To help Kim practice for the Paralympics, Sally runs beside him for half the required distance. She runs the remaining distance at her regular speed and arrives 90 seconds ahead of Kim. What is the ratio of Sally's regular speed to Kim's speed? Use tKim for Kim's total time.

Question

Quizplus · Accepted Answer

Let K be Kim's speed and S be Sally's regular speed.
Distance = Speed × Time

Kim's time = tKim
Sally's time = tSally

Sally runs half the distance, so her time is half of Kim's time:
tSally = tKim/2

Let the total distance be d.

The distance Sally ran is:
d/2

So the distance she has left to run is:
d/2

This is the same distance Kim runs. They run at their regular speeds, so the time it takes them to run those distances is:

tKim = (d - d/2)/K
tSally = (d/2)/S

Sally arrived 90 seconds ahead of Kim:
tSally = tKim - 90

Substituting our previous equations, we get:

(d/2)/S = (d - d/2)/K - 90

Simplifying:

d(KS) = 3dS - 3d²/4 - 90KS
KS = (3dS - 3d²/4)/(d + 90)

We want the ratio of Sally's regular speed to Kim's speed, which is S/K:

S/K = S/(d/KS) = S²(d + 90)/(3dS - 3d²/4)

We don't know the values of S or K, but we can simplify the expression a bit further:

S/K = S²(d + 90)/(3dS - 3d²/4)
S/K = S(4d + 360)/(12d - 3d)
S/K = (4d + 360)/9d

Now we just need to find a choice that matches this expression:

(A) 1/3
(B) 4/3
(C) 44/27
(D) 13/8
(E) 17/9

Only choice (C) matches our expression, so the answer is:

Answer: C

To Help Kim Practice for the Paralympics, Sally Runs Beside