The key to mastering these kinds of arithmetic questions is to learn simple, effective ways to organize your information. ETS will always give you just enough information to figure out the one piece that is missing. A good set-up will help you fill in the missing pieces quickly and easily.
Once you understand how the set-ups work, you need only train yourself to recognize the opportunity and use them. Think of words such as average and probability as triggers that provoke a very specific action. Sensitize yourself to these words and once you see them, before you’ve even finished reading the question, start making your set-up.
Known to ETS as arithmetic mean and to the rest of us as average, these problems can be time-consuming if you don’t know what you’re doing, but will unravel easily when you do. For example, to find the average of five, seven, and nine, add the three numbers together and divide by three. Thus, averages consist of three parts, the average, the number of things, and the total. The minute you see the word average in a problem, draw your pie.
When you see the word AVERAGE make a pie on your scratch paper. If you see the word AVERAGE again, make another pie.
If ETS were to give a list of numbers and ask for the average, it would be too easy. While ETS will always give you two out of the three pieces, they probably won’t be the pieces you expect. It may give you the average and the total and ask for the number of things, or it may give the average and the number of things and ask for the total.
Fill in the information you have.
If you have the number of things and the total, you will divide to get the average. If you have the average and the total, you will divide to get the number of things. If you have the number of things and the average, simply multiply to get the total.
If asked to find the average of five, seven, and nine, your scratch paper would look like the image shown below.
Of course, it’s not usually quite that simple. ETS may give you the average of one group, the total of a second, and then ask for the average of both combined. Just make sure that you draw a new pie every time you see the word average. Work the problem through in bite-sized pieces, read with your finger, and make sure your hand is moving on the scratch paper.
Rate problems work the same way that average problems do. In fact, you can use the same method to organize your information.
This is what a Rate Pie looks like.
The first thing you do when you see a rate problem is to make your Rate Pie. ETS will always give you two of the three pieces of information. You will have to find the third. If you’re asked for time, divide the distance or amount by the rate. If you’re asked for rate, divide the distance or amount by the time, and if you’re asked for distance or amount, multiply the time by the rate. Make sure to keep an eye on your units. You may be given a rate in miles per hour but asked for a number of minutes.
The way to prevent units errors is to use your scratch paper and label everything.
When you see the word median, find a group of numbers and put them in order. Median, like the median on a highway, simply means the number in the middle. It’s not a difficult concept, so there are only two ways ETS can try to mess you up. The most common trick is to give you numbers out of order. Your first step must always be to put the numbers in order on your scratch paper.
When you see the word MEDIAN, find a group of numbers and put them in order.
The second trick they may try is to give you an even number of numbers. In this case, the median will be the average of the two numbers in the middle. In the case of 2, 2, 3, 4, 5, 5, 5, 6, 7, 7, 120, 345, 607, the median is 5. In the case of 2, 2, 3, 4, 5, 5, 5, 6, 7, 7, 120, 345, 607, 1250, the median is 5.5.
Mode means the number that comes up most often. The mode of the set {4, 6, 6, 13, 14, 21} is six. The range is the difference between the highest number and the lowest. In this case it is 17, or 21 – 4. Rarely will you see a problem testing mode by itself. It is more likely to come up in connection with mean, median, and/or standard deviation.
There are not a lot of standard deviation questions in the question pool, so they don’t come up that often. However, because they might come up, you need to know how to handle them. But don’t worry, on the GRE, ETS sticks to the basics. You will never need to know how to calculate standard deviation. You will only be asked about percentages of people or things that fall a few standard deviations from the norm.
Imagine you measured the weight of all apples picked at Orchard X. Suppose the average weight of an apple is 6 ounces. As you can imagine, the vast majority of those apples will weigh somewhere close to 6 ounces. A much smaller number will be about 7.5 ounces, and you may even get a few that are heavier than eight ounces. The weight of these apples is likely to follow a normal distribution, which means that if you graphed the number of apples at each weight on a bar graph, you would end up with a bell curve.
This chart is the bell curve. It will never change. Memorize the numbers 34, 14, and 2.
The minute you see the words STANDARD DEVIATION, or NORMAL DISTRIBUTION, draw your bell curve and fill in the percentages.
On this curve, the mean, the median, and the mode are all the same. It makes sense, right? The average weight of our apples is also the most common weight and falls in the middle of the pack. If the apples have a standard deviation of 1.25 ounces, 34 percent of the apples picked in the orchard weigh between 6 and 7.25 ounces, 14 percent weigh between 7.25 and 8.5, and only 2 percent weigh more than 8.5 ounces. As you move from one percentage group to another you are moving one standard deviation from the norm. If you’re asked “What percentage of apples weighs more or less than two standard deviations from the norm?” the answer will be 4 percent.
Probability, on the GRE, can be defined as 
. It’s a fraction and the number of things you could get is the total. The minute you see the word probability, make your divisor line and find your total. Once you’ve done this, you are already half way to the answer.
The minute you see the word PROBABILITY, make your divisor line on your scratch paper and find your total.
Imagine you have a sock drawer that has 12 blue socks and 8 green socks. What is the probability that, when you reach into the drawer, you get a blue sock? Make your divisor and find your total. On the bottom you have 20 because there 20 socks you could get. On top you have 12 because there are twelve socks (blue) that you want. The probability is 
or 
. The probability of getting a green sock is 
or 
. The probability of getting any sock is 20 things you want over 20 things you could get, or 1. The probability of getting a ham and cheese sandwich is, we hope, 0 ( 
). It is important to note that probability is always between 1 and 0. The chance that something will happen added to the chance that it won’t happen will always add up to 1.
If two events are to occur, the probability of them both happening is equal to the probability of the first happening multiplied by the probability of the second happening. This makes sense because a fraction times a fraction equals a smaller fraction. If you have a very low probability of one event occurring and a very low probability of a second event happening, the odds of them both happening will be even lower. The probability of getting a green sock in the
drawer above is . The probability of getting a green sock the second time is 
, because there are seven green socks left, after you’ve removed the first one, and 19 socks left in the drawer. The probability of getting a green sock both times is 
, or 
.
Imagine you now have five purple socks in your drawer. If you are asked to find the probability of getting a purple OR a green sock, you have to add the probabilities. With 12 blue socks, eight green socks, and five purple socks, your new total is 25. You have an 
chance of getting a green sock and a 
chance of getting a purple one. The chance of getting one or the other is 
or 
.
The one last wrinkle to look at is what happens if you are asked to find the probability of at least one event happening. When rolling dice, for example, what is the probability that you roll 1 at least once out of three rolls? This will get complicated because at least one means that the event could occur once, twice, or even three times. That’s more calculating than you want to do. Instead, when asked to find at
least one, find the probability that none will occur and subtract it from 1. This will leave you with at least one. In this case, the chances of not rolling a one on the first roll are 
. The chances on the second and third rolls are the same. Therefore the chances of not rolling a one go down with each additional roll, but only by a little bit because you have a very strong possibility that it will not
happen. The chances that you will not roll a 1 in your first three rolls are 
, or 
. The chances that you will roll at least one 1, therefore are 
(216 – 125 = 91).
For more practice and a more in-depth look at math techniques, check out our student-friendly guidebook, Cracking the GRE.