# Problem Solving: Find the time for the second bus to catch up

by Katie
(USA)

##### Question
A bus leaves a station at 8:00 a.m. and averages 30 mi/h. Another bus leaves the same station following the same route two hours after the first and averages 50 mi/h. When will the second bus catch up to the first?
STEP 1:   We are given that the first bus leaves a station at a rate of 30 mi/h at 8.00 am and that the second bus leaves the same station two hours later at a rate of 50mi/h. Assume a variable for the time taken by the first bus to travel, say, t.

STEP 2:    Recall the formula: In this case, the distances are the same, so we just need to set the rate × time equal for both the cases. STEP 3:    We can see that 50 is distributed over the difference t – 2. To remove the parentheses in this expression, first multiply 50 by t – 2. To remove the parentheses in this expression, first multiply t. We get 50t. Multiply 50 by -2 next. The result that you get is 30t = 50t – 100.

STEP 4:    Now, we have to isolate the variable terms on one side and the constant terms on the other side.

Subtract 50t from both sides of the obtained expression and simplify. STEP 5:    Isolate the variable t, by dividing both sides of the obtained expression by -20 This is the time of travel for the first bus till the second bus catches up with it.

STEP 6:   Now you have to find the time at which the second bus catch up to the first bus. We know that the first bus leaves the station at 8 00 am. So we have to add 5 hrs to the starting time of the first bus:

8.00 am + 5 hours = 1.00 pm

The second bus catches up to the first by 1.00 pm.

### Comments for Problem Solving: Find the time for the second bus to catch up

Average Rating     Jan 12, 2019 Rating     Another Method NEW by: Peter I think this can also be done by trig. I did a similar thing to calculate 'catch-up' distance which also gives time. If it's of interest?

 May 19, 2015 Rating     Reply to comment from Jan 27, 2015 by: Dante The reason for -2 is because of the 2 hours that have elapsed since the first bus has left.

 Jan 27, 2015 Rating     Clarify please by: Anonymous Why do you -2. I see how r * t = r*t. But how do you know to -2 Thank you for the great explanation. It helped

 Dec 14, 2014 Rating     Finally by: Anonymous Thanks. I've been having trouble with these problems and after looking through many websites, this was the only one that actually explained the steps taken to solve the problem.

iPod Touch & iPhone