The formula F = 3/2C + 32 is used to convert a temperature from degrees Celsius, C, to degrees Fahrenheit,...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{F = \frac{3}{2}C + 32}\)
• Find: Express C in terms of F (solve for C)

2. INFER the approach

• To isolate C, I need to "undo" the operations performed on C
• The equation shows: C is multiplied by 3/2, then 32 is added
• To undo these operations, I must work in reverse order: subtract 32 first, then divide by 3/2 (or multiply by its reciprocal 2/3)

3. SIMPLIFY by subtracting 32 from both sides

\(\mathrm{F = \frac{3}{2}C + 32}\)

\(\mathrm{F - 32 = \frac{3}{2}C}\)

4. SIMPLIFY by multiplying both sides by 2/3

• Since C is multiplied by \(\mathrm{\frac{3}{2}}\), I multiply both sides by the reciprocal \(\mathrm{\frac{2}{3}}\)
• \(\mathrm{\frac{2}{3}(F - 32) = \frac{2}{3} \times \frac{3}{2}C}\)
• \(\mathrm{\frac{2}{3}(F - 32) = C}\)

Answer: A. \(\mathrm{C = \frac{2}{3}(F - 32)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly distribute the fraction \(\mathrm{\frac{2}{3}}\) over the expression \(\mathrm{(F - 32)}\)

Instead of keeping \(\mathrm{(F - 32)}\) as a single quantity, they distribute: \(\mathrm{\frac{2}{3}(F - 32) = \frac{2}{3}F - \frac{2}{3}(32) = \frac{2}{3}F - \frac{64}{3}}\)

But then they might incorrectly write this as \(\mathrm{\frac{2}{3}F - 32}\), confusing the constants.

This may lead them to select Choice B (\(\mathrm{\frac{2}{3}F - 32}\)).

Second Most Common Error:

Missing conceptual knowledge: Students don't recognize that to undo multiplication by \(\mathrm{\frac{3}{2}}\), they need to multiply by its reciprocal \(\mathrm{\frac{2}{3}}\)

Instead, they might think they should multiply by \(\mathrm{\frac{3}{2}}\) again, leading to: \(\mathrm{\frac{3}{2}(F - 32) = C}\)

This may lead them to select Choice D (\(\mathrm{\frac{3}{2}(F - 32)}\)).

The Bottom Line:

This problem requires careful attention to the order of operations when solving equations and understanding that reciprocals are used to "undo" multiplication. The key insight is recognizing that parentheses must be preserved when multiplying by fractions.