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
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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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