CHAPTER 12

Algebra on the GMAT

If x + y = 2 and x2 − xy − 10 − 2y2 = 0, what does x − 2y equal?

Above is a typical Problem Solving algebra question. In this chapter, we’ll look at how to apply the Kaplan Method to this question, discuss the algebra rules being tested, and go over the basic principles and strategies that you want to keep in mind on every Quantitative question involving algebra. But before you move on, take a minute to think about what you see in this question and answer some questions about how you think it works:

PREVIEWING ALGEBRA ON THE GMAT

There won’t be that many problems that involve only algebra—maybe 20 percent of the Quantitative section. But a majority of the questions on the Quantitative section will involve algebra in some way. This makes algebra a necessary skill on the GMAT—mastering some frequently tested algebra concepts could yield large improvements on your Quant score. This chapter will guide you through the basics and give you practice applying algebra concepts to GMAT questions ranging from the simple to the most advanced.

What Mathematical Concepts Are Being Tested in This Question?

For those who find algebra intimidating, this question can seem complicated and abstract at first. Unlike a word problem, which gives you a story to relate to, this question directly measures your ability to use algebraic rules to manipulate unknown terms, or variables. The first step of the Kaplan Method for Problem Solving is to analyze the question stem, and often the best way to start is to “inventory” the question to determine which math skills you will have to use to solve it.

Here you see two algebraic equations, one of which has variables raised to the second power. If you recognized that factoring quadratic equations will be important here, congratulations! If not, don’t worry—you will learn and practice those topics later in this chapter. You are being asked to solve for the algebraic expression x – 2y, so you can anticipate that you will manipulate these two equations to isolate (in other words, put by itself on one side of the equal sign) the expression x – 2y.

What Does the Format of the Answer Choices Tell You about This Question?

This may seem like a somewhat obvious deduction, but it's a crucial—and potentially reassuring—one: the answer choices here are all numbers, so this means it is possible to solve for a single numerical value for x – 2y. The GMAT can’t ask you a trick question, so you can rest assured that there must be a way to use combination or substitution on these two equations to answer the question. You can think of it like a puzzle, and the testmakers have given you all the pieces; it's up to you to put them together.

Do You Need to Know the Exact Values of x and y to Answer This Question?

A common mistake that test takers make in approaching a question like this one is to think they need to solve for exact values of x and y. While it's true that values for those two variables would enable you to solve for the expression x – 2y, they are not necessary—in fact, they may be time-consuming or even impossible to solve for. For instance, if x = 8 and y = 3, you would come up with the same value for the expression (x – 2y = 2) as you would if x = –10 and y = –6. An infinite number of values are possible for x and y that would all yield the same result for this expression. Stay focused on the specific task and keep an open mind about how you can use the given information to find the correct answer; often the best path to the solution won’t be the most obvious one.

What GMAT Core Competencies Are Most Essential to Success on This Question?

Here the testmaker rewards Pattern Recognition; certain algebraic concepts are tested over and over on the GMAT, and you will learn and practice the most important ones in this chapter. On this question, the test taker who recognizes the opportunity to factor the quadratic equation will answer this question efficiently and accurately. Those test takers who are not as comfortable with the algebraic structures that appear frequently on the GMAT could spend a long time with a question like this one just figuring out where to begin.

Here are the main topics we’ll cover in this chapter:

Handling GMAT Algebra

Now let's apply the Kaplan Method to the algebra question you saw earlier:

If x + y = 2 and x2 − xy − 10 − 2y2 = 0, what does x − 2y equal?

Let's use the Kaplan Method for Problem Solving to find the most efficient approach to this problem. Don’t worry if the math is unfamiliar to you; you will have an opportunity to learn it later in the chapter. For now, focus on the basics of applying the Kaplan Method to a GMAT algebra problem.

Step 1: Analyze the Question

The GMAT will reward you for recognizing patterns. Here, you are given two variables and two equations, one of which is not linear. When you see an equation that resembles one of the quadratic patterns you know, try factoring it using reverse-FOIL. Odds are the problem will be greatly simplified.

Step 2: State the Task

You need to solve for x – 2y. Notice that you are asked for the value of an expression, not an individual variable. Don’t waste time solving for all the variables individually if it's not necessary.

Step 3: Approach Strategically

If you add 10 to both sides of the second equation, you get x2 – xy – 2y2 = 10. Now the left side looks like an expression that you can factor with reverse-FOIL. Since you could get the x2 term by multiplying x and x, you know both factors will contain x:

image

Next, determine what two factors multiplied together will equal –2y2. You need either –2y and y or 2y and –y. Since the coefficient of the xy term is negative, choose –2y and y. Then the sum of the outer and inner products will give you –xy. Thus, the factorization of x2 – xy – 2y2 is

image

Now, notice that both factors of x2 – xy – 2y2 appear in the question stem. You are told that x + y = 2, and you are asked to find the value of x – 2y.

You can find the value of x – 2y by returning to the equation you’ve already factored. You know that (x + y)(x – 2y) = 10. Since x + y = 2, you can replace (x + y) with 2, giving you 2(x – 2y) = 10, or (x – 2y) = 5. So (D) is the correct answer.

Did you get to 2(x – 2y) = 10 and then go further to 2x – 4y = 10 or to x = 5 + 2y? If so, you lost sight of what was asked. You may find it helpful to write down what's asked on your noteboard so it's always in your line of sight.

Step 4: Confirm Your Answer

Reread the question stem to check that you have answered the right question. Here, confirm that you have solved for x – 2y and not x or y. When you have done that, you can move on.

Now let's look at each of the six most important algebra topics that show up on the GMAT Quantitative section, starting with translating words into expressions and equations.

TRANSLATING WORDS INTO EXPRESSIONS AND EQUATIONS

Often the most frustrating aspect of word problems is the odd way in which information is presented. Don’t get frustrated. Just break down the information into small pieces and take things one step at a time. Word problems can usually be translated from left to right, but not always. Say you see this sentence: “There are twice as many dollars in George's wallet as the amount that is five dollars less than the amount in Bill's wallet.” Instead of trying to translate it into math all in one go, approach it piecemeal.

Whenever possible, choose letters for your variables that make sense in the context of the problem. You could start by calling the amount in George's wallet G and the amount in Bill's wallet B. Now, think about the relationship between the two amounts: G is not compared to B but to 5 dollars less than B, or (B – 5). You can now say that G is twice as large as (B – 5). So if you were to set them equal to each other, you’d have to multiply (B – 5) by two. The equation is G = 2(B – 5).

If you had tried to translate it in the order it's written, you might have come up with something like 2G = 5 – B, and you can be sure that there’d be a wrong answer choice waiting to take advantage of that. So think carefully before you translate.

The hardest part of word problems is the process of taking the English sentences and extracting the math from them. The actual math in word problems tends to be the easiest part. The following translation table should help you start dealing with English-to-math translation.

Word Problems Translation Table

English

Math

equals, is, was, will be, has, costs, adds up to, the same as, as much as

=

times, of, multiplied by, product of, twice, double, by

×

divided by, per, out of, each, ratio

÷

plus, added to, and, sum, combined

+

minus, subtracted from, smaller than, less than, fewer, decreased by, difference between

−

a number, how much, how many, what

x, n, etc.

