Math Workout for the GMAT

Part III Content and Strategy Review

Chapter 8 Data Sufficiency 2

In Chapter 1, you learned about the data sufficiency format and the “AD or BCE” method that you should use to tackle it. In this chapter, you will see some more sophisticated techniques that will help you with specific types of data sufficiency questions.

SIMULTANEOUS EQUATIONS

In Chapter 6, you learned how to solve simultaneous equations that involve two variables. Here’s the general rule for solving simultaneous equations: You need as many distinct equations as you have variables. For example, to solve for two variables, you need two distinct equations. To solve for three variables, you need three distinct variables.

For data sufficiency questions, you don’t actually need to solve the equations; you just need to realize when you have enough information to do so. The key is recognizing equations and variables. In some cases, such as the following example, it’s relatively easy.

1.If x + y = 7, what is x ?

(1) x − y = 1

(2) y + z = 11

Statement (1) is sufficient because you now have two distinct equations. That’s enough to solve for both variables and answer the question. Narrow it down to (A) and (D). Statement (2) is not sufficient, because it introduces a third variable and you only have two equations. You can’t solve it, so (A) is the correct answer.

In other cases, identifying the variables and the equations becomes more difficult. Look at this next example.

2.Angela goes shopping. If she buys a total of 7 hats and belts, how many hats did she buy?

(1) Angela buys one more hat than she does belts.

(2) Angela buys a total of 11 belts and skirts.

This question has two variables: the number of hats and the number of belts. It also has one equation, hats + belts = 7. Statement (1) has one equation, hats − 1 = belts. Statement (2) has one equation, belts + skirts = 11, and one more variable, the number of skirts. It’s essentially the same as question 1, just disguised as a word problem. The correct answer is also A. If you can spot the variables and equations in word problems, data sufficiency questions will become much easier.

As mentioned above, to solve for the variables in simultaneous equations, you must have distinct equations. In other words, the equations must not be multiples of each other. For example, the equation 2x + 2y = 4 is a multiple of the equation x + y = 2. (If you multiply both sides of the equation x + y = 2 by 2, you end up with the first equation, 2x + 2y = 4.)

Here’s an example:

3.What is the value of x ?

(1) 2x + 4y = 16

(2) 6x + 12y = 48

Statement (1) is only one equation with two variables. Therefore, it’s not sufficient to solve for x. So, narrow your answers to (B), (C), and (E). Statement (2) gives you only one equation with two variables, so it is also not sufficient, and you can eliminate choice (B). At first glance you may think the answer is (C), but don’t be so hasty. Although you have two equations, the second equation is simply the first equation multiplied by 3. Since it is therefore a multiple of the first equation, the two equations are not distinct. Therefore, the answer is (E).

OVERLAPPING RANGES

For a statement to answer the question, it must provide a single value for the number asked. If a statement narrows down the possibilities to a few numbers, but not just one, it is not sufficient to answer the question. However, if each statement narrows the possibilities to a small set and there is only one value in common, then those statements together are sufficient. Look at the following example:

1.What is x ?

(1) x is an odd integer between 0 and 10.

(2) x is a multiple of 5.

Statement (1) tells you that x is 1, 3, 5, 7, or 9; however, that’s not a single value, so you can’t answer the question. Narrow the answers to (B), (C), and (E). Statement (2) tells you that x is 5, 10, 15, 20, 25, or some other multiple of 5. However, that’s not a single value. Eliminate (B). Both statements together are sufficient. The only value they share is 5, so x must equal 5. Choose (C).

DRILL 1

Answers can be found on this page.

1.What is the value of even integer n ?

(1) is an integer.

(2) 20 < n < 50

2.If a student’s total cost for a semester’s tuition, fees, and books was $11,250, how much was his cost for books that semester?

(1) The cost for fees was 15 percent of the cost for tuition.

(2) The combined cost for books and fees was 25 percent of the cost for tuition.

