There are two types of distance problems: catch up and miles apart. A "catch up" problem has two people leaving someplace at different times going the same direction and the question asks how long it will take for the second person to catch up. A "miles apart" problem has two people leaving the same place but going in opposite directions and then asks how long will it take for them to be a certain number of miles apart.

Let's try an example:

Bob leaves home to go to work and is traveling 35 mph. Larry realizes Bob left his briefcase at home and leaves one hour later going 45 mph. How long will it take Larry to catch up to Bob?

First, which kind of problem is it: catch up or miles apart? (If you can't figure that part out, reread the problem and look for either term.) Our problem is a "catch up" problem because it asks: How long will it take Larry to catch up with Bob?

There are two ways to look at these problems. One is through the actual distance and rate, the other is by looking at the relative distance and rate. Both ways are right and get you to the answer.

For the first way let's use a table to help organize our information.

We can input what we know for their rates (Bob=35, Larry=45).

Time is what the question is asking, so it will be our variable: x. The trick here is to remember that Larry is traveling one hour less than Bob so we need to put (x-)1 in his box.

The formula for distance is D=rt, so we will multiply each rate times the time to get their respective distances.

Since they both left the same place and when they catch up to one another they will be at the same place, their distances are equal. So your equation sets up like this:

35x=45(x-1) now just solve

35x=45x-45

-10x=45

x=4.5

Be careful here. We need to remember what x was representing. It was telling us how long Bob was driving. The question asks how long it took Larry to catch up. Larry drove for one hour less than Bob so our answer is 3.5 hours.

For the second method, using relative rates and distances we need to understand what relative rates actually are. On the freeway when you are driving 65mph and pass someone going 60mph your relative speed is 5mph. Between the two of you that is the difference of speed. In our problem the relative speed between Bob and Larry is 10mph. (Bob=35, Larry=45; 45-35=10) So let's say that after Bob has driven one hour (the time when Larry leaves) Bob has driven 35 miles and then pulls over and sits still. Larry is now driving at 10mph to catch up to him. (Really, he is driving 10mph faster than Bob is.) If Bob is 35 miles ahead, how long will it take Larry to drive the 35 miles going 10mph? Distance=rate * time.

35=10t

t=3.5

So, Larry drives for 3.5 hours to catch up to Bob.

You can use either method to get to the answer. Use whichever one makes the most sense to you.