Take a look at the following example:

image

Start by breaking the problem down into smaller, more manageable pieces:

Always take the time to make sure you are translating the problem correctly. Improper translation will cost you points.

Translation Exercise

Answers follow the exercises

Translate the following sentences into algebra. When names are used, use the first letter of each person's name as the appropriate variable.

  1. w is x less than y.

  2. The ratio of 3x to 8y is 5 to 7.

  3. The product of x decreased by y and one-half the sum of x and twice y.

  4. Mike's score on his geometry test was twice Lidia's score.

  5. Samantha is 4 years older than Jeannette.

  6. Jamie is 5 years older than Charlie was 3 years ago.

  7. Giuseppa's weight is 75 pounds more than twice Jovanna's weight.

  8. Luigi has 17 fewer dollars than Sean has.

  9. In 5 years, Sandy will be 4 years younger than twice Tina's age.

  10. The sum of Richard's age and Cindy's age in years is 17 more than the amount by which Tim's age is greater than Kathy's age.

  11. If Mack's salary were increased by $5,000, then the combined salaries of Mack and Andrea would be equal to three times what Mack's salary would be if it were increased by one-half of itself.

Translation Exercise: Answers

  1. w = y – x

  2. image

  3. image

  4. M = 2L

  5. S = J + 4

  6. J = (C – 3) + 5

  7. G = 2J + 75

  8. L = S – 17

  9. S + 5 = 2(T + 5) – 4

  10. R + C = (T – K ) + 17

  11. image

In-Format Question: Translating Words into Expressions and Equations on the GMAT

Now let's use the Kaplan Method on a Problem Solving question dealing with translating words into expressions and equations:

Charles's and Sarah's current ages are C years and S years, respectively. If 6 years from now, Charles will be at least as old as Sarah was 2 years ago, which of the following must be true?

Step 1: Analyze the Question

This problem may look complicated at first, but it's really just asking you to translate the English sentence into math. Once the word problem has been translated, you can apply basic algebra to simplify the statement to match the correct answer choice.

Step 2: State the Task

You need to translate the word problem into math.

Step 3: Approach Strategically

Look at the word problem and work from left to right to extract your algebraic statements. Rather than picking x and y for your variables, use C for Charles's age and S for Sarah's age:

Six years from now, Charles's age will be C + 6.

Two years ago, Sarah's age was S – 2.

Now connect these two algebraic statements. You are told that in 6 years, Charles will be at least as old as Sarah was 2 years ago. The phrase “will be at least as old as” implies that Charles could be exactly the same age as Sarah 2 years ago, or he could be older. Therefore, use the “greater than or equal to” sign between the two statements:

image

The correct answer is (E).

Step 4: Confirm Your Answer

This translation directly matches (E), but be careful to check that the variables are in the correct order and that you have used the correct inequality sign.

TAKEAWAYS: TRANSLATING WORDS INTO EXPRESSIONS AND EQUATIONS

 

Practice Set: Translating Words into Expressions and Equations on the GMAT

Answers and explanations at end of chapter

  1. Peter read P books last year, and Nikki read N books last year. If Peter read 35 more books than Nikki last year, which of the following reflects this relationship?

    • P > 35N

    • P < N – 35

    • P > N + 35

    • P = N – 35

    • P = N + 35

  2. The youngest of 4 children has siblings who are 3, 5, and 8 years older than she is. If the average (arithmetic mean) age of the 4 siblings is 21, what is the age of the youngest sibling?

    • 17

    • 18

    • 19

    • 21

    • 22

ISOLATING A VARIABLE

A term is a numerical constant or the product of a numerical constant and one or more variables. Examples of terms are 5, 3x, 4x2yz, and 2ac.

An algebraic expression is a combination of one or more terms. Terms in an expression are separated by either + or – signs. Examples of expressions are 3xy, 4ab + 5cd, and x2 – 1.

In the term 3xy, the numerical constant 3 is called a coefficient. In a simple term such as z, 1 is the coefficient. A number without any variables is called a constant term. An expression with one term, such as 3xy, is called a monomial; one with two terms, such as 4a + 2d, is a binomial; one with three terms, such as xy + z – a, is a trinomial. The general name for expressions with more than one term is polynomial.

Substitution

Substitution is a method that you can employ to evaluate an algebraic expression or to express an algebraic expression in terms of other variables.

Example: Evaluate 3x2 – 4x when x = 2.

Replace every x in the expression with 2 and then carry out the designated operations. Remember to follow the order of operations (PEMDAS):

image

image

Example: Express image in terms of x and y if a = 2x and b = 3y.

Here, replace every a with 2x and every b with 3y:

image

Operations with Polynomials

All of the laws of arithmetic operations, such as the commutative, associative, and distributive laws, are also applicable to polynomials.

Commutative Law:

image

Associative Law:

image

Note that the process of simplifying an expression by subtracting or adding together those terms with the same variable component is called combining like terms.

Distributive Law:

image

The product of two binomials can be calculated by applying the distributive law twice.

Example:

image

You multiply the First terms first, then the Outer terms, then the Inner terms, and finally the Last terms. A simple mnemonic for this is the word FOIL. You will learn more about applying FOIL to binomials in the section on quadratic equations later in this chapter.

Factoring Algebraic Expressions

Factoring a polynomial means expressing it as a product of two or more simpler expressions.

Common Monomial Factor: When there is a monomial factor common to every term in the polynomial, it can be factored out by using the distributive law.

Example: 2a + 6ac = 2a(1 + 3c)

Here, 2a is the greatest common factor of 2a and 6ac.

Making problems look more complicated than they are by distributing a common factor is a classic GMAT trick. Whenever algebra looks scary, check whether common factors could be factored out.

Equations

An equation is an algebraic sentence that says that two expressions are equal to each other. The two expressions consist of numbers, variables, and arithmetic operations to be performed on these numbers and variables.

Solving Equations

To solve for some variable, you can manipulate the equation until you have isolated that variable on one side of the equal sign, leaving any numbers or other variables on the other side. Of course, you must be careful to manipulate the equation only in accordance with the equality postulate: whenever you perform an operation on one side of the equation, you must perform the same operation on the other side. Otherwise, the two sides of the equation will no longer be equal.

The steps for isolating a variable are as follows:

  1. Eliminate any fractions by multiplying both sides by the least common denominator.

  2. Put all terms with the variable you’re solving for on one side by adding or subtracting on both sides.

  3. Combine like terms.

  4. Factor out the desired variable.

  5. Divide to leave the desired variable by itself.

Linear, or First-Degree, Equations

These are equations in which all the variables are raised to the first power (there are no squares or cubes). To solve such an equation, you’ll need to perform operations on both sides of the equation in order to get the variable you’re solving for alone on one side. The operations you can perform without upsetting the balance of the equation are addition and subtraction, as well as multiplication or division by a number other than 0. At each step in the process, you’ll need to use the reverse of the operation that's being applied to the variable in order to isolate the variable.

Example: If 4x − 7 = 2x + 5, what is x?

1.   Put all the terms with the variable on one side of the equation. Put all constant terms on the other side of the equation.

