GMAT Math : Algebra

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Example Questions

Example Question #946 : Data Sufficiency Questions

The volume of a fixed mass of gas varies inversely with the atmospheric pressure, in millibars, acting upon it, given that all other conditions remain constant.

At 12:00, a balloon was filled with exactly 100 cubic yards of helium. What its current volume?

Statement 1: The atmospheric pressure at 12:00 was 109 millibars.

Statement 2: The atmospheric pressure is now 105 millibars.

Possible Answers:

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

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

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

Explanation:

You can use the following variation equation to deduce the current volume:

$V_{1} P _{1} = V_{2} P _{2}$

or, equivalently,

$V_{2} = \frac{V_{1} P _{1} }{P _{2}}$

To find the current volume $V_{2}$ , you therefore need three things - the initial volume $V_{1}$ , which is given in the body of the question; the initial pressure $P _{1}$ , which you know if you are given Statement 1; and the current pressure, $P _{2}$ , which you know if you are given Statement 2. Just substitute, and solve.

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Example Question #14 : Dsq: Solving Equations

is a rational number. True or false: N = 7

Statement 1: $N^{2} + 4N + 100 = 18N + 51$

Statement 2: |N - 14| = N

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.

EITHER STATEMENT ALONE provides 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.

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

$N^{2} + 4N + 100 = 18N + 51$

$N^{2} + 4N + 100- 18 N - 51 = 18N + 51 - 18 N - 51$

$N^{2} -14N +49 =0$

$(N-7)^{2} = 0$

N-7 = 0

N= 7

Assume Statement 2 alone. If |N - 14| = N , then either N - 14 = N or N - 14 =- N , so we can examine both scenarios.

Case 1:

N - 14 = N

N - 14- N = N - N

-14 = 0

This is identically false, so we dismiss this case.

Case 2:

N - 14 =- N

N - 14 + N + 14 =- N + N + 14

2N = 14

N = 7

Since each equation has only 7 as a solution, either statement alone is sufficient to identify N = 7 as a true statement.

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Example Question #14 : Dsq: Solving Equations

is a real number. True or false: is positive.

Statement 1: $3^{N} > \frac{1}{3}$

Statement 2: The arithmetic mean of 100 and is positive.

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.

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.

Correct answer:

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

Explanation:

Assume both statements are true. We show that we cannot determine for certain whether or not is positive.

Case 1: N = 0

$3^{N} = 3^{0} = 1 > \frac{1}{3}$ , satisfying the condition of Statement 1.

The arithmetic mean of 0 and 100 is half their sum, which is $\frac{0+100}{2} = 50$ , a positive number; the condition of Statement 2 is satisfied.

Case 2: N = 2

$3^{N} = 3^{2} = 9 > \frac{1}{3}$ , satisfying the condition of Statement 1.

The arithmetic mean of 2 and 100 is half their sum, which is $\frac{2+100}{2} = 51$ , a positive number; the condition of Statement 2 is satisfied.

Therefore, may or may not be positive.

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Example Question #16 : Dsq: Solving Equations

True or false: |X| = 9

Statement 1: $X^{2}+ 27 = 12X$

Statement 2: $X^{2}+6X = 27$

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.

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 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.

Correct answer:

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

Explanation:

The absolute value of a positive number is that number; the absolute value of a negative number is the positive opposite.

Assume Statement 1 alone.

$X^{2}+ 27 = 12X$

$X^{2}+ 27- 12X= 12X - 12X$

$X^{2}- 12X + 27= 0$

(X-9)(X-3) = 0

Either X - 9 = 0 , in which case X = 9 , and

or X-3= 0 , in which case X = 3 .

Either |X| = 9 or |X| = 3 .

Assume Statement 2 alone.

$X^{2}+6X = 27$

$X^{2}+6X - 27 = 27 - 27$

$X^{2}+6X - 27 = 0$

(X+9)(X-3) = 0

Either X + 9 = 0 , in which case X = -9 , and

or X-3= 0 , in which case X = 3 .

Either |X| = 9 or |X| = 3 .

Therefore, neither statement alone proves or disproves that |X| = 9 . But if both statements are true, then it must hold that X = 3 , since this is the only solution of both statements. Therefore, |X| = 9 can be proved false.

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Example Question #17 : Dsq: Solving Equations

is a real number. True or false: N = -3

Statement 1: |N+ 4| = 1

Statement 2: $2^{N} = 0.125$

Possible Answers:

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.

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.

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:

Assume Statement 1 alone. The absolute value of a positive number is that number; the absolute value of a negative number is the positive opposite. As can be seen from substitution, there are two solutions to the equation |N+ 4| = 1 - N = -3 and N = -5

If N = -3 , the equation becomes

|N+ 4| = 1

|-3+ 4| = 1

|1| = 1 , a true statement.

