Math Workout for the GRE, 3rd Edition (2013)

Chapter 6. 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:

Question 11 of 20

Rhoda deposited money into her savings account for four consecutive months: March, April, May, and June. For each of the last three months, the amount she saved each month was double the amount of the previous month. If the amount she saved in June was $227.50 more than the amount she saved in March, how much did she save in March?

$260.00

$130.00

$97.50

$65.00

$32.50

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 4 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).

Let’s 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 answer choice (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-world topics. You should know exactly what to do when any of these topics come up, so you don’t get lost in all the extra word problem garbage.

As always, focus on learning what you need to recognize in a problem to know what to start writing on your scratch paper.

THE THREE M’S

When it comes to crunching numbers, there are three statistical terms that every GRE student should recognize and distinguish from each other.

· The mean, or “arithmetic mean,” is just another word for the average.

· The median is the middle number in a list of numbers.

· The mode is the term that occurs most frequently in a list of numbers.

To remember these last two terms, you can think that (1) the median of a highway is in the middle of the highway, and (2) if you say the word most as if you have a terrible head cold, it comes out sounding like mode.

One final term you should know is range; the range of a set of numbers is the difference between the greatest and smallest numbers in the set.

The Mean (Average)

You calculate the average value of a list of numbers by finding their total value and dividing by the quantity of numbers in that list. To find the average of 12, 29, 32, 8, and 19, for example, add them all up (12 + 29 + 32 + 8 + 19 = 100) and divide by the number of terms (five). The answer is , or 20.

Question 9 of 20

A basketball player scores 12 points during her first game, 29 during her second, 32 in her third, 8 in her fourth, and 19 in her fifth game. What was her average (arithmetic mean) score per game during that week?

14

17

20

25

33

Here’s How to Crack It

From our previous work, we know that the answer is (C). However, it’s worth pointing out that you can eliminate a few answer choices right away. In particular, choice (E) stands out, because the average of a list of numbers cannot be greater than the greatest number in the list. To solve problems like these, you can use the Average Pie.

The Average Pie

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

Trigger: The word
“average.”

Response: Draw an
average pie for every time
the word average appears
in the question.

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:

 Question 14 of 20

During the six weeks between Thanksgiving and New Year’s Eve, an average (arithmetic mean) of 12.6 million people rode mass transit in City X each week.

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

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, 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

Question 1 of 4

Farmer Brown took note of the quantity of eggs produced by the hens on his farm and compiled the chart above. What was the average (arithmetic mean) number of eggs that his hens produced each day?

43

44

45

46

47

Question 2 of 4

On average, each laying hen lays 255 eggs per year. If Farmer Jones has 21 hens, how many eggs can she expect her hens to lay over the course of a year?

Question 3 of 4

During a particularly productive month, Farmer Green’s hens laid a total of 572 eggs. If each hen laid an average (arithmetic mean) of 22 eggs that month, how many hens did Farmer Green have that month?

18

22

26

32

45

Question 4 of 4

Over a certain month, the average (arithmetic mean) of eggs laid among 12 hens was 9. If Farmer Dobbs removes the eggs laid by the top three producers, the average of the remaining hens is 7.

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Explanations for Averages Quick Quiz

1. During the seven days, the hens produced a total of 301 eggs. The Average Pie, therefore, looks like this:

The average value equals 301 ÷ 7, or 43. The answer is (A).

2. You know the number and the average, so set up your Average Pie like this:

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).

Question 8 of 20

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

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 listed 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.

Trigger: The word
“median” appears
in the problem.

Response: Put the list of
numbers in order and find
the middle number.

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

Question 12 of 20

Set D = {the first 15 positive integers}

Set F = {the prime elements in Set D}

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Here’s How to Crack It

Set D contains 15 elements (numbers 1 through 15), so the median is the middle number, or 7. The prime elements in Set D are 2, 3, 5, 7, 11, and 13, so the median is the average of the middle two numbers, 5 and 7. Their average is 6, so the answer is (A).