VALUE DATA SUFFICIENCY AND THE PIECES OF THE PUZZLE

The answer to a value data sufficiency problem revolves around which pieces of information are sufficient to answer the question posed. If you can determine which pieces of information are missing, you can more easily recognize which statement(s) contain(s) those pieces of information. Identify the value for which the question asks and determine which numbers you need to get there. It’s like putting together a jigsaw puzzle.

Look at this next example:

1.If Steve owns red, blue, and green marbles in the ratio of 2 : 3 : 5, respectively, how many blue marbles does he own?

(1) Steve owns 20 red marbles.

(2) Steve owns a total of 100 red, blue, and green marbles.

Before you even look at the statements, try to figure out what information you’ll need to answer the question. You know that a ratio plus any of the actual values will allow you to find all the other actual values. So look for actual values as you check each statement.

Statement (1) contains an actual value, so it is sufficient. Narrow the choices to (A) and (D). Statement (2) has an actual value, so it is sufficient. The answer must be (D).

This approach works well with any data sufficiency problem involving a formula of some sort. For example, many geometry questions involve some sort of formula. Look at the following example:

2.In triangle ABC above, the length of AB is 6. What is the area of triangle ABC ?

(1) The length of BC is 10.

(2) The length of AC is 8.

To calculate the area of a triangle, you need to know the base and the height. The question provides the height, so the missing piece is the base, AC. A statement that gives you the length of AC or allows you to calculate it is sufficient.

Statement (1) gives you the length of BC. Because ABC is a right triangle, you can use any two sides and the Pythagorean theorem to find the length of the third side. With Statement (1) you have two sides so you can find AC and calculate the area. (Note: You may also have noticed that this triangle is a right triangle with its sides in the ratio 3 : 4 : 5. So you could calculate the length of AC without using the Pythagorean theorem, if you like.) Narrow the choices to (A) and (D). Statement (2) tells you the length of AC. You can calculate the area, so choose (D).

The pieces-of-the-puzzle approach is extremely useful for simplifying a tough data sufficiency question. By identifying the particular thing you need, you can focus on that while looking at the statements, rather than thinking through the whole process of solving the problem.

YES/NO QUESTIONS

Most of the data sufficiency questions you’ve seen so far ask for a specific value such as “What is x?” However, you will see some questions that ask not for a numerical value but for a “Yes” or “No” answer. For example, look at the question below:

1.Is x an odd integer?

(1) x is divisible by 3.

(2) x is divisible by 2.

The question doesn’t ask for the value of x. Instead it asks a yes/no question. A sufficient answer is either “Yes, x is odd,” or “No, x is not odd.” It’s important to realize that “No” is an acceptable answer. The insufficient answer is “I can’t tell from the information.”

In the above question, look at Statement (1). If x is divisible by 3, it could be odd (such as 3, 9, or 15) or even (such as 6, 12, or 18). From Statement (1) you cannot tell for certain whether x is odd or not. You can’t answer the question, so narrow the choices to (B), (C), and (E). From Statement (2), you know that x is even, so the answer to the question is “No, x is not odd.” You have answered the question, so choose (B). Remember: “No” is a sufficient answer.

Yes/No questions are among the most difficult on the Math section, because they are very tricky. As you look at each statement, be sure to consider all of the possibilities that are consistent with the information in that statement. Considering only the obvious numbers will generally lead you directly to a trap answer.

2.If ABC and DEF are triangles, and the length of AB is greater than that of DE, is the area of ABC greater than that of DEF ?

(1) The length of BC is greater than that of EF.

(2) The length of AC is greater than that of DF.

Start by considering Statement (1). You know that two sides of ABC are bigger than two sides of DEF. It makes sense that ABC could be greater in area than DEF. Just imagine stretching DEF in all directions and that could be ABC. But is ABC necessarily greater in area than DEF? A lot depends on that unknown third side for each triangle. Suppose that AC is tiny, so that ABC is a long, narrow wedge, as shown in ABC below. The area of DEF could be bigger than that of ABC in such a case, so you can’t tell for certain whether or not ABC is bigger. “I can’t tell” is insufficient, so write down (B), (C), and (E).

