: B. \(\mathrm{k = 4.2x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize this as a proportional relationship problem and instead try to find a pattern by looking at differences rather than ratios.

They might calculate \(\mathrm{x_2 - x_1 = 9.0 - 4.0 = 5.0}\) and \(\mathrm{k_2 - k_1 = 37.7 - 16.7 = 21.0}\), then incorrectly think the relationship is \(\mathrm{k = x + 16}\) or some other linear (but not proportional) form. This leads to confusion when checking against the answer choices and often results in guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly find that \(\mathrm{k/x = 4.2}\) but then incorrectly write this as \(\mathrm{x = 4.2k}\) instead of \(\mathrm{k = 4.2x}\).

This conceptual confusion about which variable depends on which (mixing up independent and dependent variables) leads them to select Choice C (\(\mathrm{x = 4.2k}\)).

The Bottom Line:

This problem tests whether students can recognize proportional relationships from data tables and correctly express them in equation form. The key insight is that when ratios are constant, you have a proportional relationship of the form \(\mathrm{y = mx}\).