2.   Combine like terms.

3.   Divide to leave the desired variable by itself.

 image

You can easily check your work when solving this kind of equation. The answer you arrive at represents the value of the variable that makes the equation hold true. Therefore, to check that it's correct, you can just substitute this value for the variable in the original equation. If the equation holds true, you’ve found the correct answer. In the previous example, you got a value of 6 for x. Replacing x with 6 in your original equation results in 4(6) – 7 = 2(6) + 5, or 24 – 7 = 12 + 5, which simplifies to 17 = 17. That's a true statement, so 6 is indeed the correct value for x.

Equations with Fractions

The GMAT loves to make algebra problems look harder than they need to be by using fractions. Whenever you see a fraction in an algebra question, always get rid of the fraction as your first step. Let's see how to solve such a problem.

Example: Solve image

1.   Eliminate fractions by multiplying both sides of the equation by the least common denominator (LCD). Here the LCD is 30.

image

2.   Put all terms with the variable on one side by adding or subtracting on both sides. Put all constant terms on the other side of the equation.

image

3.   Combine like terms.

image

4.   Divide to leave the desired variable by itself.

image

Literal Equations

On some problems involving more than one variable, you cannot find a specific value for a variable; you can only solve for one variable in terms of the others. To do this, try to get the desired variable alone on one side and all the other variables on the other side.

Example: In the formula image solve for N in terms of P, R, T, and V.

1.   Eliminate fractions by cross multiplying.

image

2.   Remove parentheses by distributing.

image

3.   Put all terms containing N on one side and all other terms on the other side.

image

4.   Factor out the common factor N.

image

5.   Divide by (VT – P) to get N alone.

image

Note: You can reduce the number of negative terms in the answer by multiplying both the numerator and the denominator of the fraction on the right-hand side by –1.

image

In-Format Question: Isolating a Variable on the GMAT

Now let's use the Kaplan Method on a Problem Solving question dealing with isolating a variable:

If image, then n =

Step 1: Analyze the Question

Follow the steps for isolating a variable carefully and methodically. In this question, the variable you need to isolate is buried under two fractions. When working with fractions, you can often simplify the work by one of two strategies: (1) multiplying by a common denominator or (2) cross multiplying the equation. For this question, use cross multiplication to isolate n.

Step 2: State the Task

Your task is to bring the variable n to one side of the equation and the rest of the terms to the other side.

Step 3: Approach Strategically

Begin by cross multiplying the equation, which results in this equation:

image

Now simplify the equation:

image

Combine like terms:

image

The correct answer is (E).

Step 4: Confirm Your Answer

You can check your calculations by simply plugging n = 6 into the original equation. If the expression on the left side is equal to 2, the calculations are correct. You could have also used Backsolving, but since there isn’t an easy way to know whether you must select an answer choice that is larger or smaller than the one you started with, algebra is the most efficient way to get the answer here.

TAKEAWAYS: ISOLATING A VARIABLE

 

Practice Set: Isolating a Variable on the GMAT

Answers and explanations at end of chapter

  1. If image and if image, what is the value of a in terms of b?

    • image

    • image

    • image

    • image

    • image

  2. If image and image, then y =

    • −2 − 3x2

    • x2 − 2

    • 2x2 − 2

    • x − 2 − x2

    • x − 2 + x2

  3. Susan weighs m pounds more than Anna does, and together they weigh a total of n pounds. Which of the following represents Anna's weight in pounds?

    • image

    • image

    • image

    • 2n − m

    • n − 2m

QUADRATIC EQUATIONS

The term quadratic refers to a mathematical expression in the form ax2 + bx + c, where a, b, and c are constants and a does not equal zero. The algebraic rules you learned for solving a linear equation still apply to quadratics, but there are also some specific things to know about handling quadratic equations. On the GMAT, you will find that questions dealing with quadratics use the same patterns over and over.

Factoring Quadratic Expressions

Earlier in this chapter, you learned that two binomials can be multiplied together by applying the distributive law twice; in other words, you use FOIL (multiplying the First terms first, then the Outer terms, then the Inner terms, and finally the Last terms) to “expand” the expression. The result is often a quadratic expression in the form ax2 + bx + c.

Example:

image

Factoring is the reverse of this process of expanding the expression. Factoring can be thought of as applying the FOIL method backward, or using “reverse-FOIL.” Below you will learn how to factor the most common forms of quadratic expression on the GMAT.

Polynomials of the Form x2 + bx + c

Many quadratic polynomials can be factored into a product of two binomials. The product of the first term in each binomial must equal the first term of the polynomial. The product of the last term of each binomial must equal the last term of the polynomial. The sum of the remaining products must equal the second term of the polynomial.

Example: x2 − 3x + 2

Using reverse-FOIL, you can factor this into two binomials, each containing an x term. Start by writing down what you know.

image

In each binomial on the right, you need to fill in the missing term. The product of the two missing terms will be the last term in the polynomial: 2. The sum of the two missing terms will be the coefficient of the second term of the polynomial: −3. Try the possible factors of 2 until you get a pair that adds up to –3. There are two possibilities: 1 and 2 or –1 and –2. Since (–1) + (–2) = –3, you can fill –1 and –2 into the empty spaces.

Thus, x2 − 3x + 2 = (x − 1)(x − 2).

If the coefficient of the constant (the last term) is negative, then your binomials will have different signs (one + and one –). If the coefficient of the constant is positive, then your binomials will both have the same sign as the coefficient in the middle term (two +'s or two –'s). Check out the previous example: both binomials have a minus sign because of the –3 coefficient of the term −3x.

Note: Whenever you factor a polynomial, you can check your answer by using FOIL to obtain the original polynomial.

Classic Quadratics

The process of reverse-FOIL works on the quadratics you will find on the GMAT. However, there also exist some patterns called “classic quadratics,” which you can factor more quickly by recognizing the pattern than by using reverse-FOIL.

Difference of Two Perfect Squares

The difference of two squares can be factored into the product of two binomials: a2 − b2 = (a + b)(a − b).

In the case of equations written as the difference of two squares, you don’t need to go through all the steps of factoring using reverse-FOIL; just recognize the pattern.

Example: d2 − 16 = (d + 4)(d − 4)

Polynomials of the Form a2 + 2ab + b2

This type of polynomial is also a pattern that appears frequently enough on the GMAT to warrant memorization. Any polynomial of this form is equivalent to the square of a binomial. Notice that according to FOIL, (a + b)2 = a2 + 2ab + b2.

Factoring such a polynomial using reverse-FOIL is just reversing this procedure.

Example: x2 + 6x + 9 = (x)2 + 2(x)(3) + (3)2 = (x + 3)2

Polynomials of the Form a2 − 2ab + b2

Any polynomial of this form is equivalent to the square of a binomial as in the previous example. Here though, the binomial is the difference of two terms: (a − b)2 = a2 − 2ab + b2.

Example: x2 − 10x + 25 = (x)2 − 2(x)(5) + (5)2 = (x − 5)2

The GMAT uses these three “classic quadratics” over and over. Review them thoroughly and look out for them as you practice. Notice that all the forms begin and end in a perfect square:

image

Picking up on these patterns will save you lots of work, especially in the difficult questions. Imagine seeing this expression show up on the test:

image

This may look daunting at first glance. But if you realize that 169 is 132, and 4x4 is (2x2)2, you can then make the educated guess that this polynomial factors as (2x2 + 13)2. You then only have to confirm that 52x2 = 2 × 2x2 × 13 (which it is) to start making the problem easier.