Three M’s Quick Quiz

List G = (4, 8, 5, 9, 8, 3, 8, 5, 4)

Question 1 of 5

What is the average (arithmetic mean) of List G ?

Question 2 of 5

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 3 of 5

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 4 of 5

What is the range of List G?

Question 5 of 5

The addition of which of the following numbers leaves the median of List G unchanged?

Indicate all such numbers.

1

3

4

5

6

7

9

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 fifth and sixth elements both 5, effectively keeping the median at 5. Therefore, any of the first four values (1, 3, 4, and 5) works.

RATIOS AND PROPORTIONS

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:

Trigger: The word “ratio”
appears in the problem.

Response: Draw a Ratio
Box on your scratch paper.

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:

Question 7 of 20

In a certain used-car showroom, the ratio of hardtop cars to convertibles is 7 : 2. If there are 16 convertibles, how many cars are in the showroom?

2

7

9

56

72

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, we 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:

Finish the box by multiplying down each of the other columns.

We now know that there are 56 hardtops and 16 convertibles, for a total of 72 vehicles so the answer is choice (E).

It’s Expandable!

Not all ratio questions have just two parts, and you can expand your Ratio Box to include as many parts as necessary. Here’s how that works:

Question 9 of 20

A contractor mixes cement, water, sand, and gravel in a ratio of 1 : 3 : 4 : 2 by weight to make concrete. How many pounds of sand are needed to make 28 pounds of concrete?

2.8

5.6

8.4

11.2

13.6

Here’s How to Crack It

The formula has four parts, so create a column for each one, like this:

The total is 1 + 3 + 4 + 2, or 10 parts. If the actual amount is 28 pounds, then the multiplier is 28 ÷ 10, or 2.8. Don’t bother filling in every column; all you need is the amount of sand. Place the 2.8 in the sand column, multiply by 4, and you get 11.2 pounds. The answer is (D).

Ratios Quick Quiz

Question 1 of 4

The ratio of boys to girls in a hospital maternity ward is 3 : 4. Which of the following CANNOT be the number of babies in the ward?

28

35

38

42

63

Question 2 of 4

If there are only tulips and daffodils in a garden and the ratio of tulips to daffodils is 3 : 5, what percent of the flowers in the garden are daffodils?

37.5%

40%

60%

62.5%

133.3%

Question 3 of 4

At a certain restaurant, the ratio of line cooks to waiters is 2 : 3 and the ratio of waiters to busboys is 4 : 3.

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 4 of 4

If there are 120 workers in an office, each of the following could be the ratio of men to women EXCEPT

2 : 1

3 : 1

4 : 1

6 : 1

7 : 1

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 percent (use your calculator if you haven’t already memorized this), 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).

Proportions

Proportions are related to ratios, because a ratio between two elements is proportional to the actual values. If you know the ratio of two quantities and you have to extrapolate that ratio onto some actual values, you’ll probably end up writing a proportion.

The Setup

Proportions are comparisons of two things in a fixed ratio. There are two very important considerations to keep in mind when you set proportions up.

• Make sure your elements are consistent: If you’re comparing miles per hour and you decide that miles are in the numerator, make sure they’re always in the numerator.

• Make sure your units are consistent: If you’re comparing distances, and one distance is given in feet while the other is in inches, convert one of those distances so that the units are the same.

Once your proportion is ready, you can cross-multiply and solve for the missing value.

Question 5 of 20

If a train maintains a constant speed of 80 miles per hour, how far does the train travel in 24 minutes?

Here’s How to Crack It

The first ratio compares miles and hours, so your first instinct might be to set up a proportion like this:

WRONG:

The elements are aligned, but the units aren’t consistent because the question mentions 24 minutes, not hours. To solve this, convert 1 hour to 60 minutes, and you’ll be ready to cross-multiply:

60x = 80 × 24

60x = 1,920

x = 32

Proportions Quick Quiz

Question 1 of 4

A muffin recipe that calls for 2 cups of flour yields 25 muffins. If Yvonne needs to make 40 muffins, how much flour should she use?