Statement (2) sets up a very similar situation. Two sides of ABC are bigger than two sides of DEF, but the third side of each is completely unknown. By the same reasoning you used with Statement (1), Statement (2) is insufficient. Cross off (B).

Now put both statements together. Each side of ABC is bigger than the corresponding side of DEF. It seems natural to think that the area of ABC is bigger than that of DEF. That certainly could be true, but does it have to be true? No. Again, you need to imagine all of the possibilities. ABCcould be very thin, even if all three sides are relatively long, as shown below.

In the extreme case, the area of ABC is almost zero. Statements (1) and (2) together are still insufficient, even though your initial response says otherwise, so the answer is (E). The key is to acknowledge the obvious possibility, but then to explore the counterintuitive possibilities.

Plugging In for Yes/No Data Sufficiency

The Plugging In technique can be applied to Yes/No data sufficiency questions. It is very helpful in sorting out whether a particular statement leads always to “Yes” or always to “No” or to “I can’t tell.” If a fact statement for a Yes/No question contains a variable, try plugging in various values for that variable to see whether you get “Yes” or “No” in answer to the question. Be certain that you plug in only numbers that are consistent with the statement under consideration and with any information in the initial setup.

3.Is xy < 0 ?

(1) x < 0

(2) y = x²

With Statement (1), you know that x is negative. However, you don’t know anything about y. If you plug in x = −1 and y = −2, then xy = 2, which is a “No” answer. On the other hand, you could plug in x = −1 and y = 3 to get xy = −3, which is a “Yes” answer. Statement (1) is inconclusive, so write down (B), (C), and (E).

With Statement (2), you know that y = x². If x = −2, then y = 4 and xy = −8; that’s a “Yes.” However, x could be positive. You need to forget Statement (1) for the moment. If x = 2, then y = 4 and xy = 8, which is a “No” answer. Statement (2) overall leads to “I can’t tell,” so eliminate (B).

With both statements, you know that x is negative and y = x². If x = −2, then y = 4 and xy = −8, which leads to “Yes.” No matter what negative number you plug in for x, y will be positive, making xy negative. Since you always get “Yes,” both statements together are sufficient, and the correct answer is (C).

Plugging in numbers will be most effective when you select numbers carefully. Think about what types of numbers or combinations lead to “Yes” answers versus those that lead to “No” answers. Then see whether both types of numbers are consistent with the statement you are considering. In the previous example, the issue was positive/negative, so you needed to see whether you could plug in both positive and negative numbers at each step.

DRILL 2

Answers can be found on this page.

1.Is it true that x < y ?

(1) 5x < 5y

(2) xz < yz

2.If p is a positive integer, is p even?

(1) 4p is even.

(2) p + 2 is odd.

Comprehensive Data Sufficiency 2 Drill

Answers can be found on this page.

Remember!

For Data Sufficiency problems in this book, we do not supply the answer choices. The five possible answer choices are the same every time.

1.What is the value of n ?

(1) n² + 5n + 6 = 0

(2) n² – n − 6 = 0

2.If x and y are integers, does x = y ?

(1) xy = y²

(2) x² = y²

3.Kevin buys beer in bottles and cans. He pays $1.00 for each can of beer and $1.50 for each bottle of beer. If he buys a total of 15 bottles and cans of beer, how many bottles of beer did he buy?

(1) Kevin spent a total of $18.00 on beer.

(2) Kevin bought 3 more cans of beer than bottles of beer.

4.What is the area of the circle with center O above?

(1) The circumference of the circle is 12π.

(2) n = 6

5.Pete works at three part-time jobs to make extra money. What are his average earnings per hour?

(1) Pete earned $500 for 20 hours at the first job, $150 for 10 hours at the second job, and $100 for 5 hours at the third job.

(2) Pete earned an average of $25 per hour at the first job, $15 per hour at the second job, and $20 per hour at the third job.

6.If p is an integer, is p positive?

(1) pq > 0 and qr < 0

(2) pr < 0

7.If x is an integer, what is the value of x ?

