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**File comment:** Check out the rate problem!
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When we start reading this problem, the first part that seems important is this: „speed of 4 mph“. This phrase tells us that we deal with a problem that has something to do with rates. Why do we know that? Easy rates problems tell you explicitly that they are rates problems by including some phrase like “apples per hour”, “words per minute” or “gallons per second”. You have some kind of quantity per some kind of time. In our problem, this quantity is a little more hidden because mph = miles per hour. Here we go again: quantity per time!

Now is the time when you start writing “R x T = D” on your paper because you know that you will have to calculate one of the parameters. The next sentence tells us that there is a horse B with the rate of 8 mph that goes into the OPPOSITE direction of horse A. Do not worry about the 5 hours so far because we still do not know what to do with it and it just requires too much brain power to remember the details about it.

A MUST-DO HABIT

When you start reading the last sentence, stop after this part: “For how many hours will horse A be running”. This is the time when you write “T of A = ?” on the top right corner of your paper and circle it. This is a top priority habit that you absolutely should obey! Always write what the question is asking for on the top right corner of your paper and before you answer, check if your result matches the question.

This has two positive effects: 1. When you write down what you have to find out, you focus much more on solving this particular problem rather than trying some formulas and seeing what comes out.

2. Before you answer your question, always check if your answer matches with the question you wrote on the top right of your paper. Did you calculate the correct part of the RTD chart (work, rate or time)? Did you do it for the correct person, horse, pipe, etc (horse A or horse B)?

The reason why I stress this point is that about 30 % of wrong answers in this kind of question come from oversight. In this example, test taker force you to calculate the time T for horse B first because it is much easier. Many test takers forget that they should have to calculate time T for horse A and go straight to the answer choices. Since the test takers know that test takers will make this mistake, they included a wrong answer that matches the time T for horse B. Of course test takers are happy to see their calculated solution and they pick the wrong answer.

To sum up, we have two horses running in a circular track towards each other. Horse A starts earlier than horse B and will therefore have run more miles than horse B but horse B is faster than horse A. The horses have to pass each other twice and put another 12 miles between them. How long does it take horse A to do that?

Remember that the question tells us that the arena is circular. When the test takers add such kind of detail that specifies the shape of an item, it is ALWAYS important. By this time, you should draw a picture of the arena and mark the directions in which the horses run, to better understand the problem.

IMPORTANT CONCEPT EXAMPLE

We will approach things chronically. This first that happened is that horse A starts running with 4 mph for five hours - nothing else happened. You can insert these values in a RTD chart and calculate the distance D that horse A has run so far: 4 x 5 = 20. Horse A has run 20 miles before horse B even starts running.

The next question is: When will they meet each other for the first time? To answer this question, I will introduce you to a very important concept. Imagine, your favorite friend and you stand across each other with a distance of exactly 10 meters. Let us assume that each of you two’s step length is exactly 1 meter. If both of you make a step towards each other, how far away do you stand? The answer is 8 because each of you shortened the distance by 1 meter. If each of you makes another step forward, the distance will be only 6 meters.

If a question would be like this: A and B are standing across each other with a distance of 10 meters. Each of them starts at the same time moving towards the other with a rate of 1 meter per minute. After how many minutes do they meet? After one minute, the distance is 8. After two minutes the distance is 6 and so forth. You just have to add the two rates: 1 + 1 = 2. This tells us with which rate you move TOWARDS each other. After each minute, the distance gets shortened by 2.

R x T = D

2 x T = 10

T = 10/2

T = 5

After 5 minutes you will meet each other!

IMPORTANT CONCEPT APPLIED TO THIS PROBLEM

When look at this problem, the situation is very similar to the one with your friend: You have to horses running towards each other. It does not matter that they run in a circular area in terms of rates. They could also run slalom and it would not influence the rates. However, the track has of course influence on the distance D calculation - but we come back to that in a minute.

If we add the rate of horse A and horse B, we see that they are moving TOWARDS each other with a rate of 12 mph. Next we need to calculate the distance they have to run. We know that the arena is circular, so we have to calculate the circumference of a circle. The formula is 2 x radius x π. In our case this means: 2 x 12 x π = 24 π.

If the horses would both start at the starting point at the same time and run towards each other until they meet once, this would be the distance to cover. However, our horses have to pass each other twice => we have to multiply the circumference by 2 = 48 π. Furthermore, our horses have to pass each other twice AND put another 12 miles between them. So, we have to add 12 miles: 48 π + 12. Now comes the tricky part that is easy to forget: We are currently calculating the distance of the arena that BOTH horses are running TOWARDS each other. So, both horses have to run at the same time. That way they move to each other with the rate of 12 mph, we calculated earlier.

However, for the first five hours, horse A was running alone! So, by the time BOTH horses are starting to run, horse A is already 20 miles closer to horse B. Therefore, we have to subtract these 20 miles: 48 π + 12 - 20 = 48 π - 8.

To make things simpler, consider this: When horse B starts running, the horses have to cover less distance than the whole circumference. So, to meet for the first time the need to cover the distance of one circumference minus the distance that horse A is closer to horse B: 24 π - 20. After that they have to go the whole circumference to meet each other. Of course, horse B goes more miles in this round than horse A because it is faster. But this does not matter because we are considering the rate they approach each other, not singularly. Even if horse A stops after the first meeting, horse B would have to go the full circumference. The faster horse A moves, the less distance horse B has to cover.

In the second round, they have to cover the full distance of 24 π. After this second meeting, they still have to put another 12 miles between them, as the question demands: 12.

In total this means the following:

Round 1: 24 π - 20

Round 2: 24 π

Round 3: 12

Total: 48 π -8

Insert this value in the RTD chart:

R x T = D

12 x T = 48 π - 8

T = (48 π - 8)/ 12

T = 4 π - 2/3

DO NOT JUMP TO THE ANSWER CHOICE NOW

Check the right corner of your paper. Did we calculate the number of hours from the perspective of hors A or B? Right, from horse B’s perspective. Why? Because we assumed that horse A just happened to be 20 miles closer in round one for some reason. That is ok from the perspective of horse B - it does not care how horse A came to this spot or why!

But from horse A’s perspective, you have to give it credit for the extra miles it did in the five hours when horse B was still having a nap. When horse B does not care how horse A came 20 miles closer, horse A of course did invest extra time to get there. You have to add this time to the time we calculated earlier. It is also logical, right? Horse B started 5 hours later, so of course it took horse B exactly this amount of time less to meet because horse A has done all the work.

So, we just need to add 5 hours: 4 π - 2/3 + 5 = 4 π + 4.3!

Notice that if we had not checked our top right corner, we might have went for the wrong answer A when in fact answer B is the correct one!

The correct answer is B!
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