Anita created a batch of green paint by mixing 2 ounces of blue paint with 3 ounces of yellow paint....

1. TRANSLATE the problem information

• Given information:
  • First batch: 2 ounces blue paint, 3 ounces yellow paint
  • Second batch: 5 ounces blue paint, unknown yellow paint
  • Must use same ratio of blue to yellow

• What this tells us: We need to maintain the ratio \(2:3\) (blue:yellow)

2. INFER the approach

• Since we must maintain the same ratio, we can set up a proportion
• The ratio \(\frac{\mathrm{blue}}{\mathrm{yellow}}\) must be the same in both batches
• This gives us: \(\frac{\mathrm{first\ batch\ blue}}{\mathrm{first\ batch\ yellow}} = \frac{\mathrm{second\ batch\ blue}}{\mathrm{second\ batch\ yellow}}\)

3. TRANSLATE this relationship into an equation

• Set up the proportion: \(\frac{2}{3} = \frac{5}{x}\)
• Here x represents the unknown amount of yellow paint for the second batch

4. SIMPLIFY by solving the proportion

• Cross multiply: \(2 \times x = 3 \times 5\)
• This gives us: \(2x = 15\)
• Divide both sides by 2: \(x = \frac{15}{2} = 7.5\) ounces

5. INFER which answer choice matches

• We need 7.5 ounces of yellow paint
• Check each choice:
  • A: 5 ounces (not equal to 7.5)
  • B: \(3 + 3 = 6\) ounces (not equal to 7.5)
  • C: \(1.5 \times 3 = 4.5\) ounces (not equal to 7.5)
  • D: \(1.5 \times 5 = 7.5\) ounces ✓

Answer: D. 1.5 times the amount of blue paint used in the second batch

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread the answer choices and think they need to find a specific number of ounces rather than understanding the relationship being described.

Many students correctly calculate that 7.5 ounces of yellow paint is needed, but then select Choice A (Exactly 5 ounces) because they think the answer should match the amount of blue paint used, not realizing they need to identify which choice describes 7.5 ounces.

Second Most Common Error:

Poor INFER reasoning: Students set up the proportion incorrectly by not maintaining consistent order in their ratios.

They might write \(\frac{2}{3} = \frac{x}{5}\) instead of \(\frac{2}{3} = \frac{5}{x}\), thinking "blue over yellow equals unknown over 5." This gives them \(x = \frac{10}{3} \approx 3.33\) ounces, leading them to select Choice B (3 ounces more than the first batch) since 3.33 is close to \(3 + 3 = 6\).

The Bottom Line:

This problem requires not just solving for a numerical value, but recognizing how that value relates to other quantities in the problem. The trickiest part is understanding that the answer choices describe relationships, not just numbers.