a man drives 108 km at an average rate. If he had driven 5km/hr. faster, he would have made the trip in 15 hours less time. How fast did he drive?
You know that $\displaystyle Time = \frac {Distance}{Speed}$.
Distance $\displaystyle = 108 $ km
Speed $\displaystyle = s$ km/h
If he had driven $\displaystyle 5$ km/hr faster, he would have reached in $\displaystyle 15$ hours less time.
That means... Actual Time taken $\displaystyle -$ Time taken while driving $\displaystyle 5$ km/h faster$\displaystyle = 15$ hours.
$\displaystyle \Rightarrow \frac {108}{s} - \frac {108}{s + 5} = 15 $
Now express as a single fraction:
$\displaystyle \Rightarrow \frac {108 (s + 5) - 108s}{s (s + 5)} = 15$
Now multiply both sides by the denominator, $\displaystyle s (s + 5)$.
$\displaystyle \Rightarrow 108 (s + 5) - 108s = 15s (s + 5)$
$\displaystyle \Rightarrow 108s + 540 - 108s = 15s^2 + 75s $
$\displaystyle \Rightarrow 540 = 15s^2 + 75s $
$\displaystyle \Rightarrow 15s^2 + 75s - 540 = 0 $
Divide throughout by $\displaystyle 15$:
$\displaystyle \Rightarrow s^2 + 5s - 36 = 0 $
Now factorise:
$\displaystyle \Rightarrow (s + 9) (s - 4) = 0 $
$\displaystyle \Rightarrow$ Either $\displaystyle s = -9$(discard) or $\displaystyle s = 4 $.
Check: So he was driving at $\displaystyle 4$ km/h.
Time taken to drive at $\displaystyle 4$ km/h$\displaystyle = \frac {108}{4} = 27 $ hours.
Time taken to drive at $\displaystyle 9$ km/h$\displaystyle = \frac {108}{9} = 12 $ hours.
Difference in times $\displaystyle = 27 - 12 = 15 $ hours.
at above poster.
I hope that helps.
ILoveMaths07.