Speed and distance problems are among the most complained about problems on the GMAT. Numerous clients come to us and say they have difficulty with speed and distance problems, word problems, or work rate problems. So we’re going to look at a particularly difficult one and see just how easy it can be with the right approach.
1. The Two Cars Problem
Two cars are travelling in the same direction. Car A is travelling at a speed of 58 miles per hour and car B is travelling at 50 miles per hour. If car A is 20 miles behind car B, how long will it take car A to pass car B by 8 miles?
A) 2 hours
B) 1½ hours
C) 4 hours
D) 5½ hours
E) 3½ hours
In this problem, we have two cars – car A and B. Car A begins 20 miles behind car B and needs to catch up. Our immediate DSM (Default Solving Mechanism) is to dive in and create an equation for this and that’s exactly what we don’t want to do.
These types of problems are notorious for being algebraically complex, while conceptually simple. If you hold on to the algebra, rather than getting rid of it, you’re going to have a hard time.
2. Speed and Distance Problems – Solution Paths
In this problem, we’re going to build up solution paths. We’re gonna skip the algebra entirely. We’re going to take a look at an iterative way to get to the answer and then do a conceptual scenario, where we literally put ourselves in the driver’s seat to understand how this problem works. So if we want to take the iterative process we can simply drive the process hour-by-hour until we get to the answer.
3. Iterative Solution Path
We can imagine this on a number line or just do it in a chart with numbers. Car A starts 20 miles behind car B – so let’s say ‘A’ starts at mile marker zero and ‘B’ starts at 20. After one hour ‘A’ is at 58, ‘B’ is at 70 and the differential is now -12 and not -20. After the second hour ‘A’ is at 116, ‘B’ is at 120. ‘A’ is just four behind ‘B’. After the third hour ‘A’ has caught up! Now it’s 4 miles ahead. At the fourth hour it’s not only caught up but it’s actually +12, so we’ve gone too far. We can see that the correct answer is between three and four and our answer is three and a half.
Now let’s take a look at this at a higher level. If we take a look at what we’ve just done we can notice a pattern with the catching up: -20 to -12 to -4 to +4. We’re catching up by 8 miles per hour. And if you’re self-prepping and don’t know what to do with this information, this is exactly the pattern that you want to hinge on in order to find a better solution path.
You can also observe (and this is how you want to do it on the exam) that if ‘A’ is going 8 miles an hour faster than ‘B’, then it’s catching up by 8 miles per hour. What we care about here is the rate of catching up, not the actual speed. The 50 and 58 are no different than 20 and 28 or a million and a million and eight. That is, the speed doesn’t matter. Only the relative distance between the cars and that it changes at 8 miles per hour.
Now the question becomes starkly simple. We want to catch up 20 miles and then exceed 8 miles, so we want to have a 28-mile shift and we’re doing so at 8 miles an hour. 28 divided by 8 is 3.5.
4. Speed and Distance Problems – Conceptual Scenario Solution Path
You might ask yourself what to do if you are unable to see those details. The hallmark of good scenarios is making them personal. Imagine you’re driving and your friend is in the car in front of you. He’s 20 miles away. You guys are both driving and you’re trying to catch up. If you drive at the same speed as him you’re never going to get there. If you drive one mile per hour faster than him you’ll catch up by a mile each hour. It would take you 20 hours to catch up. This framework of imagining yourself driving and your friend in the other car, or even two people walking down the street, is all it takes to demystify this problem. Make it personal and the scenarios will take you there.
Thanks for the time! For other solutions to GMAT problems and general advice for the exam check out the links below. Hope this helped and good luck!