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Difficulty: MediumRatios, Rates, and Proportions

A delivery truck completes a trip consisting of two consecutive legs. The ratio of the distance of the first leg to the distance of the second leg is 3:13 : 1. The ratio of the truck's constant speed on the first leg to its constant speed on the second leg is 3:23 : 2. If the average speed of the truck for the entire trip is 40 miles per hour40\text{ miles per hour}, what was the constant speed of the truck on the second leg, in miles per hour?

  1. A
    16 mph16\text{ mph}
  2. B
    25 mph25\text{ mph}
  3. 30 mph30\text{ mph}Answer
  4. D
    32 mph32\text{ mph}
  5. E
    45 mph45\text{ mph}

Answer

30 mph30\text{ mph}
The total distance traveled is 3d+d=4d3d + d = 4d. The time required for the first leg is 3d3v=dv\frac{3d}{3v} = \frac{d}{v}, and the time for the second leg is d2v\frac{d}{2v}. Combining these yields a total time of 3d2v\frac{3d}{2v}. Setting total distance over total time equal to the given average speed gives 4d3d2v=8v3=40 mph\frac{4d}{\frac{3d}{2v}} = \frac{8v}{3} = 40\text{ mph}, which yields v=15v = 15. Consequently, the constant speed on the second leg is 2v=30 mph2v = 30\text{ mph}.

Step-by-Step Solution

1
Represent the distances and speeds of both legs using ratio multipliers
Distance leg 1 = 3d3d, distance leg 2 = dd, total distance = 4d4d. Speed leg 1 = 3v3v, speed leg 2 = 2v2v.
Ratios 3:13:1 for distance and 3:23:2 for speed allow expressed quantities in terms of single parameters dd and vv.
2
Calculate the travel time for each leg using time = distance / speed
Time for leg 1: t1=3d3v=dvt_1 = \frac{3d}{3v} = \frac{d}{v}. Time for leg 2: t2=d2vt_2 = \frac{d}{2v}.
Average speed depends on total distance divided by total time.
3
Compute total time for the entire trip
Ttotal=t1+t2=dv+d2v=3d2vT_{\text{total}} = t_1 + t_2 = \frac{d}{v} + \frac{d}{2v} = \frac{3d}{2v}.
Finding a common denominator of 2v2v allows combining the two time fractions.
4
Set up the average speed equation and solve for multiplier v
\text{Average Speed} = \frac{4d}{\frac{3d}{2v}} = \frac{8v}{3} = 40 \implies v = 15.
Dividing total distance by total time gives the average speed expression.
5
Calculate the speed of the second leg
Speed of leg 2 = 2v=2(15)=30 mph2v = 2(15) = 30\text{ mph}.
The question specifically asks for the constant speed on the second leg.

Key Concept

Average rate over multi-leg trips is total distance divided by total time, not the simple arithmetic mean of speeds.
Estimated Time:1m 30s
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Topics

Ratios, Rates, and Proportions

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