GMAT Math : Algebra

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Example Question #81 :Algebra

The first two terms of a sequence, $a_{1}$ and $a_{2}$ , are given specific values, which you are not given. Each successive term is equal to the sum of the two preceding terms. For example, the fifth term is equal to the sum of the third and fourth terms.

Calculate $a_{1}$ .

Statement 1: $a_{5} = 18$

Statement 2: $a_{8} = 76$

Possible Answers:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Suppose the first and second terms of the sequence areand, respectively. The first eight terms are

A + B

B + (A+B ) , or A + 2B

(A+B )+ (A+2B ) , or 2A + 3B - this is the fifth term.

Continuing to add:

3A+5B

5A + 8B

8A+13B - the eighth term.

From Statement 1 alone, we get that

2A + 3B = 18

From Statement 2 alone, we get that

8A+13B = 76

Both statements are equations in two variables, so neither statement alone tells us the value of either. But if we know both statements together, we have a system of equations that can be solved as follows:

2A + 3B = 18

-4(2A + 3B )= -4( 18)

-8A -12B = -72

$\underline{\; \; 8A+13B = 76}$

B = 4

2A + 3 (4) = 18

2A +12 = 18

2A = 6

A = 3 - this is $a_{1}$ .

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Example Question #82 :Algebra

Evaluate $a_{10}$ .

Statement 1: $a_{9} = 123$ and $a_{11} = 322$

Statement 2: The arithmetic mean of $a_{8}$ and $a_{9}$ is 99.5.

Possible Answers:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. By the rule, $a_{9}+ a_{10} = a_{11}$ , so by substitution:

$123+ a_{10} =322$

$a_{10} =322 - 122 = 199$ .

Assume Statement 2 alone. The arithmetic mean of two numbers is half their sum, so

$\frac{a_{8}+a_{9}}{2} = 99.5$

$\frac{a_{8}+a_{9}}{2} \cdot 2 = 99.5 \cdot 2$

$a_{8}+a_{9} = 199$

By the rule, $a_{8}+a_{9} = a_{10}$ , so

$a_{10} = 199$ .

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Example Question #83 :Algebra

True or false: $\left \{ a_{n} \right \}$ . n = 1, 2, 3,... is an arithmetic sequence.

Statement 1: $a_{4} - a_{3} = a_{2} -a_{1}$

Statement 2: $a_{6}- a_{5} = a_{8} - a_{7}$

Possible Answers:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Explanation:

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

The two statements together demonstrate that two such differences are equal to each other, and that two other such differences are equal to each other. No information, however, is given about any other differences, of which there are infinitely many. Therefore, the question of whether the sequence is arithmetic or not is unresolved.

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Example Question #84 :Algebra

Give the first term of an arithmetic sequence $a_{n}$ .

Statement 1: The arithmetic mean of $a_{2}$ and $a_{3}$ is 24.

Statement 2: The common difference of the sequence is 10.

Possible Answers:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Each term of an arithmetic sequence is the preceding term plus the same number, the common difference.

Assume Statement 1 alone, and examine the sequences:

12, 20, 28,...

9, 19, 29,...

Both sequences are arithmetic; the first has common difference 8, the second, common difference 9. In both sequences, the arithmetic mean of the second and third terms - half their sum - is $\frac{20+28}{2} = \frac{19+29}{2} = 24$ . However, the first term differs.

Assume Statement 2 alone. The common difference alone is not enough to determine the first term, as evidenced in these two sequences:

10, 20, 30,...

11,21, 31, ...

both of which have common difference 10.

Now assume both statements. The arithmetic mean of $a_{2}$ and $a_{3}$ is 24, so

$\frac{a_{2}+a_{3}}{2} = 24$

$\frac{1}{2}a_{2}+ \frac{1}{2}a_{3} = 24$

Also, the common difference is 10, so

$a_{3}-a_{2} = 10$

These two equations form a two-by-two linear system which can be solved as follows:

$\left (\frac{1}{2}a_{2}+ \frac{1}{2}a_{3} \right )\cdot 2 = 24 \cdot 2$

$a_{3}+a_{2} = 48$

$\underline{a_{3}-a_{2} = 10}$

$2a_{3} \; \; \; \; \; \; = 58$

$2a_{3} \div 2 = 58 \div 2$

$a_{3}= 29$

$a_{2} = a_{3} - 10 = 29 - 10 = 19$

$a_{1} = a_{2} - 10 = 19 - 10 = 9$

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Example Question #85 :Algebra

Function

Above is the graph of a function.

