Question 1171389
.



            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.



<pre>
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  {{{400/x}}}  hours.

The travel time by the faster cyclist is  {{{400/(x+5)}}}  hours.



The difference of their travel times is  half an hour

    {{{400/x}}} - {{{400/(x+5)}}} = {{{1/2}}}.



        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

    {{{x[1,2]}}} = {{{(-5 +- sqrt((-5)^2 + 4*2000))/2}}} = {{{(-5 +- sqrt(8025))/2}}} = {{{(-5 +- 89.58)/2}}}.


The formula gives two roots, but we accept only positive root

    x = {{{(-5 + 89.58)/2}}} = {{{84.58/2}}} = 42.29 km/h.


<U>ANSWER</U>.  the slower cyclist rate is  42.29 km/h;  the faster cyclist rate is  47.29 km/h.
</pre>

Solved.


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