(1) x is the square of an integer.

(2) 0 < x < 5

8.If m and n are integers, is mn ≤ 6 ?

(1) m + n = 5

(2) 1 ≤ m ≤ 3 and 2 ≤ n ≤ 4

9.If 2x + 3y = 11, what is the value of x ?

(1) 5x − 2y = 18

(2) 6y − 22 = −4x

10.Is quadrilateral ABCD above a square?

(1) AB = BC

(2) Angle ABC is a right angle.

11.At a certain baseball game, each of the spectators is either a Bullfrogs fan or a Chipmunks fan, and no one is both. What is the ratio of Bullfrogs fans to Chipmunks fans among spectators at the baseball game?

(1) The number of Chipmunks fans among the spectators is 20% greater than the number of Bullfrogs fans.

(2) The total number of spectators at the baseball game is 4,400.

12.What is the value of x + y ?

(1) x – y = 70

(2) x = 170 – y

13.A pizzeria serves pizzas in three sizes: small, medium, and large. On Tuesday, the pizzeria served a total of 280 pizzas. How many large pizzas did the pizzeria serve on Tuesday?

(1) On Tuesday, the pizzeria served 25% more small pizzas than medium pizzas.

(2) On Tuesday, the number of medium pizzas served by the pizzeria was 80% of the number of large pizzas.

14.If ab ≠ 0, is < ?

(1) a = b

(2) b > 0

15.If x is a positive integer, is x² − 1 divisible by 3 ?

(1) x is even.

(2) x is divisible by 3.

16.If m and n are positive integers and mn = 30, what is the value of m + n ?

(1) is an integer.

(2) is an integer.

17.What is the value of ?

(1) a = and c = .

(2) b = c and a = 4c.

18.Is 3 ^{p − 2} greater than 1,000 ?

(1) 3 ^{p + 1} < 54,000

(2) 3 ^p < 3 ^{p − 1} + 2,000

19.In right triangle JKL, shown above, what is the length of KL ?

(1) The length of JK is 5.

(2) y = 2x

Challenge!

Take a crack at this high-level GMAT question.

20.If f is a positive integer, is an integer?

(1) g is an integer and f = (g + 1)(g − 1).

(2) g is an integer and is an integer.

ANSWERS AND EXPLANATIONS

Drill 1

1.CLook at Statement (1). is an integer such as 1, 2, 3, 4, 5, 6, 7, and so on. So n must be an even perfect square such as 4, 16, 36, and so on. That’s not enough to answer the question, so narrow the choices to (B), (C), and (E). Look at Statement (2). n could be any even integer between 20 and 50, such as 22, 24, 26, and so on. That’s not enough to answer the question, so eliminate (B). Try Statements (1) and (2) together. The only number on both lists is 36, so n = 36. You can answer the question, so choose (C).

2.CThe question itself provides one equation and contains three variables. Look at Statement (1). This gives you a second equation. However, you need three equations to solve for three variables. You can’t answer the question, so narrow the choices to (B), (C), and (E). Look at Statement (2). This provides another equation. However, you still have only two equations for three variables. You can’t answer the question, so eliminate (B). Try Statements (1) and (2) together. Now you have three distinct equations and you can solve for all three variables. You can answer the question, so choose (C).

Drill 2

1.AStart with Statement (1). If you divide both sides of the inequality by 5, you get x < y. So the answer to the question is “Yes.” Narrow the choices to (A) and (D). Look at Statement (2). The problem here is that you don’t know what kind of number z is. If you plug in a positive number, such as z = 4, you get x < y when you divide both sides by z. If you plug in a negative number, such as z = −3, then you get x > y when you divide both sides by z. Remember: You have to flip the inequality sign when you multiply or divide by a negative number. You can’t answer the question, so choose (A).

2.BLook at Statement (1). p could be either an odd number, such as p = 3, or an even number, such as p = 2. Both numbers fit the statement, so you can’t answer the question. Narrow the choices to (B), (C), and (E). Look at Statement (2). You can plug in odd numbers such as p = 3. However, you can’t plug in even numbers because they won’t fit the statement. So p is odd and the answer is “No.” You can answer the question, so choose (B).