1

2

2

3

5

Question 2 of 4

A greenskeeper wants to fertilize the eighteenth hole of a golf course, which has an area of 12,000 square feet. If the fertilizer he wants to use suggests using one bag for every 10 square yards, how many full bags are required?

48

133

134

266

1,200

Question 3 of 4

If 75 percent of a certain number is 1,200, then what is 10 percent of that number?

90

120

160

240

480

Question 4 of 4

Arturo saves $1,490 in 8 months.

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Explanations for Proportions Quick Quiz

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

When you cross-multiply, your equation becomes 25x = 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, 90x = 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 75x = 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, and the answer is (C).

Naturally, statistics can get a little more complicated, but we’ll wait until Chapter 8 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.

Math in the Real World Drill

Question 1 of 16

A machine punches x plates per hour for 4 hours and then y plates per hour for 2 hours. Which of the following is an expression for the average number of plates punched per hour by the machine for the entire 6 hours?

2x + 4y

4x + 2y

Question 2 of 16

Twenty bottles contain a total of 8 liters of apple juice. If each bottle contains the same amount of apple juice, how much juice, in liters, is in each bottle?

Question 3 of 16

If 12 equally priced melons cost a total of $9.60, then what is the cost of 9 of these melons?

$7.00

$7.20

$8.00

$8.45

$8.65

Question 4 of 16

If 2a = 3b = 4c = 72, then what is average (arithmetic mean) of a, b, and c ?

39

26

24

18

9

Question 5 of 16

Jenny notices a consistent, increasing pattern in the number of geese she sees flying over her house each day. If she sees 5 geese on Monday, 8 geese on Tuesday, 11 geese on Wednesday, and 14 geese on Thursday, and the next week starts on Sunday, on which of the following days of the next week will she see a prime number of geese if the pattern continues?

Indicate all such values.

Sunday

Monday

Tuesday

Thursday

Friday

Saturday

Question 6 of 16

Set B contains only positive, even integers. Which of the following could be the median of set B ?

Indicate all such values.

−2

0

1

3

3.5

4

Question 7 of 16

A car manufacturer has 2,992 forklifts, which is approximately one forklift for every 48.9 employees. Which of the following is the closest approximation, in thousands, of the number of employees employed by the manufacturer?

60

100

150

175

300

Question 8 of 16

A recipe for 4 loaves of bread requires cups of sugar. If Chris wants to make 2 loaves of bread, which of the following calculations yields the amount of sugar he needs?

Indicate all such values.

×

× 2

÷

÷ 2

Question 9 of 16

V is a sequence of numbers in which every term after the first two is the average of the two previous terms. If the first term x is 16 more than the second term y, then which of the following represents the fifth term of the sequence, in terms y ?

y − 2

y

y + 2

y + 4

y + 6

Question 10 of 16

Set A contains only even integers. Which of the following CANNOT be the median of set A ?

−2

−1

0

0.5

1

Question 11 of 16

A violinist needs 2 hours to tune a violin made in the twentieth century. To tune violins made before the twentieth century, the violinist needs twice as long, and for violins made after the twentieth century, she needs half as long. Which of the following groups of violins could she tune in 6 hours?

Indicate all such groups.

Two violins made before the twentieth century

Three violins made during the twentieth century

One violin made during the twentieth century, one made before, and two made after

Two violins made after the twentieth century, and one made before

Two violins made after the twentieth century, and two made during the twentieth century

Question 12 of 16

2g = 6h

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 13 of 16

If 6 students have an average (arithmetic mean) score of 88 on an exam, and one of those students scored a 93 on the exam, what is the average score on this exam for the other 5 students?

84

85

86

87

89

Question 14 of 16

Let H be a sequence, h₁, h₂, h₃ … h₄, such that each term after the first is two less than one-third of the previous term. If the fourth term in the sequence is 0, which of the following is the sum of the first and fifth term of the sequence?

