A photometric sensor reports a reading u in sensor units, and the corresponding luminous intensity b, in lumens, is given...
GMAT Algebra : (Alg) Questions
A photometric sensor reports a reading \(\mathrm{u}\) in sensor units, and the corresponding luminous intensity \(\mathrm{b}\), in lumens, is given by \(\mathrm{b = \frac{9(u - 120)}{5} + 75.5}\), where \(\mathrm{u \gt 0}\). For a certain measurement, the luminous intensity is 525.5 lumens. What is the corresponding sensor reading \(\mathrm{u}\)?
1. TRANSLATE the problem information
- Given information:
- Formula: \(\mathrm{b = \frac{9(u - 120)}{5} + 75.5}\)
- Known luminous intensity: \(\mathrm{b = 525.5}\) lumens
- Need to find: sensor reading u (as integer)
- What this tells us: We need to substitute the known b-value and solve for u
2. SIMPLIFY by setting up the equation
- Substitute \(\mathrm{b = 525.5}\) into the formula:
\(\mathrm{525.5 = \frac{9(u - 120)}{5} + 75.5}\)
3. SIMPLIFY to isolate the fraction term
- Subtract 75.5 from both sides:
\(\mathrm{525.5 - 75.5 = \frac{9(u - 120)}{5}}\)
\(\mathrm{450 = \frac{9(u - 120)}{5}}\)
4. SIMPLIFY to eliminate the fraction
- Multiply both sides by 5:
\(\mathrm{450 \times 5 = 9(u - 120)}\)
\(\mathrm{2250 = 9(u - 120)}\)
5. SIMPLIFY to solve for (u − 120)
- Divide both sides by 9:
\(\mathrm{2250 \div 9 = u - 120}\) (use calculator if needed)
\(\mathrm{250 = u - 120}\)
6. SIMPLIFY to find the final answer
- Add 120 to both sides:
\(\mathrm{u = 250 + 120 = 370}\)
Answer: 370
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make sign errors or arithmetic mistakes during the multi-step algebraic process, particularly when dealing with the fraction \(\mathrm{\frac{9(u - 120)}{5}}\).
Common mistakes include:
- Incorrectly distributing the multiplication by 5
- Making errors when dividing 2250 by 9
- Forgetting to add back the 120 at the end
This leads to confusion and incorrect final answers that don't match the expected integer result.
Second Most Common Error:
Poor TRANSLATE reasoning: Students may attempt to solve for b instead of u, or incorrectly set up the substitution by confusing which variable is given and which needs to be found.
This causes them to work backwards through the problem or set up an entirely different equation, leading to answers that are completely off-track.
The Bottom Line:
This problem requires careful algebraic manipulation through multiple steps while maintaining accuracy. The key challenge is keeping track of the correct sequence of inverse operations to "undo" the original formula.