SOLUTION: Please help, I have been trying to figure this out for 3 days. I know I should use x for Elizabeth's speed and x+5 for Rick's speed but I am having trouble coming up with the equa

Question 7621: Please help, I have been trying to figure this out for 3 days. I know I should use x for Elizabeth's speed and x+5 for Rick's speed but I am having trouble coming up with the equation to solve.
Elizabeth and her husband Rick bicycled cross-country together. One morning, Elizabeth rode 40 miles. By traveling only 5 miles per hour faster and putting in one more hour, Rick covered twice the distance Elizabeth covered. What is the speed of each cyclist?
Answer by longjonsilver(2297) (Show Source): You can put this solution on YOUR website!
As you say,
let Elizabeth speed = x (in miles)
let Rick's speed = x+5

Elizabeth rode 40 miles, Rick rode 80 miles

Elizabeth's time = t (in hours)
Rick's time = t+1
========================================================================
we want to find their speeds, but we also do not know their times, so we will get an equation for each based upon speed = distance/time and then re-arrange for t and then equate, to get rid of t. This will just leave one equation with the unknown, x, which we will solve.

That is the theory :-)

Right then, first Elizabeth:

speed = distance/time
x = 40/t --> t = 40/x

Now Rick:

speed = distance/time
x+5 = 80/(t+1)
t+1 = 80/(x+5)
t = 80/(x+5) - 1

so we have . Now it is just algebraic manipulation...

. Now cross multiply

put this into the quadratic formula to give .

This means that x = 27.8 mph or x=7.192mph.

For a person on a cycle, i would suggest that 7.192mph is the answer required. Then Rick's speed would be 12.192mph.

Please check my working though, just in case i have made an error somewhere.

jon.

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