Question 1204036

Let's start with the distance formula:
{{{d = r*t}}} where {{{d}}} = distance traveled, {{{r}}} = rate (speed), and {{{t}}} = time of travel.  

We can write two such equations, one for Steve (1) and the other for his brother (2):

{{{d[1] = r[1]*t[1]}}}......eq. (1) 
and

{{{d[2] = r[2]*t[2]}}}.......eq.(2)

Now when Steve's brother on the motor bike overtakes Steve on his bicycle, each will have traveled the same distance, so:

 {{{d[1] = d[2]}}}  

and, since Steve's brother set out two hours after Steve did (10 am - 8 am), then Steve's time is two hours more than his brother's time, and, of course, the speeds of both are given, so:

{{{t[1] = t[2]+2}}} 

Now we have enough information to complete the two equations:

{{{d = 18*(t[2]+2)}}}
{{{d = 54*(t[2])}}} 

Since {{{d = d}}} we'll set these two equations equal to each other to get:

{{{18*(t[2]+2) = 54*t[2]}}}........ Solve for {{{t[2]}}}

{{{18*t[2]+36 = 54*t[2]}}}......... Subtract {{{18*t[2]}}} from both sides.

{{{36 = 36*t[2]}}} ........Divide both sides by {{{36}}}.

{{{t[2] = 1}}} but we want Steve's time ({{{t[1]}}}, so

{{{t[1] = t[2]+2}}}....... Substitute {{{t[2] = 1}}}

{{{t[1] = 1+2}}}

{{{highlight(t[1] = 3)}}}

Steve had traveled for {{{3}}} hours when his brother overtook him.