A recipe calls for flour and sugar in the ratio 4:1. If f represents the amount of flour used, which...

1. TRANSLATE the problem information

• Given information:
  • Flour and sugar are in the ratio \(\mathrm{4:1}\)
  • \(\mathrm{f}\) = amount of flour used
  • Need to find: total amount of flour and sugar combined

• What this tells us: For every 4 parts flour, there is 1 part sugar

2. INFER the relationship between flour and sugar amounts

• Since the ratio is \(\mathrm{4:1}\), flour is 4 times the amount of sugar
• If flour = \(\mathrm{f}\), then sugar = \(\mathrm{\frac{f}{4}}\)
• Strategy: Find sugar amount, then add flour + sugar

3. SIMPLIFY the total calculation

• Total = flour + sugar = \(\mathrm{f + \frac{f}{4}}\)
• Convert to common denominators:
  \(\mathrm{f + \frac{f}{4}}\)
  \(\mathrm{= \frac{4f}{4} + \frac{f}{4}}\)
  \(\mathrm{= \frac{5f}{4}}\)

Answer: B) \(\mathrm{\frac{5f}{4}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what the ratio \(\mathrm{4:1}\) means in relation to the variable \(\mathrm{f}\).

Many students think "ratio \(\mathrm{4:1}\)" means sugar = \(\mathrm{4f}\) instead of sugar = \(\mathrm{\frac{f}{4}}\). They incorrectly reason that since the ratio shows "4 to 1," the sugar amount should involve multiplying by 4. This leads to a total of \(\mathrm{f + 4f = 5f}\).

This may lead them to select Choice D (\(\mathrm{5f}\)).

Second Most Common Error:

Poor INFER reasoning: Students correctly find that sugar = \(\mathrm{\frac{f}{4}}\) but then think the total should be \(\mathrm{\frac{f}{4}}\) (just the sugar amount) rather than flour + sugar.

They get confused about what "total amount" means and only consider one component.

This may lead them to select Choice A (\(\mathrm{\frac{f}{4}}\)).

The Bottom Line:

Ratio problems require careful translation of the proportional relationship and clear understanding of what quantity each variable represents. The key insight is recognizing that if flour is the larger quantity (the "4" in \(\mathrm{4:1}\)), then when flour = \(\mathrm{f}\), the smaller quantity (sugar) must be \(\mathrm{f}\) divided by the ratio factor.