The equation above can be used to calculate the distance d, in miles, traveled by a car moving at a...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
The equation above can be used to calculate the distance \(\mathrm{d}\), in miles, traveled by a car moving at a speed of \(\mathrm{55}\) miles per hour over a period of \(\mathrm{t}\) hours. For any positive constant \(\mathrm{k}\), the distance the car would have traveled after \(\mathrm{9k}\) hours is how many times the distance the car would have traveled after \(\mathrm{3k}\) hours?
1. TRANSLATE the problem information
- Given information:
- Distance formula: \(\mathrm{d = 55t}\) (speed of 55 mph for t hours)
- Need to compare distance after 9k hours vs distance after 3k hours
- Want to find: "how many times" larger one distance is than the other
2. INFER the approach
- This is asking for a ratio between two distances
- Since both distances use the same formula with different time values, I can calculate each distance separately then find their ratio
- The "k" will likely cancel out since it appears in both expressions
3. SIMPLIFY by calculating each distance
- Distance after 9k hours: \(\mathrm{d_1 = 55(9k) = 495k}\) miles
- Distance after 3k hours: \(\mathrm{d_2 = 55(3k) = 165k}\) miles
4. SIMPLIFY to find the ratio
- Ratio = \(\mathrm{\frac{d_1}{d_2} = \frac{495k}{165k}}\)
- The k values cancel: \(\mathrm{\frac{495k}{165k} = \frac{495}{165}}\)
- Simplify: \(\mathrm{495 \div 165 = 3}\)
Answer: A. 3
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students misunderstand what "how many times" means and think about the relationship additively rather than multiplicatively.
They might reason: "9k hours is 6k hours more than 3k hours, so the distance must be 6 times larger." This leads them to select Choice B (6).
Second Most Common Error:
Poor TRANSLATE reasoning: Students incorrectly include the constant k in their final answer, thinking that since k appears in the problem, it must appear in the solution.
They correctly find the ratio as 3 but then think they need to multiply by k, leading them to select Choice C (3k).
The Bottom Line:
This problem tests whether students understand proportional relationships and can distinguish between additive and multiplicative comparisons. The key insight is recognizing that in a proportional relationship like \(\mathrm{d = 55t}\), when time triples (from 3k to 9k), distance also triples.