\(\mathrm{D = T - \frac{9}{25}(100 - H)}\)The formula above can be used to approximate the dew point D, in degrees...

1. TRANSLATE the problem information

• Given formula: \(\mathrm{D = T - \frac{9}{25}(100 - H)}\)
• Goal: Express H in terms of D and T (solve for H)

2. INFER the approach

• We need to isolate H by systematically undoing the operations in the original equation
• Work backwards through the operations: subtraction, then multiplication by fraction

3. SIMPLIFY through algebraic manipulation

• Start: \(\mathrm{D = T - \frac{9}{25}(100 - H)}\)
• Subtract T from both sides: \(\mathrm{D - T = -\frac{9}{25}(100 - H)}\)
• Multiply both sides by -25/9: \(\mathrm{-\frac{25}{9}(D - T) = 100 - H}\)
• Subtract 100 from both sides: \(\mathrm{-\frac{25}{9}(D - T) - 100 = -H}\)
• Multiply both sides by -1: \(\mathrm{\frac{25}{9}(D - T) + 100 = H}\)

4. SIMPLIFY to final form

• \(\mathrm{H = \frac{25}{9}(D - T) + 100}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when multiplying by negative fractions

Students often lose track of negative signs, especially when multiplying both sides by -25/9. They might forget that multiplying by a negative flips signs, leading them to get 25/9(D - T) - 100 instead of 25/9(D - T) + 100.

This may lead them to select Choice B (\(\mathrm{H = \frac{25}{9}(D - T) - 100}\))

Second Most Common Error:

Poor SIMPLIFY reasoning: Incorrectly handling the (D - T) vs (D + T) terms

Some students make algebraic errors early in the process, particularly when moving T to the left side. Instead of getting D - T, they might incorrectly manipulate to get D + T, carrying this error through the remaining steps.

This may lead them to select Choice C (\(\mathrm{H = \frac{25}{9}(D + T) + 100}\)) or Choice D (\(\mathrm{H = \frac{25}{9}(D + T) - 100}\))

The Bottom Line:

This problem tests careful algebraic manipulation with attention to signs. The multiple negative operations create opportunities for sign errors that directly correspond to the incorrect answer choices.