A factory produces a blend of coffee beans consisting of arabica and robusta beans. In every batch of the blend,...

1. TRANSLATE the problem information

• Given information:
  • Arabica to robusta ratio = \(7:3\)
  • Total weight increases by 40 pounds
  • Need to find: increase in arabica weight

• What this tells us: For every 7 units of arabica, there are 3 units of robusta in the blend.

2. INFER the proportion structure

• Key insight: When the total batch size changes, each component must change proportionally to maintain the same ratio.
• Strategy: Find what fraction of the total blend is arabica, then apply that fraction to the increase.

3. SIMPLIFY the fraction calculation

• Total parts in ratio: \(7 + 3 = 10\) parts
• Arabica represents: 7 parts out of 10 total parts = \(\frac{7}{10}\)
• Arabica increase = \(\frac{7}{10} \times 40 = 28\) pounds

Answer: C) 28 pounds

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the ratio as meaning arabica gets 7 pounds and robusta gets 3 pounds for every 10-pound increase.

They calculate: \(\frac{7}{10} \times 40 = 28\), but think this means 7 pounds per 10, so they compute \(7 \times 4 = 28\). While this gives the right answer, the reasoning is flawed. More commonly, they think arabica should get 7 pounds out of every 40, leading to 7 pounds total increase.

This may lead them to select Choice A (12 pounds) after confusing the ratio mechanics.

Second Most Common Error:

Inadequate INFER reasoning: Students don't recognize that the ratio must be maintained proportionally when total weight changes.

They might think: "If we add 40 pounds total, and the ratio is 7:3, then arabica gets more" but fail to establish the proportional relationship correctly. Some students divide 40 by the ratio numbers directly (\(40 \div 7 \approx 5.7\)) or try other non-proportional approaches.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can translate ratio language into fractional relationships and apply those fractions to new quantities while maintaining proportional thinking.