GRE Math: Averages, Ratios, and Proportions, Study Guides, Projects, Research of Mathematics

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Chapter 7

Math in the Real World

TRYING TO RELATE

The most common critique against the GRE is that it tests topics that have no direct connection to the topics most graduate school students will use. So the Educational Testing Service, makers of the GRE, decided not to test too much abstract math. Instead, they try to make questions have actual, real-life connections.

Of course, for people who spend their entire day writing standardized tests, “real-life situations” means something very different than it does for most normal people. For the GRE, this is a real-world problem:

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Here’s How to Crack It A typical, real-world situation that we’ve all been in. You save and save, but then forget how much you have in your account. When you ask your bank how much you have, you are informed that you put $227.50 more in your account this month than you did three months ago. “But how much do I actually have in my account?” you ask. The bank teller types some numbers into her computer, answers “you put double the amount into your account each month for the past four months. Since that answers your question, thank you for banking with Oblique Bank, and remember to sign up for an Obfuscation Checking Account,” and hangs up.

You mean you’ve never been in that situation? But it’s a perfectly common situation.

Okay, so it’s not a common situation at all. However, it is a common style of GRE question, and one that you may have seen before. Notice all those numbers in the answers, and how the question is asking for a specific amount? You may have recognized the opportunity to PITA. If so, good for you. If not, feel free to look over Chapter 5 to learn more about Plugging In the Answers.

Write down A B C D E on your scratch paper, copy the answers, and label the column “March.” Start with (C). If she saved $97.50 in March, then she saved double that in April ($195), doubled again in May ($390), and doubled once more in June ($780). The amount she saved in June was ($780 – $97.50) = $682.50 more than what she saved in March, which is way more than $227.50. Because (C) is too big, cross off (C), (B), and (A).

Try (D). If she saved $65 in March, she saved $130 in April, $260 in May, and $520 in June. She therefore saved ($520 – $65) = $455 dollars more in June than in March, which means (D) is too large as well. Cross off (D), and pick (E), the only answer left.

Fictional World Examples

So the questions on the GRE won’t actually apply to the real world. However, they’ll frequently use certain real-

All average problems involve three quantities—the Total value, the Number of things, and the Average value of those things. You can relate them in a diagram we call the Average Pie, which looks like this:

The Average Pie helps you visualize the relationship between the three numbers. It also helps you organize your thoughts by giving you three discrete compartments in which to put your information.

In order to solve the previous problem, you would add up the elements to get 100, recognize that there were five numbers, and place that information in the Average Pie like this:

When you divide 100 by 5, you see that the answer is 20.

Try another example:

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Here’s How to Crack It Here you have two of the three elements that go in the Average Pie. You know the Average and the Number of things, and you’re looking for the Total, so set up your Average Pie like this:

To find the Total, multiply the two bottom numbers: 6 × 12.6 = 75.6 million, which is slightly larger than the 75 million in Quantity B. The answer is (A).

Averages Quick Quiz

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Farmer Jones can expect to get 21 × 255, or 5,355 eggs over the course of a year.

3. This time, you know the total and the average, so the Average Pie looks like this:

The number of hens equals 572 ÷ 22, or 26 hens. The answer is (C).

4. Average is used three times, so set up three Average Pies and then start filling in what information is known to solve for what is not. If 12 hens lay an average of 9 eggs apiece, then the total number of eggs is 12 × 9, or 108. Once the top three egg-layers are disqualified, there are 9 hens left that lay an average of 7 eggs. They account for 7 × 9, or 63 of the eggs. The top three layers must therefore account for 108 − 63, or 45 eggs. The average value among the top three is 45 ÷ 3, or 15.

Because 15 is greater than 11, the answer is (A).

Sequences: A Helpful Hint

The GRE likes to make certain average questions seem more difficult and time-consuming than they are by having them involve huge sequences of numbers. The good news is that if the elements in a list are evenly spaced, there’s a lot less work involved than you might think.

The average of any sequence of evenly spaced elements is either

• the middle number (if the number of elements is odd); or
• the average of the middle two numbers (if the number of elements is even).

Quantity A

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Here’s How to Crack It The decoy answer is (B), because it looks like 21 numbers would lead to a greater answer than 20 numbers. Keep in mind that because there are no variables, you can eliminate (D).

The first 20 even numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, and 40. Once you list them all out, you might panic at the thought of having to add them all up and divide by 20. However, these numbers are evenly spaced, and there are 20 of them. Therefore, the average value is the average of the middle two numbers, 20 and 22. This average is 21.

Calculate Quantity B in a similar way. There are 21 terms in the sequence, so the middle number, the 11th, is the average. If you count along the sequence of odd numbers, the 11th number is 21. Therefore, the answer is (C).

The Median

As you know, the median of a list of numbers is the middle value when the numbers are placed in order of increasing size. One of the most common places to find median values is in a grad-school brochure, which often displays its “median” GRE score.

Once again, the number of elements in the list is important. Once you’ve ordered them from least to greatest, the median will be either the middle value (if the number of elements is odd) or the average of the middle two values (if the number of elements in the set is even).

