Question:The equation \(\mathrm{p = \frac{4(h-125)}{7} + 28}\) gives the atmospheric pressure p, in kilopascals, at an altitude of h meters...
GMAT Algebra : (Alg) Questions
The equation \(\mathrm{p = \frac{4(h-125)}{7} + 28}\) gives the atmospheric pressure p, in kilopascals, at an altitude of h meters above sea level, where \(\mathrm{h \gt 0}\). If the atmospheric pressure at a certain altitude is 44 kilopascals, what is the corresponding altitude, in meters, above sea level?
1. TRANSLATE the problem information
- Given information:
- Pressure equation: \(\mathrm{p = \frac{4(h-125)}{7} + 28}\)
- Atmospheric pressure at unknown altitude: \(\mathrm{p = 44}\) kilopascals
- Need to find: altitude h in meters
- What this tells us: We need to substitute \(\mathrm{p = 44}\) into the equation and solve for h.
2. INFER the solution approach
- Since we know the pressure value and have the equation relating pressure to altitude, we can substitute and solve.
- Strategy: Replace p with 44, then use inverse operations to isolate h.
3. SIMPLIFY by substitution and algebraic manipulation
- Substitute \(\mathrm{p = 44}\):
\(\mathrm{44 = \frac{4(h-125)}{7} + 28}\)
- Subtract 28 from both sides:
\(\mathrm{16 = \frac{4(h-125)}{7}}\)
- Multiply both sides by 7:
\(\mathrm{112 = 4(h-125)}\)
- Divide both sides by 4:
\(\mathrm{28 = h-125}\)
- Add 125 to both sides:
\(\mathrm{h = 153}\)
Answer: 153 meters
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make sign errors or calculation mistakes during the multi-step algebraic process.
For example, when subtracting 28 from both sides, they might write 44 + 28 = 72 instead of 44 - 28 = 16, or when adding 125 at the end, they might get h = 28 - 125 = -97. This leads to confusion and incorrect answers that don't make physical sense.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret what to substitute, perhaps trying to solve for p instead of h, or not recognizing that "atmospheric pressure is 44 kilopascals" means \(\mathrm{p = 44}\).
This causes them to set up the wrong equation or get stuck on how to begin the problem, leading to guessing.
The Bottom Line:
This problem tests whether students can systematically work through multi-step algebraic manipulations while maintaining accuracy. The key is careful execution of each inverse operation in sequence.