Solving Quadratic Equations

If the expression ax2 + bx + c is set equal to zero, there is a special name for it: a quadratic equation. Since it is an equation, you can find the value or values for x that make the equation work. You can do so by using the factored form of the equation obtained through reverse-FOIL.

Example: x2 − 3x + 2 = 0

To find the solutions, or roots, start by factoring using reverse-FOIL. Factor x2 − 3x + 2 into (x − 1)(x − 2), making the quadratic equation

image

You now have a product of two binomials that is equal to zero. When does a product of two terms equal zero? The only time that happens is when at least one of the terms is zero. If the product of (x − 1) and (x − 2) is equal to zero, that means either the first term equals zero or the second term equals zero. So to find the roots, you just need to set the two binomials equal to zero. That gives you

image

Solving for x, you get x = 1 or x = 2. As a check, you can plug in 1 and 2 into the equation x2 − 3x + 2 = 0, and you’ll see that either value makes the equation work.

In-Format Question: Quadratic Equations on the GMAT

Now let's use the Kaplan Method on a Problem Solving question dealing with quadratic equations:

If (a − 3)2 = 5 − 10a, then which of the following is the value of a?

Step 1: Analyze the Question

When you are asked to solve for a variable that is squared, the most efficient solution is typically to factor the equation into two binomials. Remember that in order to factor the equation, you must bring all of the terms to one side of the equation and have only zero on the other side.

Step 2: State the Task

You need to solve for the value of a. To do this, you’ll need to first simplify the equation in the question stem to fit the standard quadratic equation format needed to factor (ax2 + bx + c = 0); then determine the factors for the equation.

Step 3: Approach Strategically

You are given the equation in the form (a − 3)2 = 5 − 10a.

You can simplify the equation by subtracting 5 from both sides and adding 10a to both sides:

image

Next, multiply out the far left expression using FOIL and simplify:

image

Now use reverse-FOIL to factor the quadratic expression on the left:

image

To solve for a, set each of the binomials equal to zero. In this case, there is only one binomial, so a + 2 = 0, and a = −2. Choice (B) is correct.

You can also use Backsolving, although this may not be the most efficient solution, since it would be difficult to know after testing an incorrect answer choice whether the right answer should be larger or smaller, and you could end up testing several choices before finding the correct one. Even in such situations, Backsolving can still help you find the right answer when you aren’t sure how to solve algebraically. Let's say you start with choice (B).

image

In this case, you got lucky and found the correct answer on the first try. If you hadn’t, you could then proceed to test the other choices (in order of their manageability) until you found one that works.

Step 4: Confirm Your Answer

You can plug your calculated value for a into the equation in the question stem to confirm that the calculations are correct.

TAKEAWAYS: QUADRATIC EQUATIONS

 

Practice Set: Quadratic Equations on the GMAT

Answers and explanations at end of chapter

  1. Is n negative?

    • (1)  n2 + n − 6 = 0

    • (2)  n2 + 6n + 9 = 0

  2. Which of the following expressions CANNOT be equal to 0 when x2 − 2x = 3?

    • x2 − 6x + 9

    • x2 − 4x + 3

    • x2 − x − 2

    • x2 − 7x + 6

    • x2 − 9

  3. If x2 − 2x − 15 = (x + r)(x + s) for all values of x, and if r and s are constants, then which of the following is a possible value of r − s?

    • 8

    • 2

    • −2

    • −3

    • −5

SYSTEMS OF LINEAR EQUATIONS

In general, if a problem has multiple variables and you want to find unique numerical values for all of the variables, you will need as many distinct equations as you have variables. If you are given two distinct equations with two variables, you can combine the equations to obtain a unique solution set. Don’t be intimidated by calculating values, since usually—though not always—the GMAT will give integer answers for the variables. Focus instead on looking for opportunities to combine equations. The GMAT rewards those who find clever combinations with quick solutions.

Note that the word “distinct” means that each equation must provide new, different information. In other words, each additional equation must contain information you couldn’t have derived using the equation(s) you already have.

Approach 1

Isolate one variable in one equation. Then plug the expression that it equals into its place in the other equation.

Example: Find the values of m and n if m = 4n + 2 and 3m + 2n = 20.

1.   You know that m = 4n + 2. Substitute 4n + 2 for m in the second equation.

image

2.   Solve for n.

image

3.   To find the value of m, substitute 1 for n in the first equation and solve.

image

Approach 2

Add or subtract whole equations from each other to eliminate a variable.

Example: Find the values of x and y if 4x + 3y = 27 and 3x − 6y = −21.

There's no obvious isolation/substitution to be done, since all variables have coefficients. But if you multiplied the first equation by 2, you’d be able to get rid of the y's.

Distribute:

image

Now add the new equation to the second equation, carefully lining them up to combine like terms:

image

Divide by 11: x = 3

This value can now be substituted back into either equation to yield the other value.

image

In-Format Question: Systems of Linear Equations on the GMAT

Now let's use the Kaplan Method on a Problem Solving question dealing with systems of linear equations:

If x + y = 5y − 13 and x − y = 5, then x =

Step 1: Analyze the Question

You have two distinct linear equations and two variables, so you will be able to solve for x. Remember that there are two techniques for solving distinct linear equations: (1) substitution and (2) combination. Since these equations look easy to simplify, substituting one equation into the other will be the most efficient approach here.

Step 2: State the Task

Substitute one linear equation into the other to eliminate y and solve for x.

Step 3: Approach Strategically

Simplify the first equation:

image

Simplify the second equation:

image

Substitute the second equation into the first and solve for x:

image

Choice (A) is correct.

Step 4: Confirm Your Answer

You can confirm your work by substituting x = 11 into either equation to find a value for y. This value for y can then be substituted into the other equation to ensure that x = 11.

TAKEAWAYS: SYSTEMS OF LINEAR EQUATIONS

 

Practice Set: Systems of Linear Equations on the GMAT

Answers and explanations at end of chapter

  1. A theater charges $12 for seats in the orchestra and $8 for seats in the balcony. On a certain night, a total of 350 tickets were sold for a total cost of $3,320. How many more tickets were sold that night for seats in the balcony than for seats in the orchestra?

    • 90

    • 110

    • 120

    • 130

    • 220

  2. Hallie has only nickels, dimes, and quarters in her pocket. Nickels are worth $0.05, dimes are worth $0.10, and quarters are worth $0.25. If she has at least 1 of each kind of coin and has a total of $2.75 in change, how many nickels does she have?

    • (1)  She has a total of 21 coins, with twice as many dimes as nickels.

    • (2)  She has $1.50 in quarters.

  3. Jacob is now 12 years younger than Michael. If 9 years from now Michael will be twice as old as Jacob, how old will Jacob be in 4 years?

    • 3

    • 5

    • 7

    • 15

    • 19

SPECIAL CASES IN SYSTEMS OF LINEAR EQUATIONS

On the GMAT, it is not always necessary to solve for each variable to answer a question. Some questions will ask you to solve for a sum, difference, or other relationship between variables rather than for the variables themselves. In such a case, rather than attempting to solve for both x and y, you will solve for values for expressions such as (x + y) or (x − y), the average of x and y, or the ratio between the two variables. Questions involving special cases reward test takers who seek out opportunities to use Critical Thinking and Pattern Recognition in GMAT questions.