Explanations for Three M’s Quick Quiz

1. The sum of all of the elements in List G is 4 + 8 + 5 + 9 + 8 + 3 + 8 + 5 + 4, or 54. There are nine elements, so the average value is 54 ÷ 9, or 6.
2. When the elements are arranged in order, List G looks like this: (3, 4, 4, 5, 5, 8, 8, 8, 9). The middle value is 5, so the answer is (C).
3. The element that occurs most often is 8, which is greater than the median (5). The answer is (A).
4. The least value in List G is 3, and the greatest value is 9. Therefore, the range of the list is 9 − 3, or 6.
5. The addition of a tenth number to the list means that the new median is the average of the fifth and sixth elements. Any number that is less than or equal to 5 makes both the fifth and sixth elements 5, effectively keeping the median at 5. Therefore, any of the first four values (1, 3, 4, and 5) works.

RATIOS

Ratios are a lot like fractions and decimals, with one important difference: Fractions and decimals compare parts to the whole, while ratios are more concerned with comparing two or more parts that together don’t necessarily represent the whole. Most of the time, ratios are denoted with a colon, as in “the ratio of boys to girls in the classroom was 4 : 3.” This means that for every four boys in the room, there were three girls. The actual number of boys is therefore a multiple of 4, the number of girls is a multiple of 3, and the number of children is a multiple of 7.

Most of us have been trained to use algebra when solving ratio questions, but the Ratio Box lets you throw algebra out the window.

The Ratio Box

Rather than deal with variables when you encounter a ratio problem, you can use the Ratio Box to organize your data in nice little columns:

The Ratio Box is a great tool because it lets you compare the parts within the whole at a glance and it clearly relates the ratio (along the top row) to the actual number of elements you have (the bottom row). Here’s how to use it.

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Here’s How to Crack It Set up your Ratio Box, label your Parts columns, and enter all the information you know:

Notice that the first thing to do with the numbers in the ratio row is to put their sum in the far-right column. Now, from the top row, you know that for every 9 cars, 7 of them are hardtops and 2 are convertibles.

The next step is to make the connection between the ratio of the cars and the actual number of cars, which are separated by a multiplier. The link is in the convertible column; there are 16 actual convertibles, and the ratio value is

2. Therefore, the multiplier for the whole box is 16 ÷ 2, or 8. Enter 8 across the entire multiplier row, like this:

and you get 11.2 pounds. The answer is (D).

Ratios Quick Quiz

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Explanations for Ratios Quick Quiz

1. When you fill in the Ratio Box with the given ratio, the sum of the two parts is 7.

Therefore, the number of babies in the maternity ward must be a multiple of 7. Each of the answer choices except 38 is a multiple of 7. The answer is (C).

2. If the ratio of tulips to daffodils is 3 : 5, your Ratio Box looks like this:

Because the total is 8, the fractional amount of daffodils is 5 out of 8, or. Because is equivalent to

62.5%, the answer is (D).

3. Line up the ratios like this, so that you can compare all three parts at once.

There are two separate ratios listed here, and the only way to compare line cooks and busboys is to relate them to a common number of waiters. If you multiply the two numbers you have for waiters, you get 12— a lowest common denominator. From here, you can convert the ratio of line cooks to waiters from 2 : 3 to 8 : 12, and the ratio of waiters to busboys from 4 : 3 to 12 : 9. This makes the ratio of line cooks to busboys 8 : 9. There are more busboys than line cooks, so the answer is (B).

4. When you fill in the Ratio Box, the value in the Whole column of the ratio row must be a factor of 120. Because 6 + 1 = 7, and 7 is not a factor of 120, the answer is (D).

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Explanations for Proportions Quick Quiz

1. Set up the proportion that compares cups of flour to muffins.

When you cross-multiply, your equation becomes 25 x = 80, and x = 3.2. Because this is equivalent to 3 ,

the answer is (D). Note: Just because you can use your calculator doesn’t mean you should forget all about

fractions, which can still appear in answer choices.

2. First, an important conversion: Because one yard equals three feet, one square yard is the same as nine square feet. Therefore, if each bag fertilizes 10 square yards, it fertilizes 90 square feet. Set up your proportion, comparing bags to square feet of coverage:

When you cross-multiply, 90 x = 12,000 and x = 133.3. Because he has to buy full bags, he must buy 134 for complete coverage. The answer is (C).

3. You can find the value of the number, but you really don’t need to. Just set up the proportion that compares percentages to numbers.

From this proportion, you can cross-multiply to find that 75 x = 12,000, and x = 160. The answer is (C).

4. Even though the quantities are proportional, there’s no need to set up a proportion. If Arturo wants to save 3.5 times the amount, he’ll need 3.5 times more time. When you multiply 8 by 3.5, you get 28 months. The two quantities are equal, so the answer is (C).

Naturally, statistics can get a little more complicated, but we’ll wait until Chapter 9 to delve into the world of combinations, probability, and standard deviation. In the meantime, here are some more practice questions about ratios, proportions, and real-life math situations.

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