rate times time equals distance

Question

a slow train leaves springfeild at 7 am going 43 mph. a fast train leaves springfeild going the same direction at 10:30 going 56 mph. at what time will the second train catch up to the first train? how many miles away from springfeild will the trains be when they catch up?

Answer 1

Hi, Claudia.

 

For this problem, the slow train has a 3.5 hour head start on the other train. So, at 10:30a, it will have traveled (3.5)(43) miles already. Then, every hour it will travel 43 miles. The other train, at 10:30a will have traveled 0 miles (because it just started its commute), and every hour thereafter it will travel 56 miles. Since, we do not know how many hours it will take for them to catch up, we will let that quantity be x hrs. Since, we already calculated the head start for the slow train. x is the number of hours after 10:30a.

 

The distance from Springfield of the slow train after x hours is 43x + (3.5)(43) miles; the distance traveled by the fast train in the same amount of time (x hours) is 56x. We are trying to find the time (x) when these two distances are equal

 

So, we have the equation

43x + (3.5)(43) = 56x

 

Let's solve for x:

1. Multiply (3.5) & (43)

(3.5)(43) = 150.5, so...

43x + 150.5 = 56x

 

2. Subtract 43x from both sides to get x on the right side and the constant on the left side

43x - 43x + 150.5 = 56x - 43x

150.5 = 13x

 

3. Divide both sides by 13 to get x (the number of hours from 10:30a at which point the two trains will be caught up)

150.5/13 = 13x/13

 

Oddly, here we get a decimal: about 11.76... Which is about 11.75 hours or 11 hours and 45 minutes. So, in about 11 hours and 45 minutes from 10:30a (which is around 10:15p), the buses should be caught up.

 

In order to find out how many miles away from Springfield the trains will be at this point, plug in the value you got for x into either the left or the right side (it doesn't matter which one, because they both are equal - make sure you do not plug in an estimated value) and that will tell you the distance of the trains from Springfield.

 

Answer 2

distance = rate x time

 

slow train travels for x hours

fast train travels for (x - 3.5) hours

 

distance traveled by slow train = 43x

distance traveled by fast train = 56(x - 3.5)

 

We need to find when the two distances are equal

 

43x = 56(x - 3.5)

43x = 56x - 196

-13x = -196

x = 15.08

 

The second train will catch up with the first train after the second train travels 15.08 - 3.5 = 11.58 hours

 

The distance traveled is 43x = 43(15.08) = 648.44 miles

Answer 3

d_fast = 56t

d_slow = 43(t + 3.5)

The distances are equal

56t = 43(t + 3.4)

 

Can you solve for t and answer?