SOLUTION: A runner traveled 3 miles in the same time that a cyclist traveled 10 miles. The speed of the cyclist was 14 mph greater than that of the runner. What was the cyclists speed? Th

Question 1060146: A runner traveled 3 miles in the same time that a cyclist traveled 10 miles. The speed of the cyclist was 14 mph greater than that of the runner. What was the cyclists speed?
The answer is supposed to be 20. I have tried everything I can think of but keep coming up with decimals.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

"What was the cyclists speed?" is the ultimate question here. Since it's unknown and we want to find it, let's make

x = cyclists speed in miles per hour.

The runner is 14 mph slower than the cyclist, so the runner's speed is x-14 miles per hour.

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We will use the formula

distance = rate*time

"rate" is another term for "speed". In short,

d = r*t

The distance for the runner is 3 miles. So d = 3 for the runner. The value of t is unknown. Let's just leave it as t for now.

So
d = r*t
turns into
3 = (x-14)*t
after replacing the variables with the expressions set up earlier.

Similarly, for the cyclist, this equation
d = r*t
turns into
10 = x*t

We can solve for t in that other equation to get

10 = x*t
10/x = t
t = 10/x

Then let's plug that into the first equation

3 = (x-14)*t
3 = (x-14)*(10/x) ... replace t with 10/x

Now isolate x

3 = (x-14)*(10/x)
3*x = (x-14)*(10/x)*x ... multiply both sides by x
3x = 10(x-14)
3x = 10x - 140
3x - 10x = -140
-7x = -140
x = 20 ... divide both sides by -7

So that's why the cyclist's speed is 20 mph.

Let me know if this helps or not. Thank you.

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