Special cases in systems of linear equations appear most frequently in problems involving sums, differences, averages, and ratios. You will see them appear more often in Data Sufficiency questions than in Problem Solving questions, because Data Sufficiency questions require you to recognize whether you have enough information to determine a relationship. Don’t assume that you can’t answer a question just because there are more variables present than there are equations. By simplifying the equation, you may be able to cancel out one or more variables entirely.

In-Format Question: Special Cases in Systems of Linear Equations on the GMAT

Let's look at a Data Sufficiency question that involves special cases in systems of linear equations.

Step 1: Analyze the Question Stem

First, look at your question stem and recognize that this is a special situation. You are not asked for the value of x or y; rather, you are asked for the value of x − y. Next, determine whether there is enough information to find the value of x − y. Keep in mind that you do not have to know the values of each of the variables x and y to be able to find the value of x − y. Because the question stem does not provide any information for finding the value of x − y, look at the statements.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) is 3x + 3y = 31. There is no way to rearrange this equation to find the value of x − y. Therefore, Statement (1) is insufficient. You can eliminate (A) and (D).

Statement (2) is 3x − 3y = 13. Factoring 3 from the left side of this equation, you have 3(x − y) = 13. Dividing both sides of this equation by 3, you have image, and Statement (2) is sufficient to find the value of x − y. Choice (B) is correct.

TAKEAWAYS: SPECIAL CASES IN SYSTEMS OF LINEAR EQUATIONS

 

Practice Set: Special Cases in Systems of Linear Equations on the GMAT

Answers and explanations at end of chapter

  1. If the average (arithmetic mean) of a and b is 45 and the average of b and c is 70, what is the value of c − a?

    • 25

    • 50

    • 90

    • 140

    • It cannot be determined from the information given.

  2. What is the value of p?

    • (1)  2p − m = 5p + 2m

    • (2)  2(p − m) = 8 − 2m

  3. What is the value of x?

    • (1)  3x + 2y = 6

    • (2)  4y = 12 − 6x

FUNCTIONS AND SYMBOLISM

The GMAT uses classic function notation that you may recall from your later high school algebra classes. It also uses some untraditional notation, like ◊, ♠, or ⊗. Both types boil down to substitution.

The basic idea is that these questions ask you to substitute values or operations in a unique way described by the problem. Here's a classic function problem.

Example: What is the minimum value of the function f(x) = x2 − 1?

What this problem is telling you is that whatever number is between those parentheses gets substituted in place of x in x2 − 1. For instance, f(2) = 22 − 1 = 3.

A strategic way to solve would be to substitute whatever answer choices the GMAT gives you as the answer to the function. Some may not be possible; some will be. For example, x2 = −4 isn’t possible, since the GMAT uses only “real” numbers, and the square of a real number must be non-negative. The smallest of the possible answer choices would be the correct response.

You could solve this question logically, too. The value of the function will be smallest when x2 is smallest. Squaring a negative number produces a positive number, so x2 can never be negative. The smallest result you could get is x2 = 0. So the smallest x2 − 1 can be is 0 − 1, or −1.

Some questions offer strange symbols but work basically the same way—as rules for how to substitute.

Example: If x ♠ y = 3x − y2, then what is the value of 8 ♠ 2?

The given equation is really just a rule for substitution. Whatever is to the left of the ♠ symbol is x and should be substituted in place of x in 3x − y2. Similarly, anything to the right of ♠ is y and should be substituted for y.

image

For the most part, symbols on the GMAT define operations (i.e., the process that one or more numbers must be put through). Occasionally, the GMAT will use symbols to stand in for digits, but this isn’t as common.

In-Format Question: Functions and Symbolism on the GMAT

Now let's use the Kaplan Method on a Data Sufficiency question dealing with functions and symbolism:

The symbol ♣ represents one of the following operations: addition, subtraction, multiplication, or division. What is the value of 6 ♣ 2?

Step 1: Analyze the Question Stem

You need to determine whether there is sufficient information to find the value of 6 ♣ 2, and you are told that the symbol ♣ can stand for any one of the four operations of addition, subtraction, multiplication, or division.

Now, since there is no further information in the question stem, look at the statements.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) tells you that 0 ♣ 3 = 0. The potential operations that yield this value are:

image

So ♣ could stand for multiplication or division. However, 6 × 2 = 12 and 6 ÷ 2 = 3, so there is more than one possible answer to the question. Statement (1) is insufficient. Eliminate (A) and (D).

Statement (2) tells you that 2 ♣ 1 = 2. The potential operations that yield this value are:

image

So again, ♣ could be multiplication or division; as you saw in evaluating the first statement, the value of 6 ♣ 2 changes depending on whether the symbol represents multiplication or division. More than one answer is possible for the question, so Statement (2) is insufficient. Eliminate (B).

Taking the statements together, you still know only that ♣ could be multiplication or division. However, since each of these operations yields a different value, the two statements taken together are insufficient. Therefore, (E) is correct.

TAKEAWAYS: FUNCTIONS AND SYMBOLISM

 

Practice Set: Functions and Symbolism on the GMAT

Answers and explanations at end of chapter

  1. Let image and image, for all integers x and y. If m = 2, image is equal to which of the following?

    • image

    • 3

    • image

    • 5

    • image

    image

  2. In the multiplication problem above, each of the symbols ◊, Δ, and • represents a positive digit. If ◊ > Δ, what is the value of ◊?

    • (1)  Δ = 1

    • (2)  • = 9

  3. For all numbers a and b, the operation ▼ is defined by a ▼ b = (a + 2)(b − 3). If 3 ▼ x = −45, then x =

    • −15

    • −6

    • 3

    • 6

    • 15

  4. If the function f is defined for all x by f(x) = ax2 + bx − 43, where a and b are constants, what is the value of f(3)?

    • (1)  f(4) = 41

    • (2)  3a + b = 17

Answer Key

Practice Set: Translating Words into Expressions and Equations on the GMAT

  1. E

  2. A

Practice Set: Isolating a Variable on the GMAT

  1. A

  2. C

  3. A

Practice Set: Quadratic Equations on the GMAT

  1. B

  2. D

  3. A

Practice Set: Systems of Linear Equations on the GMAT

  1. A

  2. A

  3. C

Practice Set: Special Cases in Systems of Linear Equations on the GMAT

  1. B

  2. B

  3. E

Practice Set: Functions and Symbolism on the GMAT

  1. D

  2. B

  3. B

  4. B

Answers and Explanations

Practice Set: Translating Words into Expressions and Equations on the GMAT

1.    (E)

Peter read P books last year, and Nikki read N books last year. If Peter read 35 more books than Nikki last year, which of the following reflects this relationship?

Step 1: Analyze the Question

The sentences in this word problem need to be translated into algebraic statements so that we can determine the relationship between the number of books that Peter and Nikki have read.

Step 2: State the Task

Once the word problem has been translated, we will apply basic algebra to simplify the statement to match the correct answer choice.

Step 3: Approach Strategically

Translating the phrase “Peter read 35 more books than Nikki,” we have P = N + 35.

Step 4: Confirm Your Answer

This translation directly matches (E), but be careful to check that the variables are in the correct order.