Given: a functionwith domain [-3, 3]

True or false: $g ^{-1}$ exists.

Statement 1: For eachsuch that $-3 \le N \le 0$ , it holds that g(N) = f(N- 3) .

Statement 2: For eachsuch that $0 \le N \le 3$ , it holds that g(N) = f(N- 3) .

Possible Answers:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation:

g(x) = f(x- 3) in each statement, so the graph of the functionis the same as that of the functiontranslated three units right. However, we are restricting the domain ofto [-3, 3] . Each of the two statements examines one half of the graph. See the graph below, which divides the graph into the portion on the domain [-3, 0] (in blue; discussed in Statement 1) and the portion on the domain [0, 3] (in green, discussed in Statement 2):

Function 5

Assume Statement 1 alone. The portion of the graph ofon the domain [-3, 0] passes the horizontal line test, since no horizontal line passes through it twice. However, without knowing anything about the other half of the graph, the question about whether $g^{-1}$ exists cannot be resolved.

Assume Statement 2 alone. Notice that we can draw a horizontal line through this portion of the graph that passes through it twice - $y = \frac{1}{2}$ would work. This half of the graph alone proves that $g^{-1}$ does not exist.

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Example Question #86 :Algebra

Give the first term of an arithmetic sequence $a_{n}$

Statement 1: The eighth and ninth terms are 65 and 72, respectively.

Statement 2: $a_{2} +7 = a_{3}$

Possible Answers:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Correct answer:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

From Statement 1, the difference of two consecutive terms $a_{8}$ and $a_{9}$ is $a_{9} - a_{8 } = 72 - 65 = 7$ ; since, in an arithmetic sequence,

$a_{n} = a_{1} + (n-1)d$

we can substitute n = 9, d=7 and find $a_{1}$ .

$72= a_{1} + (9-1) \cdot 7$

$72= a_{1} +8 \cdot 7$

$72= a_{1} + 56$

$a_{1} = 72 - 56 = 16$

However, Statement 2 alone gives insufficient information helpful in finding $a_{1}$ ; for example, the sequences

1, 8, 15,...

and

2, 9, 16,...

have the characteristic that $a_{2} +7 = a_{3}$ - but $a_{1}$ differs between them.

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Example Question #87 :Algebra

Given a function, it is known that:

f(1) = 7

f(2) = 10

f(3) = 6

f(4) = 9

f(5) = 8

Doeshave an inverse?

Statement 1: The range ofis the set $\left \{6, 7, 8, 9, 10 \right \}$ .

Statement 2: The domain ofis the set $\left \{ 1, 2, 3, 4, 5 \right \}$ .

Possible Answers:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation:

A functionhas an inverse if and only if, if f(a) = f(b) , then a=b , or, equivalently, if $a \ne b$ , then $f(a) \ne f(b)$ .

Assume Statement 2 alone. If $\left \{ 1, 2, 3, 4, 5 \right \}$ is the entire domain, then f (a) cannot exist for any value ofnot in that set. Also, it can be seen that for each $a, b \in \left \{ 1, 2, 3, 4, 5 \right \}$ such that $a \ne b$ , $f(a) \ne f(b)$ . Therefore,has an inverse.

Assume Statement 1 alone. We show that the question of whetherhas an inverse cannot be answered by taking two cases.

Case 1: $\left \{ 1, 2, 3, 4, 5 \right \}$ is the entire domain. If this is true, then the range is $\left \{6, 7, 8, 9, 10 \right \}$ , and the situation described in Statement 2 exists; consequently,has an inverse.

