How long did it take Alison to overtake Alex?

Question

Exactly 20 minutes after Alex left home, his sister Alison set out to overtake him. Alex drove at 48 mph and Alison drove at 54 mph. How long did it take Alison to overtake Alex?

Answer 1

First, note that the problem gives time in minutes but speed in miles per hour.  This will be a problem if you don't convert to all the same unit!  I chose to convert 20 minutes to 20/60 = 1/3 hours.

 

Rate questions like this always use the fact that distance = rate*time.  In this case, we are looking for the time it took for Alison to overtake Alex -- meaning the time at which Alison and Alex were at the same distance.  So we want to get an equation for distance for both Alison and Alex, then set them equal.

 

Define your variable:

x = the number of hours Alison drove

 

Distance equations:

Alex's distance = (1/3 hr)(48 mph)+ x(48 mph) = 16 + 48x ==> This is how far Alex drove during his 20 minute head start (time*rate) + how far Alex drove until Alison overtook him.

 

Alison's distance = x(54) = 54x

 

Set them equal:

 

16 + 48x = 54x

16 = 6x (subtracted 48x from each side)

x = 16/6 = 8/3 (divided both sides by 6)

 

So it took Alison 8/3 hours to catch Alex.  You can convert 8/3 hours back to minutes by multiplying by 60. Then 8/3 hours = 160 minutes = 2 hours and 40 minutes.