Question 121905
Remember the formula, {{{d=rt}}}


The rate going up is then {{{r=15/t}}} and the rate coming down is 1 km/h faster and the time is a half hour less for the same 15 km, so {{{r+1=15/(t-0.5)}}}


Solve the second equation for r by adding -1 to both sides


{{{r=(15/(t-0.5))-1}}} giving us a second expression for r in terms of t.


Now since r = r, we can say:


{{{15/t=(15/(t-0.5))-1}}}


The LCD is {{{t(t-0.5)}}}, so


{{{(15(t-0.5)/t(t-.05))=(15t/t(t-0.5))-(t(t-0.5)/t(t-0.5))}}}


Now put everything on the left:


{{{(15(t-0.5)-(15t)+(t(t-0.5)))/t(t-0.5)=0}}}


Distribute, expand, and collect terms in the numerator but leave the denominator alone:


{{{(15t-7.5-15t+t^2-0.5t)/t(t-0.5)=0}}}
{{{(t^2-0.5t-7.5)/t(t-0.5)=0}}}


First consider the zeros of the denominator -- 0 and 0.5.  We need to remember these values and exclude them if they come up in the process of finding the zeros of the numerator.  For now, just realize that {{{a/b=0}}} if and only if {{{a=0}}} and {{{b<>0}}}.  Therefore, all we need to do is solve:


{{{(t^2-0.5t-7.5)=0}}}


Multiply by 2 to get rid of those pesky decimal fraction coefficients


{{{2t^2-t-15=0}}}


Conveniently, this factors:


{{{(2t+5)(t-3)=0}}}


So {{{t=3}}} or {{{t=-2.5}}}.  Neither of these values are 0 or 0.5 which would have made the denominator in the original equation go to zero, but we can exclude the negative value.  As much as it would be nice to make the clock run backwards sometimes, negative time just doesn't make sense.  So, our time up the mountain is 3 hours.


The time down the mountain, is 1/2 hour less, or 2.5 hours.  Therefore the total trip was 3 + 2.5 = 5.5 hours.