Comprehensive Data Sufficiency 2 Drill

1.CLook at Statement (1). If you factor it, you get (n + 2)(n + 3) = 0, which means that n = −2 or −3. That’s not a single value, so you can’t answer the question. Narrow the choices to (B), (C), and (E). Try Statement (2). This factors to (n + 2)(n − 3) = 0, so n = −2 or 3. This isn’t a single value, so you can’t answer the question. Eliminate (B). Try Statements (1) and (2) together. The only number that is on both lists is −2, so n = −2. You can answer the question, so choose (C).

2.CLook at Statement (1). If you plug in a positive number for y, such as y = 2, then x = y and the answer is “Yes.” Same thing if you use a negative number, such as y = −2. However, if you plug in y = 0, then x could equal anything, so “No” is a possibility. You can’t answer the question, so narrow the choices to (B), (C), and (E). Try Statement (2). If you plug in two positive numbers, such as x = 2 and y = 2, then x = y and the answer is “Yes.” If you plug in a positive number and a negative number, such as x = 2 and y = −2, then the answer is “No.” You can’t answer the question, so eliminate (B). Try Statements (1) and (2) together. The only numbers that will fit both statements are those in which x = y. You can’t use a positive and a negative because that won’t fit Statement (1). You can’t use 0 for y and something else for x, because that won’t fit Statement (2). It must be true that x = y. The answer is “Yes,” so choose (C).

3.DThe question provides two variables, bottles and cans, and one equation, bottles + cans = 15. Look at Statement (1). This provides another equation, (1 × cans) + (1.5 × bottles) = 18. You have two equations for two variables, so you can answer the question. Narrow the choices to (A) and (D). Look at Statement (2). This provides another equation, cans − 3 = bottles. You have two equations for two variables, so you can answer the question. Choose (D).

4.DYou know that the formula for the area of a circle is A = πr², so the missing piece is the radius. Look for that as you go through the statements. In Statement (1), you’re given the circumference, so you can solve for the radius by plugging C = 12π into the equation C = 2πr. You can answer the question, so narrow the choices to (A) and (D). Look at Statement (2). This gives you the radius. You can answer the question, so choose (D).

5.ATo find an average, you need to know the total and the number of elements. So the missing pieces are the total money earned and the number of hours. Look for those in the statements. In Statement (1) you can find both the total money earned and the number of hours worked. You can answer the question, so narrow the choices to (A) and (D). Look at Statement (2). You can find neither the total money earned nor the number of hours worked. Remember: Never average the averages. You can’t answer the question, so choose (A).

6.ELook at Statement (1). You could plug in positive numbers for p and q and that would make r negative. The answer would be “Yes.” Or you could plug in negative numbers for p and q and that would make r positive. In that case, the answer would be “No.” You can’t answer the question, so narrow the choices to (B), (C), and (E). Look at Statement (2). You could plug in a positive number for p and a negative number for r and the answer would be “Yes.” Or you could plug in a negative number for p and a positive number for r and the answer would be “No.” You can’t answer the question, so eliminate (B). Try Statements (1) and (2) together. You could plug in p = +, q = +, and r = –. That would fit both statements and the answer would be “Yes.” Or you could plug in p = –, q = –, and r = +. That would fit both statements and the answer would be “No.” You can’t answer the question, so choose (E).

7.ELook at Statement (1). x must be a perfect square, such as 1, 4, 9, 16, and so on. However, you can’t narrow it down to a single value. You can’t answer the question, so narrow your choices to (B), (C), and (E). Look at Statement (2). x could be 1, 2, 3, or 4. You can’t answer the question, so eliminate (B). Try Statements (1) and (2) together. Although the lists overlap, they share more than one value. x could be either 1 or 4. You can’t answer the question, so choose (E).

