A printer produces pages at a constant rate. The table shows the number of pages p the printer produces in...
GMAT Algebra : (Alg) Questions
A printer produces pages at a constant rate. The table shows the number of pages \(\mathrm{p}\) the printer produces in \(\mathrm{t}\) minutes for three runs.
| Time \(\mathrm{t}\) (minutes) | Pages \(\mathrm{p}\) |
|---|---|
| 0.6 | 18 |
| 1.0 | 30 |
| 1.7 | 51 |
Which equation expresses \(\mathrm{t}\) as a function of \(\mathrm{p}\)?
1. TRANSLATE the problem information
- Given information:
- Printer produces pages at constant rate
- Three data points: (0.6 min, 18 pages), (1.0 min, 30 pages), (1.7 min, 51 pages)
- Need equation for t as function of p
2. INFER the mathematical relationship
- Constant rate means the pages-per-minute ratio stays the same
- I can calculate this rate using any data pair: \(\mathrm{rate = pages ÷ time}\)
- Once I have the rate, I can write \(\mathrm{p = (rate) × t}\), then solve for t
3. Calculate the constant rate
- Using first data pair: \(\mathrm{18 ÷ 0.6 = 30}\) pages per minute
- Verify with second pair: \(\mathrm{30 ÷ 1.0 = 30}\) pages per minute ✓
- Verify with third pair: \(\mathrm{51 ÷ 1.7 = 30}\) pages per minute ✓
4. INFER the direct relationship
- Since rate = 30 pages per minute: \(\mathrm{p = 30t}\)
- But the question asks for t as a function of p, so I need t = something involving p
5. SIMPLIFY to solve for t
- Starting with: \(\mathrm{p = 30t}\)
- Divide both sides by 30: \(\mathrm{p/30 = t}\)
- Rewrite: \(\mathrm{t = p/30 = (1/30)p}\)
Answer: (B) \(\mathrm{t = (1/30)p}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misread which variable should be the function of which. They see the table with t values and p values, and incorrectly assume they need p as a function of t (since that's how the data is arranged).
This leads them to write \(\mathrm{p = 30t}\) and stop there, selecting Choice (D) \(\mathrm{t = 30p}\) because they flip the relationship incorrectly.
Second Most Common Error:
Poor rate calculation (SIMPLIFY): Students make arithmetic errors when calculating \(\mathrm{18 ÷ 0.6}\), getting rates like \(\mathrm{18 × 0.6 = 10.8}\) instead of \(\mathrm{18 ÷ 0.6 = 30}\).
With an incorrect rate, they might get relationships like \(\mathrm{t = (1/10.8)p}\), leading them to guess among the fractional choices or select Choice (C) \(\mathrm{t = (1/18)p}\) if they confuse 18 pages with the rate.
The Bottom Line:
This problem tests whether students can correctly interpret 'constant rate' data and manipulate the resulting equation to express the requested variable as a function of the other. The key insight is recognizing that the question specifically asks for t as a function of p, not the other way around.