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06 Aug 2026, 03:50
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A cyclist followed a two-leg route from a riverside park to a lookout point. If the second leg of the route was twice as long as the first leg, what was the cyclist’s average (arithmetic mean) speed for the entire route?
(1) The cyclist’s average (arithmetic mean) speed was 12 kilometers per hour on the first leg and 24 kilometers per hour on the second leg.
(2) The cyclist spent the same amount of time on the first leg as on the second leg.
(1) The cyclist’s average (arithmetic mean) speed was 12 kilometers per hour on the first leg and 24 kilometers per hour on the second leg.
(2) The cyclist spent the same amount of time on the first leg as on the second leg.
06 Aug 2026, 03:50
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Official Solution:
A cyclist followed a two-leg route from a riverside park to a lookout point. If the second leg of the route was twice as long as the first leg, what was the cyclist’s average (arithmetic mean) speed for the entire route?
Let the length of the first leg be \(d\) kilometers. Then the length of the second leg is \(2d\) kilometers. So the total distance is:
\(d + 2d = 3d\)
The question asks for the cyclist’s average speed for the entire route:
\(\frac{\text{total distance}}{\text{total time}}\)
(1) The cyclist’s average speed was 12 kilometers per hour on the first leg and 24 kilometers per hour on the second leg.
The time spent on the first leg was:
\(\frac{d}{12}\)
The time spent on the second leg was:
\(\frac{2d}{24} = \frac{d}{12}\)
So the total time was:
\(\frac{d}{12} + \frac{d}{12} = \frac{d}{6}\)
Therefore, the average speed for the entire route was:
\(\frac{3d}{\frac{d}{6}} = 18\) kilometers per hour
Sufficient.
(2) The cyclist spent the same amount of time on the first leg as on the second leg.
This tells us only that, because the second leg was twice as long, the cyclist’s speed on the second leg was twice the speed on the first leg. It does not give either actual speed.
For example, the speeds could have been 12 and 24 kilometers per hour, giving an overall average speed of 18 kilometers per hour.
Or the speeds could have been 20 and 40 kilometers per hour, giving an overall average speed of 30 kilometers per hour.
Not sufficient.
Answer: A
A cyclist followed a two-leg route from a riverside park to a lookout point. If the second leg of the route was twice as long as the first leg, what was the cyclist’s average (arithmetic mean) speed for the entire route?
Let the length of the first leg be \(d\) kilometers. Then the length of the second leg is \(2d\) kilometers. So the total distance is:
\(d + 2d = 3d\)
The question asks for the cyclist’s average speed for the entire route:
\(\frac{\text{total distance}}{\text{total time}}\)
(1) The cyclist’s average speed was 12 kilometers per hour on the first leg and 24 kilometers per hour on the second leg.
The time spent on the first leg was:
\(\frac{d}{12}\)
The time spent on the second leg was:
\(\frac{2d}{24} = \frac{d}{12}\)
So the total time was:
\(\frac{d}{12} + \frac{d}{12} = \frac{d}{6}\)
Therefore, the average speed for the entire route was:
\(\frac{3d}{\frac{d}{6}} = 18\) kilometers per hour
Sufficient.
(2) The cyclist spent the same amount of time on the first leg as on the second leg.
This tells us only that, because the second leg was twice as long, the cyclist’s speed on the second leg was twice the speed on the first leg. It does not give either actual speed.
For example, the speeds could have been 12 and 24 kilometers per hour, giving an overall average speed of 18 kilometers per hour.
Or the speeds could have been 20 and 40 kilometers per hour, giving an overall average speed of 30 kilometers per hour.
Not sufficient.
Answer: A
