1. word problems

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?

2. Word Problem

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.