2.    (A)

The youngest of 4 children has siblings who are 3, 5, and 8 years older than she is. If the average (arithmetic mean) age of the 4 siblings is 21, what is the age of the youngest sibling?

Step 1: Analyze the Question

We can apply the translation rules to this word problem to convert the question into a manageable algebraic equation.

Step 2: State the Task

Once the word problem has been translated, we will apply basic algebra to simplify the statement to match the correct answer choice.

Step 3: Approach Strategically

Let x be the age of the youngest sibling in years. Then the ages of the other siblings, who are 3, 5, and 8 years older than the youngest sibling, are x + 3, x + 5, and x + 8, respectively. The average age of the four siblings is 21. Therefore, the average of x, x + 3, x + 5, and x + 8 is 21.

The average formula is image. So we can write the equation image

Let's solve this equation for x:

image

The correct answer is (A).

Step 4: Confirm Your Answer

For Average questions, an easy way to confirm your answer is to plug your value for x into the average formula and see if the same answer comes out on both sides. If not, go back over your calculations.

Practice Set: Isolating a Variable on the GMAT

3.    (A)

If b ≠ 1 and if image, what is the value of a in terms of b?

Step 1: Analyze the Question

We are provided with two pieces of information: b ≠ 1 and image

Step 2: State the Task

Find the value of a in terms of b.

Step 3: Approach Strategically

Isolate the variable a:

image

Our final expression doesn’t seem to match any of the answer choices, but multiplying the expression on the right by image yields the expression image, which matches (A).

Step 4: Confirm Your Answer

We can check our answer by plugging image for a back into image. When doing so, remember that 1 − b = −1(b − 1):

image

The equality holds, so we’ve confirmed that choice (A) is correct.

4.    (C)

If x ≠ 0 and image, then y =

Step 1: Analyze the Question

For this question, we must isolate the variable y on one side of the equation. Note that when there are fractions in an equation, it is best to multiply the entire equation by a common denominator to remove the fractions.

Step 2: State the Task

Move y to one side of the equation and everything else to the other side.

Step 3: Approach Strategically

Both of the fractions in the equation have a common denominator of x. If we multiply both sides of the equation by x, all the denominators will be eliminated. Be careful on this step to multiply every term in the equation by x:

image

The correct answer is (C).

Step 4: Confirm Your Answer

When checking your calculations here, confirm that you applied the common denominator to all terms in the equation and that any negative signs were correctly transferred across the parentheses.

5.    (A)

Susan weighs m pounds more than Anna does, and together they weigh a total of n pounds. Which of the following represents Anna's weight in pounds?

Step 1: Analyze the Question

In this question, we must determine the relationship between Susan and Anna's weights. Notice that even though there are variables in the answer choices, it will be difficult to apply the Picking Numbers strategy here, because without the actual weights of the girls, we are still left with algebraic equations to solve. We must instead directly apply algebra to solve this question.

Step 2: State the Task

To solve this question, we must first translate the words into equations, then isolate Anna's weight on one side of the equation.

Step 3: Approach Strategically

Let's call Anna's weight A pounds.

Because Susan weighs m pounds more than Anna, Susan's weight is A + m pounds.

Because the sum of Anna's weight and Susan's weight is n pounds, we have the equation A + (A + m) = n.

We can now solve this equation for A:

image

The correct answer is (A).

Step 4: Confirm Your Answer

When confirming your answer, make sure that the original phrases were correctly translated and make sense.

Practice Set: Quadratic Equations on the GMAT

6.    (B)

Is n negative?

Step 1: Analyze the Question Stem

In this Yes/No question, sufficiency means determining that n is either definitely negative or definitely non-negative (zero or positive).

Step 2: Evaluate the Statements Using 12TEN

Statement (1) provides a quadratic equation in which the sign of the constant term (−6) is negative. A negative sign in this position of a quadratic equation guarantees that the quadratic has two roots—one positive and one negative. This statement is therefore insufficient. Eliminate (A) and (D).

Statement (2) provides a quadratic equation in which the sign of the constant term (9) is positive. A plus sign here means the quadratic may have one (if the quadratic is a perfect square) or two roots. In the case of two roots, the roots would either be both positive or both negative. The plus sign before the middle term (6n) indicates that the roots will both be negative. This statement is therefore sufficient to provide a definite “yes” answer to the question. Eliminate (C) and (E).

Statement (2) alone is sufficient, so the correct answer is (B).

7.    (D)

Which of the following expressions CANNOT be equal to 0 when x2 − 2x = 3?

Step 1: Analyze the Question

Pay special attention to the language used in this question stem. We must determine which of the answer choices cannot be equal to zero if the equation x2 − 2x = 3 is true. Another way to interpret this would be to eliminate the answer choices that share common factors with the equation in the question stem.

Step 2: State the Task

We must first factor the equation in the question stem. Then we’ll factor the equations in the answer choices until we find an equation that does not share a factor with the stem equation.

Step 3: Approach Strategically

Factoring the question stem

We must have all terms on one side of the equation and zero on the other. To do so, we subtract 3 from both sides of x2 − 2x = 3 and get x2 − 2x − 3 = 0.

This factors to (x − 3)(x + 1) = 0.

Now, we just check to see which answer choice has either of these factors and eliminate those that do. Because this is a “which of the following” question, let's start with (E).

(E) is x2 − 9 = (x + 3)(x − 3). Eliminate.

(D) is x2 − 7x + 6 = (x − 6)(x − 1). Correct.

Since there can be only one correct answer to this question, we can stop here, confident that each of the other answer choices will share at least one factor with the equation in the question stem. For the record, testing the other answer choices yields the following:

(C) is x2 − x − 2 = (x − 2)(x + 1). Eliminate.

(B) is x2 − 4x + 3 = (x − 3)(x − 1). Eliminate.

(A) is x2 − 6x + 9 = (x − 3)(x − 3). Eliminate.

(D) is the only answer choice without either factor, so it is correct.

Step 4: Confirm Your Answer

When factoring equations, an easy way to check your work is to expand the binomials back into the original equation by applying the FOIL technique. Be sure to do this for every equation you factor to avoid simple calculation errors.

8.    (A)

If x2 − 2x − 15 = (x + r)(x + s) for all values of x, and if r and s are constants, then which of the following is a possible value of r − s?

Step 1: Analyze the Question

At first glance, this question appears very abstract, but if we look a little deeper, we are really only asked to factor the expression on the left side of the equation and compare the binomials with the variables on the right.

Step 2: State the Task

Factor the expression on the left side of the equation.

Step 3: Approach Strategically

Factor the expression x2 − 2x − 15. We have x2 − 2x − 15 = (x − 5)(x + 3).

So (x − 5)(x + 3) = (x + r)(x + s).

So either (1) r = −5 and s = 3 or (2) r = 3 and s = −5.

If r = −5 and s = 3, then r − s = −5 − 3 = −8.

If r = 3 and s = −5, then r − s = 3 − (−5) = 3 + 5 = 8.

The possible values of r − s are −8 and 8. Only 8 appears among the answer choices. Choice (A) is correct.

Step 4: Confirm Your Answer

When factoring equations, an easy way to check your work is to expand the binomials back into the original equation by applying the FOIL technique. Be sure to do this for every equation you factor to avoid simple calculation errors.

