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Difficulty: HardRate, Time, and Distance Problems

A commuter drives from home to work along a straight route at a constant speed of vv miles per hour. On the return trip along the exact same route, heavy traffic causes the drive to take 50%50\% longer than the outbound trip. If the commuter's average speed for the entire round trip was 4848 miles per hour, what was the value of vv, in miles per hour?

  1. A
    36
  2. B
    50
  3. C
    57.6
  4. 60Answer
  5. E
    72

Answer

60 miles per hour
The total distance for the round trip is 2d2d. If outbound time is t1=dvt_1 = \frac{d}{v}, the return time taking 50%50\% longer is 1.5t1=1.5dv1.5t_1 = \frac{1.5d}{v}. The total time is 2.5t1=5d2v2.5t_1 = \frac{5d}{2v}. Dividing total distance 2d2d by total time 5d2v\frac{5d}{2v} gives average speed 4v5\frac{4v}{5}. Setting 4v5=48\frac{4v}{5} = 48 gives v=60v = 60 miles per hour.

Step-by-Step Solution

1
Define variables for distance and outbound time.
Let the one-way distance be dd miles. Outbound time is t1=dvt_1 = \frac{d}{v} hours.
Relating time to distance and speed using the standard formula t=dvt = \frac{d}{v}.
2
Express return time and return speed in terms of outbound parameters.
Return time is t2=1.5t1=1.5dvt_2 = 1.5 t_1 = \frac{1.5d}{v} hours.
The return trip takes 50% longer, so t2=t1+0.5t1=1.5t1t_2 = t_1 + 0.5 t_1 = 1.5 t_1.
3
Calculate total distance and total time for the round trip.
Total distance =2d= 2d. Total time =t1+t2=t1+1.5t1=2.5t1=2.5dv=5d2v= t_1 + t_2 = t_1 + 1.5 t_1 = 2.5 t_1 = \frac{2.5d}{v} = \frac{5d}{2v} hours.
Average speed requires total distance divided by total time.
4
Set up the average speed equation and solve for vv.
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{5d}{2v}} = \frac{4v}{5}.Givenaveragespeedis48mph:. Given average speed is 48 mph: \frac{4v}{5} = 48 \implies 4v = 240 \implies v = 60$.
Solving the linear algebraic equation yields the exact outbound speed.

Key Concept

Average speed for multi-leg journeys must always be calculated as Total Distance divided by Total Time.
Estimated Time:2m 0s
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Topics

Rate, Time, and Distance Problems

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