Case 2: $\left \{ 1, 2, 3, 4, 5 , 6 \right \}$ is the domain, and f(6) = 6 . The range is still the set $\left \{6, 7, 8, 9, 10 \right \}$ . However, f(3) = f(6) = 6 , so there exists a,b in the range such that $a \ne b$ , but f(a) = f(b) . This means thatdoes not have an inverse.

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Example Question #88 :Algebra

Given a function, it is known that:

f(1) = 7

f(2) = 10

f(3) = 6

f(4) = 9

f(5) = 8

Doeshave an inverse?

Statement 1: f(6)= 7

Statement 2: f(7) =11

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

A functionhas an inverse if and only if, if f(a) = f(b) , then a=b , or, equivalently, if $a \ne b$ , then $f(a) \ne f(b)$ .

If Statement 1 alone is assumed, then this condition is known to not be true, since f(1)=f(6)= 7 . Therefore,does not have an inverse.

If Statement 2 alone is assumed, since no two valuesandare known such that $a \ne b$ and f(a) = f(b) , it is possible forto have an inverse. However, there may or may not be other values in the domain of, any of which may be paired with range elements in the set $\left \{6, 7, 8, 9, 10 \right \}$ . Therefore, Statement 2 does not resolve the issue of whetherhas an inverse.

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Example Question #89 :Algebra

Evaluate $[f(4)+f(-4)] \cdot [g(4)+g(-4)]$ .

Statement 1:is an even function.

Statement 2:is an odd function.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Assume Statement 1 alone. Sinceis even, then by definition, for eachin its domain,

f (c) = f (-c) .

Specifically,

f (4) = f (-4)

and

$[f(4)+f(-4)] \cdot [g(4)+g(-4)] = [f(4)+f(4)] \cdot [g(4)+g(-4)] = 2 f(4) \cdot [g(4)+g(-4)]$ .

Without further information, this expression cannot be evaluated.

Assume Statement 2 alone. Sinceis odd, then by definition, for eachin its domain,

g (c) = - g (-c) .

Specifically,

g(4) = -g (-4) ,

and

$[f(4)+f(-4)] \cdot [g(4)+g(-4)] = [f(4)+f(4)] \cdot [g(4)-g(-4)] = 2 f(4) \cdot 0 = 0$ .

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Example Question #90 :Algebra

Letandbe functions, the doimains of both of which are the set of all real numbers. Is the functionodd, even, or neither?

Statement 1:is neither odd nor even.

Statement 2:is neither odd nor even.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true. A functionis odd if, for allin its domain, f(c) = -f(-c) , and even if, for allin its domain, f(c) = f(-c) . We show that knowing that neithernoris odd or even is insufficient to answer the question of whetheris odd, even, or neither by examining two scenarios.

Case 1: f (x)= x+1 and g (x) = x-1 .

f (1) = 1+1 = 2

f (-1) = -1+1 = 0

Since there exists at least one valuefor which neither f(c) = f(-c) nor f(c) = -f(-c) ,is neither odd nor even.

By a similar argument,can be shown to be neither odd nor even.

However,

$(fg) (x) = f(x)\cdot g(x) = (x+1)(x-1) = x^{2}-1$

and, for allin the domain,

$(fg) (-c) = (-c)^{2}-1 = c^{2} - 1 = (fg) (c)$ ,

makingeven.

Case 2: f (x)= x+1 and g (x) = x-2 .

Again,is neither even nor odd, andcan be similarly demonstrated to be neither as well.

$(fg) (x) = f(x)\cdot g(x) = (x+1)(x-2)$

$(fg) (2) = (2+1)(2-2) = 3 \cdot 0 = 0$

$(fg) (-2) = (-2+1)(-2-2) = -1 \cdot (-4) = 4$

Since there is at least one valuein the domain ofsuch that $fg(-c) \ne fg (c)$ ,is neither odd nor even.

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #81 :Algebra

Example Question #82 :Algebra

Example Question #83 :Algebra

Example Question #84 :Algebra

Example Question #85 :Algebra

Example Question #86 :Algebra

Example Question #87 :Algebra

Example Question #88 :Algebra

Example Question #89 :Algebra

Example Question #90 :Algebra

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