A courier travels along a straight route from Office P to Office Q at a constant speed r_1… — GRE General Test

A courier travels along a straight route from Office P to Office Q at a constant speed $r_1$ , and immediately returns along the exact same route from Office Q to Office P at a constant speed $r_2$ . The ratio of the outbound speed to the return speed is $r_1 : r_2 = 3 : 2$ . Which of the following statements must be true? Select all such statements.

The time spent on the return trip is $50\%$ greater than the time spent on the outbound trip.Cevap
B
The ratio of the time spent on the outbound trip to the time spent on the return trip is $3 : 2$ .
The average speed for the entire round trip is equal to $80\%$ of the outbound speed.Cevap
D
The average speed for the entire round trip is equal to the arithmetic average of the outbound speed and the return speed.
E
The outbound trip accounts for $60\%$ of the total travel time for the round trip.

Cevap

The statement that the return trip time is 50 percent greater than the outbound trip time, and the statement that the average speed for the entire round trip is equal to 80 percent of the outbound speed.

The return trip time is

50\%

greater than the outbound trip time because speed and time are inversely proportional over equal distances (

t_1 : t_2 = 2 : 3

). Additionally, the overall average speed is

\frac{12k}{5} = 2.4k

, which is precisely

80\%

of the outbound speed (

3k

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Express travel times in terms of distance

D

and rate multiplier

k

Let outbound speed

r_1 = 3k

and return speed

r_2 = 2k

. Outbound time

t_1 = \frac{D}{3k}

and return time

t_2 = \frac{D}{2k}

. The ratio of times

t_1 : t_2 = \frac{D/3k}{D/2k} = 2 : 3

Time equals distance divided by rate, establishing an inverse relationship between speed and time for constant distance.

Evaluate the relative difference between return time and outbound time.

Return time

t_2 = 1.5 t_1

, which means

t_2

50\%

greater than

t_1

A factor of

1.5

represents a

50\%

increase over the base value

t_1

Calculate the average speed

R_{\text{avg}}

for the round trip.

Total distance is

2D

, and total time is

t_1 + t_2 = \frac{D}{3k} + \frac{D}{2k} = \frac{5D}{6k}

. Thus,

R_{\text{avg}} = \frac{2D}{\frac{5D}{6k}} = \frac{12k}{5} = 2.4k

Average speed is defined as total distance divided by total elapsed time.

Compare

R_{\text{avg}}

to the outbound speed

r_1

and the arithmetic mean of rates.

\frac{R_{\text{avg}}}{r_1} = \frac{2.4k}{3k} = 0.80 = 80\%

. The arithmetic mean is

\frac{3k + 2k}{2} = 2.5k \neq 2.4k

. Outbound time fraction is

\frac{t_1}{t_1 + t_2} = \frac{2}{5} = 40\%

Unweighted arithmetic averaging fails for rates over equal distances because more time is spent traveling at the lower speed.

Anahtar Kavram

Inverse proportion between speed and time, harmonic mean for round-trip average speed, and part-to-whole time ratios.

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