Store A sells raspberries for $5.50 per pint and blackberries for $3.00 per pint. Store B sells raspberries for $6.50...

1. TRANSLATE the problem information

• Given information:
  • Store A prices: $5.50/pint raspberries, $3.00/pint blackberries
  • Store B prices: $6.50/pint raspberries, $8.00/pint blackberries
  • Same purchase costs $37.00 at Store A or $66.00 at Store B
  • Need to find: pints of blackberries
• Let \(\mathrm{r}\) = pints of raspberries, \(\mathrm{b}\) = pints of blackberries

2. TRANSLATE into mathematical equations

• Store A equation: \(5.50\mathrm{r} + 3.00\mathrm{b} = 37.00\)
• Store B equation: \(6.50\mathrm{r} + 8.00\mathrm{b} = 66.00\)

3. INFER the solution approach

• This is a system of two equations with two unknowns
• Elimination method will work well since we can eliminate one variable

4. SIMPLIFY using elimination method

• Multiply first equation by 6.5: \(35.75\mathrm{r} + 19.5\mathrm{b} = 240.5\)
• Multiply second equation by 5.5: \(35.75\mathrm{r} + 44\mathrm{b} = 363\)
• Subtract first from second: \(24.5\mathrm{b} = 122.5\)
• Solve: \(\mathrm{b} = 122.5 \div 24.5 = 5\)

5. Verify the answer

• If \(\mathrm{b} = 5\), then from Store A: \(5.50\mathrm{r} + 15 = 37\), so \(\mathrm{r} = 4\)
• Check Store B: \(6.50(4) + 8.00(5) = 26 + 40 = 66\) ✓

Answer: B. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to set up the correct equations from the word problem. They might confuse which store's prices go with which total cost, or mix up the variables. For example, they might write something like \(5.50\mathrm{r} + 3.00\mathrm{b} = 66.00\) (using Store A prices with Store B total).

This leads to incorrect equations and completely wrong solutions, causing them to select an incorrect answer or become confused and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the system but make arithmetic errors during the elimination process. Common mistakes include incorrectly multiplying equations by elimination factors, sign errors when subtracting equations, or division errors when solving for the final variable.

This may lead them to calculate values like \(\mathrm{b} = 8\) or \(\mathrm{b} = 4\), causing them to select Choice A (4) or Choice C (8).

The Bottom Line:

This problem requires careful translation of the word problem into mathematical language, then systematic algebraic manipulation. Students who rush through either the setup or the solving phase are likely to make errors that lead to wrong answer choices.