8.ALook at Statement (1). If you plug in m = 1 and n = 4, then mn = 4 and the answer is “Yes.” If you plug in m = 2 and n = 3, then mn = 6 and the answer is “Yes.” If you plug in m = −5 and n = 10, then mn = −50 and the answer is “Yes.” Any numbers that will fit the statement will give you a “Yes” answer. You can answer the question, so narrow the choices to (A) and (D). Look at Statement (2). If you plug in m = 1 and n = 2, then mn = 2 and the answer is “Yes.” If you plug in m = 3 and n = 4, then mn = 12 and the answer is “No.” You can’t answer the question, so choose (A).

9.AThe question provides one equation and two variables. Look at Statement (1). This gives you another equation. With two equations for two variables, you can solve for x. You can answer the question, so narrow the choices to (A) and (D). Look at Statement (2). Although this seems to give you another equation, it really doesn’t. The equation in Statement (2) is the equation from the question rearranged and multiplied by 2. If you tried to set up the simultaneous equations and solve them, everything would cancel and you’d be stuck. Remember: You need a distinct equation for each variable. You can’t answer the question, so choose (A).

10.ELook at Statement (1). It’s possible that ABCD is a square. You could make all four sides equal and make all four angles right angles. However, it’s also possible that it’s not a square. Just because AB = BC doesn’t mean that the other two sides are also the same. Don’t let the diagram fool you. It’s not necessarily drawn to scale. You can’t answer the question, so narrow the choices to (B), (C), and (E). Look at Statement (2). By itself, this doesn’t tell you much. It might be a square, but you don’t know whether the sides are all the same or whether the other angles are also right angles. You can’t answer the question, so eliminate (B). Try Statements (1) and (2) together. It might be a square, but it doesn’t have to be. The other three angles don’t have to be right angles and CD and AD might not equal AB and BC. See the diagram below. You can’t answer the question, so choose (E).

11.AStart by thinking about the pieces of the puzzle. You need the ratio of Bullfrogs fans to Chipmunks fans. Knowing both numbers would be enough, as it would be to know just the ratio itself. With Statement (1), you can’t find both numbers, because there are two variables but only one equation. However, you can find the ratio. Translate the statement to c = 120% × b, which becomes = . You can flip that upside down to get the ratio you want. Since Statement (1) is sufficient, narrow the choices to (A) and (D). With Statement (2), you get the total number of spectators, but that doesn’t help you with the number of either type of fan or the ratio of the two. Choose (A).

12.BStart by thinking about the pieces of the puzzle that you need. You could answer the question either by knowing both numbers, x and y, or by knowing just their sum. With Statement (1), you can’t find both numbers; there are two variables and only one equation. You also can’t manipulate the equation to get x + y equal to some number. Since Statement (1) is insufficient, narrow the choices to (B), (C), and (E). With Statement (2), you still can’t solve for both variables, because there is only one equation. However, you can add y to both sides to get x + y = 170. That’s sufficient to answer the question, so choose (B).

13.CFrom the initial setup, you can derive the equation s + m + l = 280. With Statement (1), you can get the equation s = 125% × m. However, you have only two equations, but three variables. You can’t solve for l, so narrow the choices to (B), (C), and (E). With Statement (2), you can derive the equation m = 80% × l. However, that’s still only two equations for three variables. Since Statement (2) is insufficient, eliminate (B). With both statements, you now have three variables for three equations. That’s enough to solve for all three variables. Choose (C).

14.CTackle Yes/No questions by plugging in various numbers. With Statement (1), both a and b could be 1, in which case the question becomes “Is < ?”and the answer is “No.” If both a and b are −1, then the question becomes “Is − < −?” and the answer is “Yes.” Statement (1) is insufficient, so narrow the choices to (B), (C), and (E). With Statement (2), knowing that b is positive doesn’t tell you anything about a. If a = 1 and b = 1, the answer is “No.” If a = −1 and b = 1, the answer is “Yes.” Since Statement (2) is insufficient, eliminate (B). With both statements, both a and b are equal and positive. If you plug in various numbers, such as a = b = 1 or a = b = 5, you’ll always get “No.” That answers the question, so choose (C).