−2

25

26

76

78

Question 15 of 16

A bag of jellybeans has red and yellow jellybeans in a ratio of c : d. If there are r red jellybeans in the bag, which of the following represents the number of yellow jellybeans in the bag?

d(c + r)

d(r − d)

Question 16 of 16

Miguel’s bowling team bowled a practice round in preparation for their upcoming league game. The team’s average (arithmetic mean) score for the practice round was 180. Miguel scored 190, Janice scored 200, and Thad scored 210. If no team member scored less than 165, and none of the remaining team members scored greater than 170, what is one possible value for the number of members on Miguel’s team? (Note: Bowling scores are always positive integers.)

EXPLANATIONS FOR MATH IN THE REAL WORLD DRILL

1. D

Plug In, and let x = 6 and y = 3. If the machine punches 6 plates per hour for 4 hours, you know that it punches 24 plates in that time. If the machine punches 3 plates per hour for 2 hours, you know that it punches 6 plates in that time. In 6 hours the machine punches a total of 30 plates. If you divide 30 by 6 you’ll get 5 plates per hour, your target answer. The only answer that matches the target is answer choice (D): = 5.

2. 0.4

Twenty bottles hold a total of 8 liters of juice. Each bottle contains the same amount, so you need to divide the amount of juice by the number of bottles. 8 ÷ 20 = 0.4.

3. B

Divide the total cost of $9.60 by 12 melons to get the per-melon cost of $0.80. Now calculate the cost of 9 melons at the same price per melon: 9 × $0.80 = $7.20.

4. B

Break the expression into three equations. 2a = 72, so a = 36. 3b = 72, so b = 24. 4c = 72, so c = 18. The average of 36, 24, and 18 is = 26. You could estimate that choice (E) is incorrect because the value of each of the three variables is greater than 9, and the average will be greater than 9.

5. A, C, and F

Once you deduce the pattern, you’ll be able to predict the number of geese for each day of the next week and select the prime numbers. The number of geese increases by 3 each day: If she saw 14 on Thursday, she’ll see 17 on Friday, 20 on Saturday, 23 on Sunday, 26 on Monday, 29 on Tuesday, 32 on Wednesday, 35 on Thursday, 38 on Friday, and 41 on Saturday. Sunday, Tuesday, and Saturday have a prime number of geese, so choices (A), (C), and (F) are correct.

6. D and F

If the set contains only positive integers, then there’s no way numbers 0 or less can be the median, so eliminate choices (A) and (B). For 1 to be the median, there has to be a number in the set less than 1, so eliminate choice (C). Don’t eliminate choice (D) just because it’s odd—remember that in a set with no unique middle term, the median is the average of the two middle terms; the set could be {2, 4}, for instance, which would have a median of 3. The same set could be extended to {2, 4, 6}, which would have a median of 4, or choice (F). Since 3.5 itself cannot be in the set, the only remaining question is whether 3.5 can be the average of two even numbers: it can’t, because 3.5 × 2 = 7, and two even numbers can’t add up to 7. Eliminate choice (E) and select choices (D) and (F).

7. C

Estimate. 2,992 is almost 3,000, and 48.9 is almost 50. The number of employees is approximately 3,000 × 50, or 150,000. Only choice (C) is close.

8. A and D

Because Chris is only making 2 loaves, he needs half as much sugar as he does for 4 loaves. You can multiply by to find how much sugar he needs, or you can divide the quantity by 2.

9. E

See variables in the answer choices, Plug In. Start by choosing your y. x is 16 more than y, so let y be some small positive integer like 2. That makes x = 18. Average the two together, and the third term in the sequence (remember, xand y are the first two terms of the sequence) is 10. Take the second and third terms, average them together, and you get 6 ((10 + 2) ÷ 2 = 6) as the fourth term. Do it one more time to get 8 ((6 + 10) ÷ 2 = 8) as the fifth term. Plug In to your answer choices, and eliminate all but choice (E).

