How Long Will It Take A Bus Traveling At 60 Miles Per Hour To Overtake A Car Traveling At 40 Mph If The Car Had A 1.5-hour Head Start?

It's the standard equation, Distance = rate * time, but in one case you give the car a 1.5 hour head start.  So let's just set up the equations and set them to the same distance when the pass each other. 

We know the general equation Distance = Rate * Time, so for each vehicle we'll add a subscript to keep them straight;  The distance is the same for both so it doesn't get a subscript.

Distance = Rate_bus * Time_bus

Distance = Rate_car * Time_car

We also know that the Time_car = Time_bus + 1.5 hours, since it had a head start, and we can substitute that into the other equation, just don't forget the parenthesis;

Distance = Rate_car * (Time_bus + 1.5)

Since the distance is the same, we'll set them both equations equal to each other;

Rate_bus * Time_bus = Rate_car * (Time_bus + 1.5)

Now let's plug in the numbers and solve

60   *   Time_bus   = 40 * Time_bus + 40 * 1.5

60 * Time_bus - 40 * Time_bus = 40 * 1.5

(60 - 40) * Time_bus = 40 * 1.5

20 * Time_bus = 60

Time_bus = 60 / 20 = 3 Hrs

Now, If we carried the units through, it would look like this;

We still plug in the numbers and solve, but we keep the units along (just don't confuse them with variables)

60 MPH  *   Time_bus   = 40 MPH * Time_bus + 40 MPH * 1.5 Hrs

60 MPH * Time_bus - 40 MPH * Time_bus = 40 MPH * 1.5 Hrs

(60 - 40) MPH * Time_bus = 40 MPH * 1.5 Hrs

20 MPH * Time_bus = 60 MPH*Hrs

Time_bus = 60 MPH*Hrs
    .     .   20 MPH

Time_bus = 3 Hrs

See how the MPH cancel and leave the answer in hours?  That gives us some assurance the equation was set up right.  Now we can also check the answer by putting it back into the original equations;

Distance = Rate_bus * Time_bus = 60 MPH * 3 Hrs  = 180 miles

Distance = Rate_car * Time_car  = 40 MPH * 4.5 Hrs  = 180 miles

So they check.  Now we know it's correct.