A farmers' market vendor charges a constant price per pound for peaches. The table shows the relationship between the weight...
GMAT Algebra : (Alg) Questions
A farmers' market vendor charges a constant price per pound for peaches. The table shows the relationship between the weight \(\mathrm{w}\), in pounds, of peaches purchased and the total price \(\mathrm{P(w)}\), in dollars.
| \(\mathrm{w}\) | 2 | 4 | 6 |
|---|---|---|---|
| \(\mathrm{P(w)}\) | 7.50 | 15.00 | 22.50 |
Which equation could define P?
\(\mathrm{P(w) = 0.375w}\)
\(\mathrm{P(w) = 1.5w}\)
\(\mathrm{P(w) = 3.75w}\)
\(\mathrm{P(w) = 15w}\)
1. INFER the relationship type
- Given information:
- Constant price per pound for peaches
- Table showing weight w and total price P(w)
- What this tells us: Since the price per pound is constant, the total price is directly proportional to the weight. This means \(\mathrm{P(w) = k \cdot w}\), where k is the price per pound.
2. SIMPLIFY to find the constant of proportionality
- Calculate the price per pound using any data point from the table:
- Using (2 pounds, $7.50): Price per pound = \(\$7.50 \div 2 = \$3.75\)
- This means \(\mathrm{k = 3.75}\), so our equation is \(\mathrm{P(w) = 3.75w}\)
3. INFER which answer choice matches
- Looking at the choices:
- (A) \(\mathrm{P(w) = 0.375w}\) → price per pound = $0.375
- (B) \(\mathrm{P(w) = 1.5w}\) → price per pound = $1.50
- (C) \(\mathrm{P(w) = 3.75w}\) → price per pound = $3.75 ✓
- (D) \(\mathrm{P(w) = 15w}\) → price per pound = $15.00
4. Verify the answer
- Check with other data points:
- \(\mathrm{3.75 \times 4 = 15.00}\) ✓ (matches table)
- \(\mathrm{3.75 \times 6 = 22.50}\) ✓ (matches table)
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Not recognizing the direct proportional relationship structure
Students might see the table and try to find patterns between consecutive values rather than understanding that "constant price per pound" means \(\mathrm{P(w) = (price\ per\ pound) \times w}\). They might calculate differences like \(\mathrm{15.00 - 7.50 = 7.50}\) or ratios between prices rather than recognizing they need price per unit weight.
This leads to confusion and guessing among the answer choices.
Second Most Common Error:
Poor SIMPLIFY execution: Calculating the wrong ratio
Students understand they need to find a rate but calculate incorrectly, such as:
- Taking \(\mathrm{w \div P(w)}\) instead of \(\mathrm{P(w) \div w}\)
- Using \(\mathrm{2 \div 7.50 = 0.267...}\) and selecting the closest value
This may lead them to select Choice A (0.375) since it's the only small decimal option.
The Bottom Line:
This problem tests whether students can translate the phrase "constant price per pound" into the mathematical concept of direct proportionality, then execute the unit rate calculation correctly.
\(\mathrm{P(w) = 0.375w}\)
\(\mathrm{P(w) = 1.5w}\)
\(\mathrm{P(w) = 3.75w}\)
\(\mathrm{P(w) = 15w}\)