Question:Temperature conversion formulas are essential in scientific applications. The relationship between temperature in degrees Celsius, C, and tem...
GMAT Advanced Math : (Adv_Math) Questions
Temperature conversion formulas are essential in scientific applications. The relationship between temperature in degrees Celsius, C, and temperature in degrees Fahrenheit, F, is given by \(\mathrm{C = (F - 32) \times \frac{5}{9}}\). If the equation is rewritten to express F in terms of C in the form \(\mathrm{F = xC + y}\), where \(\mathrm{x}\) and \(\mathrm{y}\) are constants, what is the value of \(\mathrm{x}\)? Express your answer as a fraction in simplest form.
1. TRANSLATE the problem requirements
- Given: \(\mathrm{C = (F - 32) \times \frac{5}{9}}\)
- Need: Rearrange to get \(\mathrm{F = xC + y}\) and identify the value of x
- This means we need to solve for F in terms of C
2. INFER the algebraic strategy
- We need to "undo" the operations that were applied to F
- Since F was first reduced by 32, then multiplied by 5/9, we'll reverse these operations
- Strategy: Clear the fraction first, then isolate F
3. SIMPLIFY through algebraic manipulation
- Start with: \(\mathrm{C = (F - 32) \times \frac{5}{9}}\)
- Multiply both sides by 9: \(\mathrm{9C = 5(F - 32)}\)
- Distribute the 5: \(\mathrm{9C = 5F - 160}\)
- Add 160 to both sides: \(\mathrm{9C + 160 = 5F}\)
- Divide by 5: \(\mathrm{F = \frac{9C + 160}{5}}\)
4. SIMPLIFY to standard linear form
- Separate the terms: \(\mathrm{F = \frac{9C}{5} + \frac{160}{5}}\)
- Simplify: \(\mathrm{F = \frac{9}{5}C + 32}\)
- This is now in the form \(\mathrm{F = xC + y}\) where \(\mathrm{x = \frac{9}{5}}\) and \(\mathrm{y = 32}\)
Answer: \(\mathrm{\frac{9}{5}}\) (or equivalent forms: \(\mathrm{1.8}\), \(\mathrm{1\frac{4}{5}}\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE reasoning: Students may misunderstand what "express F in terms of C" means and try to solve for C instead of F, or not recognize they need to rearrange the equation algebraically.
This leads to confusion about the goal and may cause them to attempt incorrect manipulations or give up and guess.
Second Most Common Error:
Poor SIMPLIFY execution: Students make algebraic errors during the multi-step manipulation, such as incorrectly distributing the 5, forgetting to multiply both sides by 9, or making sign errors when moving terms.
These calculation errors lead to incorrect coefficients, resulting in wrong values for x.
The Bottom Line:
This problem tests whether students can systematically reverse a series of algebraic operations. The key insight is recognizing that algebraic rearrangement follows a logical sequence - you must strategically "undo" operations in reverse order to isolate the desired variable.