A boat travels at a speed of 4 miles per hour in still water. The river has a current of...
GMAT Algebra : (Alg) Questions
A boat travels at a speed of \(\mathrm{4}\) miles per hour in still water. The river has a current of \(\mathrm{1}\) mile per hour. The boat travels upstream for \(\mathrm{u}\) hours and downstream for \(\mathrm{d}\) hours for a combined total distance of \(\mathrm{15}\) miles. Which equation represents this situation?
\(3\mathrm{u} + 5\mathrm{d} = 15\)
\(\frac{1}{3}\mathrm{u} + \frac{1}{5}\mathrm{d} = 15\)
\(\frac{1}{3}\mathrm{u} + \frac{1}{5}\mathrm{d} = 112\)
\(3\mathrm{u} + 5\mathrm{d} = 112\)
1. TRANSLATE the problem information
- Given information:
- Boat speed in still water: \(\mathrm{4~mph}\)
- River current: \(\mathrm{1~mph}\)
- Time upstream: \(\mathrm{u~hours}\)
- Time downstream: \(\mathrm{d~hours}\)
- Total distance: \(\mathrm{15~miles}\)
2. INFER how current affects boat speed
- When traveling upstream (against current): effective speed = boat speed - current
- Upstream effective speed = \(\mathrm{4 - 1 = 3~mph}\) - When traveling downstream (with current): effective speed = boat speed + current
- Downstream effective speed = \(\mathrm{4 + 1 = 5~mph}\)
3. TRANSLATE each distance using the distance formula
- \(\mathrm{Distance = speed \times time}\)
- Distance upstream = \(\mathrm{3u~miles}\)
- Distance downstream = \(\mathrm{5d~miles}\)
4. TRANSLATE the total distance condition
- Total distance = upstream distance + downstream distance
- \(\mathrm{15 = 3u + 5d}\)
- Rearranging: \(\mathrm{3u + 5d = 15}\)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students confuse the direction of current effects, thinking upstream adds current or downstream subtracts current.
They might calculate effective speeds as \(\mathrm{5~mph}\) upstream and \(\mathrm{3~mph}\) downstream, leading to the equation \(\mathrm{5u + 3d = 15}\). While this specific equation isn't among the choices, the confusion about current direction can cause students to hesitate and guess randomly.
Second Most Common Error:
Poor TRANSLATE reasoning: Students incorrectly apply the distance formula by using reciprocals of speed (thinking time = distance/speed applies directly to the setup).
Instead of using distance = speed × time, they set up (1/speed) × time = distance, creating \(\mathrm{\frac{1}{3}u + \frac{1}{5}d = 15}\). This leads them to select Choice B (\(\mathrm{\frac{1}{3}u + \frac{1}{5}d = 15}\)).
The Bottom Line:
Success requires clearly understanding how relative motion works with currents and correctly applying the distance formula. The key insight is that current always helps downstream motion and hinders upstream motion.
\(3\mathrm{u} + 5\mathrm{d} = 15\)
\(\frac{1}{3}\mathrm{u} + \frac{1}{5}\mathrm{d} = 15\)
\(\frac{1}{3}\mathrm{u} + \frac{1}{5}\mathrm{d} = 112\)
\(3\mathrm{u} + 5\mathrm{d} = 112\)