# 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 = Ratebus * Timebus

Distance = Ratecar * Timecar

We also know that the Timecar = Timebus + 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 = Ratecar * (Timebus + 1.5)

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

Ratebus * Timebus = Ratecar * (Timebus + 1.5)

Now let's plug in the numbers and solve

60   *   Timebus   = 40 * Timebus + 40 * 1.5

60 * Timebus - 40 * Timebus = 40 * 1.5

(60 - 40) * Timebus = 40 * 1.5

20 * Timebus = 60

Timebus = 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  *   Timebus   = 40 MPH * Timebus + 40 MPH * 1.5 Hrs

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

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

20 MPH * Timebus = 60 MPH*Hrs

Timebus = 60 MPH*Hrs
.     .   20 MPH

Timebus = 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 = Ratebus * Timebus = 60 MPH * 3 Hrs  = 180 miles

Distance = Ratecar * Timecar  = 40 MPH * 4.5 Hrs  = 180 miles

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

thanked the writer.
In words
The car is 1.5 hours ahead at 40 mph = 60 miles.
The difference in speed is 20 mph, so 60 miles is covered in 3 hours.

Using equations
Let t be the time in hours that the bus overtakes the car. At the point where that occurs, the bus and the car have traveled the same distance.
(1.5 + t)(40) = t(60)    (distance = speed * time for both)
1.5*40 + 40t = 60t    (distributive property, left side)
1.5*40 = 20t    (subtract 40t from both sides)
1.5*40/20 = t    (divide both sides by 20)
1.5*2 = 3 = t    (perform the arithmetic)
It will take 3 hours for the bus to overtake the car.

thanked the writer. 