10. D

Come up with possible sets that could have the medians of the answer choices. Choices (A) and (C) are eliminated because −2 and 0 are even numbers and you could easily have a set with those choices as the center number. Both −1 and 1 are a bit more difficult to eliminate, but keep in mind that if your set has two middle terms, rather than just one, the median of the set is the average of those two middle terms. So, for a median of −1, the set could be −2 and 0; for a median of 1, the set could be 0 and 2. Eliminate choices (B) and (E). Select your only remaining choice, (D).

11. B, D, and E

First, figure out how long it will take the violinist to tune each type of instrument. Violins made before the twentieth century take twice as long to tune, so they take 4 hours. Violins made after the twentieth century take half as long to tune, so they take 1 hour. Add up the total time needed to tune the violins in each answer choice. The total time for answer choice (A) is 4 + 4 = 8 hours, so choice (A) can be eliminated. For answer choice (B), the total time is 3 × 2 = 6 hours. Choice (B) is correct. For answer choice (C), 2 + 4 + 1 + 1 = 8 hours. Choice (C) can be eliminated. For answer choice (D), 1 + 1 + 4 = 6 hours. Choice (D) is correct. For answer choice (E) 1 + 1 + 2 + 2 = 6 hours. Choice (E) is correct.

12. A

Since there are variables involved, this is an opportunity to Plug In. First, simplify the equation by dividing both sides by 2 to get g = 3h. Plug in a number for h and solve for g to see the ratio. For example, if you Plug In 1 for h, then g = 3. Quantity A asks for the ratio of g to h, which is now 3 to 1. Quantity A is greater.

13. D

The average for 6 students is 88. Set up an Average Pie and use 6 for the number of things and 88 for the average. Therefore, the 6 students scored a total of 6 × 88 = 528 points. One student scored a 93, so the remaining 5 students scored a total of 528 − 93 = 435 points. Set up another Average Pie, using 435 as the total and 5 as the number of things. Divide 435 by 5 to find that these 5 students scored an average of 87 points. The answer is choice (D).

14. D

With sequence questions, PITA typically works well. This problem makes that a little more difficult by asking you for a sum of two terms, rather than the term itself (i.e. you would still have to guess about the terms, even if you attempted to begin at a certain answer choice and work from there). You are given the fourth term, so try to move back and forth from there. Each following term is one-third of the previous term minus 2. So, of 0 equals 0, and 0 − 2 equals −2, your fifth term. To find the previous terms, simply invert the sequence: add 2, then multiply by 3. The third term is (0 + 2) × 3 = 6, the second term is (6 + 2) × 3 = 24, and the first term must be (24 + 2) × 3 = 78. Add together your first and fifth terms: 78 + −2 = 76, which is choice (D).

15. B

Use the ratio box and Plug In some values. If c = 2, d = 3, and r = 20, then there will be 30 yellow jellybeans. Circle 30 as your target and check all five choices. Choices (A) and (C) are way too small, but choice (B) works. Choices (D) and (E) do not match, leaving choice (B) as your correct answer.

16. 7, 8, or 9

Use the Average Pie. Label the number of additional team members t. Assume that all of these t people scored 165. These t people therefore scored a total of 165t points. Miguel, Janice, and Thad scored a total of 600 points. Therefore the total number of points scored by the team is 165t + 600. Put this in the top segment of the pie. The total number of people on the team is t + 3, which should go in the bottom left segment. Put the average, 180, in the bottom right segment. At this point you can see that = 180 or total divided by number of things equals average. Solve algebraically. First, 165t + 600 = 180t + 540. Subtract 540 from both sides to find that 165t + 60 = 180t. Subtract 165t from both sides to find that 60 = 15t. Therefore, t = 4. Remember that t is the number of additional people on the team, and the question asks for total people, therefore if t = 4, there are 7 people on the team. The problem also gives you an integer for t if you use 170 (t = 6, and the total number of team members is 9) and 168 (t = 5, and the total number of team members is 8).