Davio bought some potatoes and celery. The potatoes cost $0.69 per pound, and the celery cost $0.99 per pound. If...

1. TRANSLATE the problem information

• Given information:
  • Potatoes cost \(\$0.69\) per pound
  • Celery costs \(\$0.99\) per pound
  • Total spent: \(\$5.34\)
  • Bought twice as many pounds of celery as potatoes
• What we need to find: pounds of celery

2. TRANSLATE word relationships into mathematical equations

• Let \(\mathrm{p}\) = pounds of potatoes, \(\mathrm{c}\) = pounds of celery
• Cost equation: \(\mathrm{0.69p + 0.99c = 5.34}\)
• Quantity relationship: \(\mathrm{c = 2p}\)

3. INFER the solution strategy

• We have two equations with two unknowns - this is a system of equations
• Since \(\mathrm{c = 2p}\) gives us c directly in terms of p, substitution is the most efficient approach
• Substitute \(\mathrm{c = 2p}\) into the cost equation to solve for p first

4. SIMPLIFY by substitution and solving

• Substitute \(\mathrm{c = 2p}\) into \(\mathrm{0.69p + 0.99c = 5.34}\):
  \(\mathrm{0.69p + 0.99(2p) = 5.34}\)
• Distribute:
  \(\mathrm{0.69p + 1.98p = 5.34}\)
• Combine like terms:
  \(\mathrm{2.67p = 5.34}\)
• Divide both sides by 2.67:
  \(\mathrm{p = 5.34 ÷ 2.67 = 2}\) (use calculator)

5. Find the final answer

• Since \(\mathrm{c = 2p}\) and \(\mathrm{p = 2}\):
  \(\mathrm{c = 2(2) = 4}\)

Answer: D. 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to correctly identify which quantity is which in the "twice as many" relationship. They might set up \(\mathrm{p = 2c}\) instead of \(\mathrm{c = 2p}\), thinking "potatoes equal twice the celery" when the problem states "twice as many pounds of celery as potatoes."

This reversal leads to the wrong equation system:

• \(\mathrm{0.69p + 0.99c = 5.34}\)
• \(\mathrm{p = 2c}\) (incorrect)

Solving this gives \(\mathrm{c = 2}\) instead of \(\mathrm{c = 4}\).
This may lead them to select Choice A (2).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the equations but make arithmetic errors during the decimal operations, particularly when combining \(\mathrm{0.69p + 1.98p = 2.67p}\) or when dividing \(\mathrm{5.34 ÷ 2.67}\).

Common calculation mistakes include getting \(\mathrm{2.77p}\) instead of \(\mathrm{2.67p}\), which leads to \(\mathrm{p ≈ 1.93}\), making \(\mathrm{c ≈ 3.86}\). Students might round this to the closest answer choice.
This may lead them to select Choice D (4) by coincidence or Choice C (2.67) if they confuse the coefficient with the answer.

The Bottom Line:

This problem tests both translation accuracy and systematic equation solving. The key challenge is correctly interpreting the "twice as many" language and maintaining precision through decimal arithmetic operations.