Question:A temperature sensor calibration relates the displayed temperature D (in degrees) to the actual temperature A (in degrees) by the...
GMAT Advanced Math : (Adv_Math) Questions
A temperature sensor calibration relates the displayed temperature D (in degrees) to the actual temperature A (in degrees) by the equation \(\mathrm{D - 15 = \frac{A - 20}{2}}\). Laboratory technicians need to express this relationship in the form \(\mathrm{A = mD + n}\), where \(\mathrm{m}\) and \(\mathrm{n}\) are constants. What is the value of \(\mathrm{m}\)?
1. TRANSLATE the problem information
- Given equation: \(\mathrm{D - 15 = (A - 20)/2}\)
- Target form: \(\mathrm{A = mD + n}\)
- Find: the value of m
2. INFER the approach
- We need to solve for A in terms of D
- This means isolating A on one side and getting all D terms on the other side
- Then we can identify the coefficient of D (which is m)
3. SIMPLIFY by eliminating the fraction
- Multiply both sides by 2: \(\mathrm{2(D - 15) = A - 20}\)
- Distribute on the left: \(\mathrm{2D - 30 = A - 20}\)
4. SIMPLIFY to isolate A
- Add 20 to both sides: \(\mathrm{A = 2D - 30 + 20}\)
- Combine like terms: \(\mathrm{A = 2D - 10}\)
5. INFER the final answer
- Comparing \(\mathrm{A = 2D - 10}\) with \(\mathrm{A = mD + n}\)
- We see that \(\mathrm{m = 2}\) and \(\mathrm{n = -10}\)
Answer: 2
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make sign errors when distributing or combining terms
Many students correctly multiply by 2 to get \(\mathrm{2(D - 15) = A - 20}\), but then incorrectly distribute to get \(\mathrm{2D + 30 = A - 20}\) instead of \(\mathrm{2D - 30 = A - 20}\). This leads to \(\mathrm{A = 2D + 30 - 20 = 2D + 10}\), giving them \(\mathrm{m = 2}\) but with the wrong constant term. While they'd still get \(\mathrm{m = 2}\) correct, the error shows weak algebraic fundamentals.
Second Most Common Error:
Poor TRANSLATE understanding: Students misinterpret what they're being asked to find
Some students solve correctly but then provide the value of n (-10) instead of m (2), or they provide both values instead of just m as requested. This shows they didn't carefully read what the question was asking for.
The Bottom Line:
This problem tests whether students can systematically manipulate a linear equation through multiple algebraic steps while maintaining accuracy. The key is treating each step as building toward the target form rather than just "doing algebra."