A cyclist and a runner start simultaneously from Point A and travel along a straight path … — GMAT

A cyclist and a runner start simultaneously from Point $A$ and travel along a straight path toward Point $B$ . The cyclist travels at a constant speed, and the runner travels at a lower constant speed. When the cyclist reaches Point $B$ , she immediately turns around and travels back toward Point $A$ at her original speed. She meets the runner at a location $12$ miles from Point $B$ .

If the cyclist had instead increased her speed by $20\%$ and the runner had decreased his speed by $10\%$ , the cyclist would have met the runner on her return trip at a location $20$ miles from Point $B$ .

What is the distance, in miles, between Point $A$ and Point $B$ ?

Answer: 60 miles

Answer

The distance between Point A and Point B is 60 miles.

The correct answer of 60 miles is obtained by setting up the ratio of distances traveled by the cyclist and the runner for both scenarios. Since time elapsed until they meet is the same for both people in each scenario, the ratio of distances equals the ratio of speeds. Setting the scaled speed ratio (4/3 times the original ratio) equal to the new distance ratio (D + 20)/(D - 20) yields the quadratic equation D^2 - 56D - 240 = 0, whose unique positive solution is D = 60.

Step-by-Step Solution

Set up the relative distance ratio for the first meeting condition.

v_c / v_r = (D + 12) / (D - 12)

Because time is equal for both participants, distance traveled is directly proportional to speed.

Adjust the speed ratio based on the given percentage changes.

v_c' / v_r' = (1.20 / 0.90) * (v_c / v_r) = (4/3) * (D + 12) / (D - 12)

Increasing speed by 20% scales it by 1.20, and decreasing speed by 10% scales it by 0.90.

Equate the adjusted speed ratio to the distance ratio for the second meeting condition.

(D + 20) / (D - 20) = 4(D + 12) / [3(D - 12)]

In the second scenario, the cyclist covers D + 20 miles while the runner covers D - 20 miles.

Expand and solve the resulting quadratic equation for D.

D^2 - 56D - 240 = 0, yielding D = 60 miles.

Cross-multiplying and simplifying yields a quadratic equation with positive root D = 60.

Key Concept

Distance-Speed Proportionality for Simultaneous Travel and Relative Motion

Estimated Time:3m 0s

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