SOLUTION: Hi, I would like to know how to solve the following word problem using rational expressions and a Distance-Speed-Time table: Two cyclists, Emaan and Daijah-Leigh, enter a 200-km

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Question 1171389: Hi, I would like to know how to solve the following word problem using rational expressions and a Distance-Speed-Time table:
Two cyclists, Emaan and Daijah-Leigh, enter a 200-km bicycle race. Emaan cycles 5 km/h faster than Daijah-Leigh, but her bicycle gets a flat tire, which takes a half-hour to repair. If Emaan and Daijah-Leigh finish the race in a tie, then determine the speed of each cyclist? Answer to one decimal place.
Sorry if I selected the wrong topic for my question, I was unsure which one I was supposed to pick.
Thanks!
Found 2 solutions by Boreal, ikleyn:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website!
d=speed*time
200-----x+5-----t-0.5 for E
200------x-------t. for D-L.
xt=200
(x+5)(t-0.5)=200=xt-0.5x+5t-2.5
the two are equal, and xt is common to both and may be removed.
0=-0.5x+5t-2.5
0.5x=5t-2.5, but x=200/t,so 0.5x=100/t
100/t=5t-2.5
100=5t^2-2.5t
0=5t^2-2.5t-100
0=2t^2-t-40
t=(1/4)(1+/- sqrt(1+320); sqrt term=17.92
positive root is (1/4)(18.92)=4.73 hours
slower cyclist had speed of 200/4.73 or 42.28 km/h
faster cyclist had speed to 47.28 km/h, and 200/47.28=4.23 hours
so 47.3 km/h and 42.3 km/h taking 4.2 hours and 4.7 hours.

Answer by ikleyn(52781)   (Show Source):
You can put this solution on YOUR website!
.

            Actually, these problems are among the most beautiful in the school Math.

            But in order for a student could see their beauty, the solution should be presented in adequate form.

Let x be the rate of the slower cyclist, in kilometers per hour. Then the rate of the faster cyclist is (x+5) km/h. The travel time spent by the slower cyclist is hours. The travel time by the faster cyclist is hours. The difference of their travel times is half an hour - = . It is, probably, the major step in the problem' solution to establish this basic equation. It is called a "time" equations, because its terms in the left side are traveled times. To solve this equation, multiply both sides by 2x*(x+5). You will get then 400(x+5) - 400x = x*(x+5) Simplify step by step 400x + 2000 - 400x = x^2 + 5x x^2 + 5x - 2000 = 0. Find the roots using the quadratic formula = = = . The formula gives two roots, but we accept only positive root x = = = 42.29 km/h. ANSWER. the slower cyclist rate is 42.29 km/h; the faster cyclist rate is 47.29 km/h.

Solved.

I hope now you can see this beauty and can learn the method.