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Difficulty: Very hardRatios, Rates, and Proportions

Two runners, Estela and Desmond, start at the same point on a circular track and run in opposite directions at constant speeds. The ratio of Estela's speed to Desmond's speed is 55 to 44. If they start at the same time, they meet for the first time after 4040 seconds. If Desmond increases his speed by 25%25\% immediately after their first meeting, how many seconds after their first meeting will they meet for the second time?

Answer: 36 seconds

Answer

36
To find the time between the first and second meetings, we determine the track length in terms of a constant kk. Given the speed ratio of 55 to 44, the speeds are 5k5k and 4k4k. The relative speed in opposite directions is the sum of these speeds, or 9k9k. In 4040 seconds, they cover the track length L=9k×40=360kL = 9k \times 40 = 360k. When Desmond increases his speed by 25%25\%, his new speed is 4k×1.25=5k4k \times 1.25 = 5k, making the new relative speed 5k+5k=10k5k + 5k = 10k. The time to meet again is the track length divided by the new relative speed: 360k/10k=36360k / 10k = 36 seconds.

Step-by-Step Solution

1
Represent the speeds of both runners using the given ratio of 55 to 44.
Estela's speed is 5k5k and Desmond's speed is 4k4k, where kk is a positive constant.
Using a common variable allows calculation of relative rates without needing their absolute speeds.
2
Calculate the initial relative speed of approach.
Relative speed = 5k+4k=9k5k + 4k = 9k.
Since they run in opposite directions around the circular track, their rates of travel add up to determine their rate of approach.
3
Determine the track circumference LL in terms of kk.
Track length L=9k×40=360kL = 9k \times 40 = 360k.
They meet for the first time when their combined distance traveled is exactly equal to one full lap of the track.
4
Calculate Desmond's updated speed after his speed increase.
Desmond's new speed = 4k×(1+0.25)=5k4k \times (1 + 0.25) = 5k.
His speed increases by 25%25\%, which corresponds to multiplying his initial speed of 4k4k by 1.251.25.
5
Find the new relative speed of approach after the first meeting.
New relative speed = 5k+5k=10k5k + 5k = 10k.
Estela's speed remains 5k5k and Desmond's speed is now 5k5k, so their sum gives the new combined rate.
6
Solve for the time elapsed between the first and second meetings.
Time t=360k10k=36t = \frac{360k}{10k} = 36 seconds.
To meet a second time, they must cover the track length of 360k360k again from their first meeting point at the new relative speed.

Key Concept

Solving multi-step rate-time-distance problems on circular tracks using ratios and percentage change.
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Topics

Ratios, Rates, and Proportions

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