15.BWith Yes/No questions, try Plugging In. For Statement (1), if x = 2, then x² − 1 = 3, which is divisible by 3. If x = 6, then x² − 1 = 35, which is not divisible by 3. Since you can get both “Yes” and “No,” Statement (1) is insufficient and you should narrow the choices to (B), (C), and (E). For Statement (2), if x = 3, then x² − 1 = 8, which is not divisible by 3. If x = 6, then x² − 1 = 35, which is not divisible by 3. For any x that is divisible by 3, the result is not divisible by 3, so the answer is “No.” That’s sufficient, so choose (B).

16.EIf mn = 30, then m and n could be any of the factors of 30: 1, 2, 3, 5, 6, 10, 15, or 30. With Statement (1), m could be 5, making n = 6 and m + n = 11. If m = 10, then n = 3 and m + n = 13. m could also be 15 or 30. Since there is more than one answer, narrow the choices to (B), (C), and (E). With Statement (2), n could be 2, 6, 10, or 30. Each possibility leads to a different value for m + n, so (2) is insufficient; eliminate (B). With both statements together, you could have m = 15 and n = 2, so m + n = 17. However, m = 5 and n = 6, with m + n = 11, also works. That’s insufficient, so choose (E).

17.CTry Plugging In. With Statement (1), if a = 1, then b = 1 and c = 1, so = 1. However, if a = 2, then b = , and c = , so = = 4. Since more than one answer is possible, Statement (1) is insufficient. Narrow the choices to (B), (C), and (E). With Statement (2), if b = c= 1, then a = 4, so = = 16. However, if b = c = 2, then a = 8, so = = 64. More than one value is possible, so eliminate (B). With both statements together, Plugging In is much more difficult. You could rely on the simultaneous equations rule of thumb. There are four equations in three variables, so that should be enough to solve for all of the variables. Algebraically, you can substitute c = into a = 4c to get a = 4 × , which becomes a² = 4. In , the b and the c will cancel, so the question is really “what is a²?” Although a can be either 2 or −2, the answer is 4 in both cases. Since there is one answer, choose (C).

18.BWith Statement (1), if you divide both sides of the inequality by 3³, or 27, you get 3^{p − 2} < 2,000. The answer could be either “Yes” or “No,” since values above and below 1,000 are possible. Narrow the choices to (B), (C), and (E). With Statement (2), subtract 3^{p − 1} from both sides to get 3^p − 3^{p − 1} < 2,000. You can factor the left side to get 3^{p − 1}(3 − 1) < 2,000 or 3^{p − 1}(2) < 2,000, which becomes 3^{p − 1} < 1,000. If you divide both sides by 3, you get 3^{p − 2} < 333, so the answer is “Yes.” Choose (B).

19.CWith right triangles, you need either two sides or one side and a special ratio (30-60-90 or 45-45-90) to find the other side(s). In Statement (1), you have only one side of the triangle and you don’t know whether the triangle fits one of the special angle patterns or not. Statement (1) is insufficient, so narrow the choices to (B), (C), and (E). In Statement (2), you know that the y angle is twice the size of the x angle, so the triangle is a 30-60-90 triangle. However, you don’t know any of the sides. Statement (2) alone is not enough, so eliminate (B). With both statements together, you have one of the sides and you know the triangle fits the 30-60-90 pattern, so you could solve for the missing side KL. Choose (C).

20.AWith Yes/No questions, try Plugging In. For Statement (1), if g = 2, then f = (2 + 1)(2 − 1) = 3. = 2, so the answer is “Yes.” If g = 5, then f = (5 + 1)(5 − 1) = 24. = 5, so the answer is “Yes” again. For any integer g (provided f is positive), you will get “Yes.” That answers the question, so narrow the choices to (A) and (D). With Statement (2), if g = 1, then f could be any positive integer. Some values for f will give “Yes” for an answer and some will give “No.” That’s insufficient, so choose (A).