Two vehicles start from the same point and travel in the same direction along a straight road. Vehicle A travels...
GMAT Algebra : (Alg) Questions
Two vehicles start from the same point and travel in the same direction along a straight road. Vehicle A travels at a rate of \(\mathrm{r_1}\) miles per hour and Vehicle B travels at a rate of \(\mathrm{r_2}\) miles per hour. After 2 hours, Vehicle A has traveled 20 miles more than Vehicle B. After 3 hours, the sum of the distances traveled by both vehicles is 210 miles. What is the value of \(\mathrm{r_1r_2}\)?
Enter your answer as an integer.
1. TRANSLATE the problem information
- Given information:
- Two vehicles start from same point, travel same direction
- Vehicle A travels at \(\mathrm{r_1}\) mph, Vehicle B at \(\mathrm{r_2}\) mph
- After 2 hours: Vehicle A traveled 20 miles more than Vehicle B
- After 3 hours: Sum of both distances = 210 miles
- Find: \(\mathrm{r_1r_2}\)
2. TRANSLATE the conditions into equations
- After 2 hours condition:
- Vehicle A distance = \(\mathrm{2r_1}\), Vehicle B distance = \(\mathrm{2r_2}\)
- "Vehicle A traveled 20 more miles": \(\mathrm{2r_1 = 2r_2 + 20}\)
- Simplifying: \(\mathrm{r_1 = r_2 + 10}\)
- After 3 hours condition:
- Vehicle A distance = \(\mathrm{3r_1}\), Vehicle B distance = \(\mathrm{3r_2}\)
- "Sum of distances = 210": \(\mathrm{3r_1 + 3r_2 = 210}\)
- Simplifying: \(\mathrm{r_1 + r_2 = 70}\)
3. INFER the solution strategy
- We have two equations with two unknowns:
- \(\mathrm{r_1 = r_2 + 10}\)
- \(\mathrm{r_1 + r_2 = 70}\)
- Substitution method will work efficiently since the first equation is already solved for \(\mathrm{r_1}\)
4. SIMPLIFY by solving the system
- Substitute \(\mathrm{r_1 = r_2 + 10}\) into \(\mathrm{r_1 + r_2 = 70}\):
\(\mathrm{(r_2 + 10) + r_2 = 70}\)
\(\mathrm{2r_2 + 10 = 70}\)
\(\mathrm{2r_2 = 60}\)
\(\mathrm{r_2 = 30}\) mph
- Find \(\mathrm{r_1}\): \(\mathrm{r_1 = 30 + 10 = 40}\) mph
- Calculate final answer: \(\mathrm{r_1r_2 = 40 \times 30 = 1200}\)
Answer: 1200
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students often struggle to correctly convert the "20 miles more" condition into an equation. They might write \(\mathrm{r_1 = r_2 + 20}\) instead of \(\mathrm{2r_1 = 2r_2 + 20}\), forgetting that this condition applies to the distances traveled (rate × time), not the rates themselves.
This fundamental translation error leads to an incorrect system of equations, causing them to get \(\mathrm{r_1 = 50}\) and \(\mathrm{r_2 = 20}\), resulting in \(\mathrm{r_1r_2 = 1000}\) instead of the correct 1200.
Second Most Common Error:
Poor SIMPLIFY execution: Even with correct equations, students may make algebraic errors when solving the system. Common mistakes include sign errors when distributing or combining like terms, leading to incorrect values for the rates.
These calculation errors typically result in confusion and guessing, as the incorrect rates don't yield any of the expected reasonable answer patterns.
The Bottom Line:
This problem requires careful attention to what quantities the given conditions actually describe - distances vs. rates - and precise algebraic manipulation of the resulting system.