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March 22, 2020

Challenge Problem 17: Three Runners Puzzle (From India’s Civil Services Exam)

There are three runners X, Y, and Z. Each runs with a different uniform speed in a 1000 meters race.

If X gives Y a start of 50 meters, they will finish the race at the same time.

If X gives Z a start of 69 meters, they will finish the race at the same time.

Suppose Y and Z are in a race. How much of a start should Y give to Z so they would finish the race at the same time?

>> Challenge Problem 15: An Amazing Result of Complicated Exponent Problem

This is adapted from India’s Civil Services Exam.

You might guess the answer is 69 – 50 = 19. That is close but it’s not correct! The reason is Y needs to give Z a 19 m start if they were running a 950 m race. But we need the answer for a 1000 m race.

We can use the formula distance = (rate)(time), which is equivalently time = (distance)/(rate). Use lower-case letters to indicate each runner’s speed (rate of running).

X and Y run the same time when X runs 1000 meters and Y runs 1000 – 50 = 950 meters. Thus we have the equation:

1000/x = 950/y
y = 0.95x

Y is 95% the speed of X.

X and Z run the same time when X runs 1000 meters and Z runs 1000 – 69 = 931 meters. Thus we have the equation:

1000/x = 931/z
Z = 0.931x

Z is 93.1% the speed of X.

Both equations equal 1000/x, so we can equate the right hand sides.

950/y = 931/z

Now multiply both sides by 1000/950 to solve for the time it takes y to run 1000 meters. This gives:

1000/y = (931/z)(1000/950)
1000/y = 980/z
z = .98y

This means Y completes 1000 meters in the time Z would complete 980 meters (or Z is 98% the speed of Y). Thus Y needs to give Z a 20 meter start.

Mind Your Decision



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