Practice Set: Systems of Linear Equations on the GMAT

9.    (A)

A theater charges $12 for seats in the orchestra and $8 for seats in the balcony. On a certain night, a total of 350 tickets were sold for a total cost of $3,320. How many more tickets were sold that night for seats in the balcony than for seats in the orchestra?

Step 1: Analyze the Question

A theater sells $12 seats in the orchestra and $8 seats in the balcony. A total of 350 seats were sold for $3,320 on a certain night.

Step 2: State the Task

Find the difference in the number of tickets sold between the balcony and the orchestra.

Step 3: Approach Strategically

With two variables to solve for and enough information to set up two distinct equations, the best approach is to set up a system of linear equations. If R = orchestra seats and B = balcony seats, then we can set up the two equations 12R + 8B = 3,320 and R + B = 350. Given the relative simplicity of the second equation, substitution is the preferred approach. Solving for B in the second equation yields B = 350 − R. Substitute this value for B into the first equation and solve for R:

image

Since R + B = 350 and R = 130, B must be 350 − 130 = 220. Watch out for trap answers (D) and (E), as the problem isn’t asking for the number of orchestra or balcony tickets. We need the difference between the two, which is 220 − 130 = 90. The correct answer is (A).

Step 4: Confirm Your Answer

We can confirm our answer by trying combination. Multiplying the second equation by 8 gives us 8R + 8B = 2,800. Subtracting this equation from the first gives us 4R = 520, or R = 130. Combination provides the same value for R and will, by extension, provide the same value for B. We’ve confirmed that choice (A) is correct.

10.   (A)

Hallie has only nickels, dimes, and quarters in her pocket. If she has at least 1 of each kind of coin and has a total of $2.75 in change, how many nickels does she have?

Step 1: Analyze the Question Stem

We can translate the information in the question stem into the equation 0.05N + 0.10D + 0.25Q = 2.75, where N, D, and Q are the numbers of nickels, dimes, and quarters, respectively. To solve for N, we need two more distinct equations containing those variables (or else some more direct information about N or 0.10D + 0.25Q).

Step 2: Evaluate the Statements Using 12TEN

Statement (1) can be translated into the equations N + D + Q = 21 and D = 2N. This gives us a total of three distinct equations and is, therefore, enough information to solve for N. We can eliminate (B), (C), and (E).

Statement (2) can be translated into the equation 0.25Q = 1.50, which simplifies to Q = 6. With Statement (2) and the question stem, we only have two distinct equations (and no special case), so we cannot answer the question.

Since Statement (2) is not sufficient, the correct answer is (A).

11.   (C)

Jacob is now 12 years younger than Michael. If 9 years from now Michael will be twice as old as Jacob, how old will Jacob be in 4 years?

Step 1: Analyze the Question

We must first translate the question into algebraic equations and then apply the techniques of combination or substitution to solve for Jacob's age.

Step 2: State the Task

Translate the question stem; then solve for Jacob's age using substitution.

Step 3: Approach Strategically

Equation for Michael's and Jacob's current ages: M = J + 12.

Equation for Michael's and Jacob's ages 9 years from now: M + 9 = 2(J + 9).

We now have two distinct equations with two unknowns that we can solve. We can substitute the first equation into the second for M, yielding this equation:

image

Thus, J = 3. So if Jacob is 3 years old now, then in 4 years, he will be 7 years old.

Step 4: Confirm Your Answer

We can plug J = 3 into the original question stem to confirm our answer. Be careful with your calculations, since (A) is a trap for those who don’t add 4 years to his current age.

Practice Set: Special Cases in Systems of Linear Equations on the GMAT

12.   (B)

If the average (arithmetic mean) of a and b is 45 and the average of b and c is 70, what is the value of c − a?

Step 1: Analyze the Question

We are given values for the average of a and b and the average of b and c.

Step 2: State the Task

Find the value of c − a.

Step 3: Approach Strategically

The average of two numbers is half their sum, so if the average of a and b is 45 and the average of b and c is 70, then a + b = 90 and b + c = 140. Using combination to subtract the first equation from the second cancels out the b term and gives you c − a = 50. The correct answer is (B).

Step 4: Confirm Your Answer

We can confirm our answer with substitution. Solving for a and c from their respective equations gives us a = 90 − b and c = 140 − b. Substituting these values into c − a gives us (140 − b) − (90 − b) = 50. We’ve confirmed that (B) is correct.

13.   (B)

What is the value of p?

Step 1: Analyze the Question Stem

The question stem gives no information about p, but it makes it clear that only information that will allow us to calculate a value is sufficient. So let's look at the statements.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) is 2p − m = 5p + 2m. We see that if we simplify this equation, no variable cancels out. We have one equation with two variables, m and p. So we cannot solve this equation for the value of either variable. Thus, there is not enough information to find the value of p. Statement (1) is insufficient. We can eliminate (A) and (D).

Statement (2) is 2(p − m) = 8 − 2m. Let's simplify this equation to see if a variable cancels out. Multiplying out the left side of the equation 2(p − m) = 8 − 2m, we have 2p − 2m = 8 − 2m. Adding 2m to both sides of this equation, we have 2p = 8. So the variable m canceled out in the simplification. We are left with one first-degree, or linear, equation with the variable p.

We know that we can solve this first-degree equation for a single possible value of p. Statement (2) is sufficient, and (B) is correct. If you see a question like this one on Test Day, don’t be too quick to assume that you need two equations to solve for exact values for the two variables—be sure to simplify the equations first.

14.   (E)

What is the value of x?

Step 1: Analyze the Question Stem

We must determine whether or not there is sufficient information to determine the value of x. Because no information is given in the question stem, let's look at the statements.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) gives us 3x + 2y = 6, which is one equation with two variables x and y. Because the number of variables is greater than the number of different equations and it is not possible to simplify, Statement (1) is insufficient. We can eliminate (A) and (D).

Statement (2) gives us 4y = 12 − 6x, which is one equation with the variables x and y. Because the number of variables is greater than the number of different equations and it is not possible to simplify, Statement (2) is insufficient. We can eliminate (B).

Now let's look at the statements taken together. It is important to keep in mind that when we have a given number of first-degree, or linear, equations, we will be able to solve for all the variables only when the equations are different. If we add 6x to both sides of the equation 4y = 12 − 6x of Statement (2), we obtain the equation 6x + 4y = 12.

Now the equations 3x + 2y = 6 and 6x + 4y = 12 are equivalent equations. If we multiply both sides of the equation 3x + 2y = 6 of Statement (1) by 2, we will obtain the equation 6x + 4y = 12, which is a rearrangement of the equation 4y = 12 − 6x of Statement (2). When we take the two statements together, we have one equation with two variables. We cannot solve this equation for the value of x (or the value of y).

Therefore, the two statements taken together are insufficient, and (E) is correct.

Practice Set: Functions and Symbolism on the GMAT

15.   (D)

Let image and image, for all integers x and y. If m = 2, image is equal to which of the following?

Step 1: Analyze the Question

This is a symbolism problem containing two definitions and a nested function. We will need to begin with the innermost function and work our way outward.

Step 2: State the Task

Find the value of image.

