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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.