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Question

Difficulty: HardRate, Time, and Distance Problems

An automated delivery van and an electric cargo scooter both travel along a straight route from Warehouse X to Warehouse Y, which are DD miles apart.

The van travels at a constant speed of 3030 miles per hour for the first half of the total distance, and at a constant speed of 6060 miles per hour for the second half of the total distance.

The scooter travels at a constant speed of 3030 miles per hour for the first half of its total travel time, and at a constant speed of 6060 miles per hour for the second half of its total travel time.

If the scooter completes the entire journey in 1515 minutes less time than the van, what is the value of DD, in miles?

  1. A
    3030
  2. B
    4545
  3. C
    6060
  4. 9090Answer
  5. E
    180180

Answer

The distance DD between Warehouse X and Warehouse Y is 9090 miles.
The option stating 9090 is correct because the van's total time is D40\frac{D}{40} hours (average speed 4040 mph across equal distances) and the scooter's total time is D45\frac{D}{45} hours (average speed 4545 mph across equal time intervals). Setting the difference D40−D45\frac{D}{40} - \frac{D}{45} equal to 14\frac{1}{4} hour yields D=90D = 90 miles.

Step-by-Step Solution

1
Calculate the total travel time for the van in terms of DD.
The van travels D2\frac{D}{2} miles at 3030 mph and D2\frac{D}{2} miles at 6060 mph. Time for first half: D/230=D60\frac{D/2}{30} = \frac{D}{60} hours. Time for second half: D/260=D120\frac{D/2}{60} = \frac{D}{120} hours. Total van time Tvan=D60+D120=3D120=D40T_{\text{van}} = \frac{D}{60} + \frac{D}{120} = \frac{3D}{120} = \frac{D}{40} hours.
For equal distance legs, average speed is calculated via harmonic mean, giving total time T=Total DistanceAverage SpeedT = \frac{\text{Total Distance}}{\text{Average Speed}}.
2
Calculate the total travel time for the scooter in terms of DD.
Let TscooterT_{\text{scooter}} be the total travel time for the scooter. It spends Tscooter2\frac{T_{\text{scooter}}}{2} hours at 3030 mph and Tscooter2\frac{T_{\text{scooter}}}{2} hours at 6060 mph. Total distance D=30(Tscooter2)+60(Tscooter2)=15Tscooter+30Tscooter=45TscooterD = 30\left(\frac{T_{\text{scooter}}}{2}\right) + 60\left(\frac{T_{\text{scooter}}}{2}\right) = 15 T_{\text{scooter}} + 30 T_{\text{scooter}} = 45 T_{\text{scooter}}. Thus, Tscooter=D45T_{\text{scooter}} = \frac{D}{45} hours.
For equal time legs, average speed is the arithmetic mean of the speeds (4545 mph).
3
Set up and solve the equation using the given time difference.
1515 minutes is equal to 1560=14\frac{15}{60} = \frac{1}{4} hours. The equation is D40−D45=14\frac{D}{40} - \frac{D}{45} = \frac{1}{4}. Finding a common denominator of 360360: 9D−8D360=14  ⟹  D360=14  ⟹  D=3604=90\frac{9D - 8D}{360} = \frac{1}{4} \implies \frac{D}{360} = \frac{1}{4} \implies D = \frac{360}{4} = 90 miles.
Equating the difference between the van's travel time and the scooter's travel time to 1515 minutes (14\frac{1}{4} hour) allows solving for DD.

Key Concept

Distinction between average speed over equal distances (harmonic mean) versus average speed over equal times (arithmetic mean).
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Topics

Rate, Time, and Distance Problems

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