Step 3: Approach Strategically

Symbolism problems may look frighteningly complicated, so it's important to remember that they are nothing more than dressed-up equations for which you are asked to plug in a value. In this question, we are given two definition equations and a nested function (a function within a function). We’re also told that m = 2. With nested functions, we must always start with the innermost equation and work outward, so begin by plugging 2 in for the variable in the circle equation: image. Now take the result, 3, and plug that into the square equation:

image

The value of image is 5, so the correct answer is (D).

Step 4: Confirm Your Answer

When working with nested functions, remember to always work from the inside out. Doing so allows us to avoid trap choices like (C), which is the result of working from the outside in.

16.   (B)

image

In the multiplication problem above, each of the symbols ◊, Δ, and • represents a positive digit. If ◊ > Δ, what is the value of ◊?

Step 1: Analyze the Question Stem

Before beginning the solution, let's note for clarity that a positive digit is one of the integers 1, 2, 3, 4, 5, 6, 7, 8, and 9. We are given in the question stem that the product of the digits ◊ and ▵ is • and that ◊ > ▵. We want to know if there is sufficient information to determine the value of the digit ◊.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) says that ▵ = 1. So the product of ◊ and 1 is •. Because the product of any number and 1 is that number, the product of ◊ and 1 is ◊. So ◊ = •. We know from the question stem that ◊ > ▵, and because Statement (1) says that ▵ = 1, we have ◊ > 1. However, we do not know the value of the digit. So ◊ could be any of the eight remaining positive digits 2, 3, 4, 5, 6, 7, 8, and 9. Because there is more than one possible value for ◊, Statement (1) is insufficient. We can eliminate (A) and (D).

Statement (2) says that • = 9. So the product of ◊ and ▵ is 9. There are two ways to write 9 as the product of two digits, 9 × 1 and 3 × 3. Because ◊ × ▵ = 9 and we must have ◊ > ▵, we must have ◊ = 9 and ▵ = 1. So ◊ has only one possible value, 9. Statement (2) is sufficient. Choice (B) is correct.

17.   (B)

For all numbers a and b, the operation ▼ is defined by a ▼ b = (a + 2)(b − 3). If 3 ▼ x = −45, then x =

Step 1: Analyze the Question

Symbolism questions can appear abstract at first, but by directly applying the definition of the symbol to the numbers provided, these questions become manageable. To work with 3 ▼ x, we will use the defining equation a ▼ b = (a + 2)(b − 3).

Step 2: State the Task

We must solve for x by substituting the values 3 and x into the definition.

Step 3: Approach Strategically

To work with 3 ▼ x, replace a with 3 and b with x in the defining equation. So 3 ▼ x = (3 + 2)(x − 3) = 5(x − 3).

Because 3 ▼ x = −45, we have the equation 5(x − 3) = −45. Let's solve this equation for x. Dividing both sides of the equation 5(x − 3) = −45 by 5, we have x − 3 = −9. Adding 3 to both sides, we have x = −6.

Step 4: Confirm Your Answer

You could confirm your work by FOILing out the expressions and plugging the correct answer choice into the equation.

18.   (B)

If the function f is defined for all x by f(x) = ax2 + bx − 43, where a and b are constants, what is the value of f(3)?

Step 1: Analyze the Question Stem

We are given the definition of a function f and are asked whether there is sufficient information to find the value of f(3). The definition of f is that for all values of x, f(x) = ax2 + bx − 43, where a and b are constants. Since the question stem asks for the value of f(3), let's substitute 3 for x into the definition of f. Doing so, we find that f(3) = a(32) + b(3) − 43 = a(9) + b(3) − 43 = 9a + 3b − 43. Thus, f(3) = 9a + 3b − 43.

We can now paraphrase the question stem as “What is the value of 9a + 3b − 43?” So we will have sufficiency if we can find values for a and b. We could also have sufficiency if we can find the value of the expression 9a + 3b − 43. Since −43 is a constant, its value will not change no matter what we find in the statements, so we can focus on 9a + 3b. Factoring out a 3, we get that 9a + 3b = 3(3a + b). Again, since 3 is a constant, we can ignore it for the purposes of determining sufficiency. We will have sufficiency if we can find the value of 3a + b, because then we will be able to find the value of 3(3a + b) − 43, which equals f(3). Now let's look at the statements.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) says that f(4)= 41. The definition of f is that f(x) = ax2 + bx − 43, where a and b are constants. Let's substitute 4 for x into the definition of f. We have that f(4) = a(42) + b(4) − 43 = a(16) + b(4)− 43 = 16a + 4b − 43. Thus, f(4)= 16a + 4b − 43. Since f(4) = 41, 16a + 4b − 43 = 41. Adding 43 to both sides of this equation, 16a + 4b = 84. Dividing both sides by 4, we have that 4a + b = 21. This is one first-degree equation that contains the two variables a and b. There is no way for us to solve this equation for the value of either a or b, and there is no way for us to solve this equation for the value of the expression 3a + b. Thus, we will not be able to find the value of f(3). Statement (1) is insufficient. We can eliminate (A) and (D).

Statement (2) gives us the value of 3a + b, which is one of the pieces of information we identified in Step 1 as being sufficient to answer the question. We know we can find the value of f(3), so we don’t need to work out all the math. If we weren’t sure whether this statement provides sufficient information, we could substitute 17 for 3a + b into the simplified version of the equation we found in Step 1: f(3) = 9a + 3b − 43 = 3(3a + b) − 43 = 3(17) − 43 = 51 − 43 = 8. Statement (2) is sufficient to solve for the value of f(3), so (B) is correct.

GMAT BY THE NUMBERS: ALGEBRA

Now that you’ve learned how to approach algebra questions on the GMAT, let's add one more dimension to your understanding of how they work.

Take a moment to try the following question. The next page features performance data from thousands of people who have studied with Kaplan over the decades. Through analyzing this data, we will show you how to approach questions like this one most effectively and how to avoid similarly tempting wrong answer choice types on Test Day.

What is the value of g?

  • (1)  f + g = 9

  • (2)  3f − 27 = −3g

  • Statement (1) ALONE is sufficient, but statement (2) is not sufficient.

  • Statement (2) ALONE is sufficient, but statement (1) is not sufficient.

  • BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

  • EACH statement ALONE is sufficient.

  • Statements (1) and (2) TOGETHER are NOT sufficient.

Explanation

This question asks you to find the value of g. Neither statement by itself can give you that value, since each contains the variable f as well.

QUESTION STATISTICS
0% of test takers choose (A)
2% of test takers choose (B)
25% of test takers choose (C)
1% of test takers choose (D)
72% of test takers choose (E)
Sample size = 4,242

It's tempting to think as one-fourth of test takers did: the statements together will be sufficient because you can rewrite Statement (1) as f = 9 − g, substitute into Statement (2), and then solve for g. But that's only tempting to a test taker who doesn’t first simplify Statement (2):

image

This is the same equation as Statement (1), so it doesn’t add any new information. Knowing that f + g = 9 isn’t enough to know the value of g, so the statements are insufficient even when combined. Choice (E) is correct.

GMAT questions rarely give you algebraic statements in their most useful form. It's a safe bet that you’ll need to simplify or re-express almost all the algebra you see on Test Day. Doing so will help you steer clear of common wrong answers and keep you on the road to a higher score.

More GMAT by the Numbers …

To see more questions with answer choice statistics, be sure to review the full-length CATs in your online resources.