If N = -5 , the equation becomes

|N+ 4| = 1

|-5+ 4| = 1

|-1| = 1 , a true statement.

Therefore, that N = -3 cannot be proved or disproved.

Assume Statement 2 alone.

$2^{N} = 0.125$ can be rewritten with the decimal expression in fraction form as

$2^{N} = \frac{1}{8}$

$2^{N} = \frac{1}{2^{3}}$

$2^{N} = 2^{-3}$

Since if two powers of the same number are equal, the exponents are equal, it follows that N = -3 .

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Example Question #18 : Dsq: Solving Equations

True or false: | N | = 5

Statement 1: N (N+10 ) = 5 (2N+5)

Statement 2: | N + 10 | = 15

Possible Answers:

BOTH STATEMENTS TOGETHER do 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.

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.

EITHER STATEMENT ALONE provides 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:

Assume Statement 1 alone. The solutions of the equation can be found as follows:

N (N+10 ) = 5 (2N+5)

$N ^{2}+10 N = 10N+25$

$N ^{2}+10 N - 10 N = 10N+25 - 10 N$

$N ^{2}=25$

$N = \pm \sqrt{25} = \pm 5$

The absolute value of a positive number is that number; the absolute value of a negative number is the positive opposite. If N = 5 , then | N | = 5 . If N = - 5 , then also. Therefore, while the value of cannot be determined for certain,either way, is a true statement.

Assume Statement 2 alone. By substitution, we can see two values of that make this true:

N = 5

| N + 10 | = 15

| 5 + 10 | = 15

|15| = 15 , which is true.

N = -25

| N + 10 | = 15

| -25 + 10 | = 15

|-15| = 15 , which is also true.

One of these two solutions has absolute value 5, but the other does not. This makes it indefinite whether | N | = 5 .

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Example Question #1 : Dsq: Solving Quadratic Equations

Consider the equation $x^{2} +Bx = C$

How many real solutions does this equation have?

Statement 1: There exists two different real numbers m, n such that m+n=B and mn=-C

Statement 2: is a positive integer.

Possible Answers:

EITHER statement ALONE is 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.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

$x^{2} +Bx = C$ can be rewritten as $x^{2} +Bx - C = 0$

If Statement 1 holds, then the equation can be rewritten as (x+m) (x+n)=0 . This equation has solution set $\left \{ -m,-n\right \}$ , which comprises two real numbers.

If Statement 2 holds, the discriminant $B^{2} -4 \cdot1\cdot(-C)= B^{2} +4C$ is positive, being the sum of a nonnegative number and a positive number; this makes the solution set one with two real numbers.

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Example Question #2 : Dsq: Solving Quadratic Equations

Let B,C be two positive integers. How many real solutions does the equation $x^{2} + Bx + C = 0$ have?

Statement 1: is a perfect square of an integer.

Statement 2: $C = \left ( \frac{B}{2} \right ) ^{2}$

Possible Answers:

EITHER statement ALONE is sufficient 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.

BOTH statements TOGETHER are insufficient 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:

The number of real solutions of the equation $Ax^{2} + Bx + C = 0$ depends on whether discriminant $B^{2} - 4A C$ is positive, zero, or negative; since A = 1 , this becomes $B^{2} - 4C$ .

If we only know that is a perfect square, then we still need to know to find the number of real solutions. For example, let C = 1 , a perfect square. Then the discriminant is $B^{2} - 4$ , which can be positive, zero, or negative depending on .

But if we know $C = \left ( \frac{B}{2} \right ) ^{2}$ , then the discriminant is

$B^{2} - 4C = B^{2} - 4 \cdot \left ( \frac{B}{2} \right ) ^{2} = B^{2} - 4 \cdot \frac{B^{2}}{4} = B^{2} - B^{2} = 0$

Therefore, $x^{2} + Bx + C = 0$ has one real solution.

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Example Question #154 : Algebra

Does the solution set of the following quadratic equation comprise two real solutions, one real solution, or one imaginary solution?

$Ax^{2} -3Ax + B = 0$

Statement 1: B = 2A

Statement 2: B = A + 2

Possible Answers:

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.

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.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER 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:

The sign of the discriminant of the quadratic expression answers this question; here, the discriminant is

$(-3A)^{2} - 4AB$ ,

$9A^{2} - 4AB$

If we assume Statement 1 alone, this expression becomes

$9A^{2} - 4AB = 9A^{2} - 4A (2A) = 9A^{2} - 8A^{2} = A^{2}$

Since we can assume is nonzero, $A^{2} > 0$ . This makes the discriminant positive, proving that there are two real solutions.

If we assume Statement 2 alone, this expression becomes

$9A^{2} - 4AB = 9A^{2} - 4A \left ( A + 2\right ) = 9A^{2} - 4A^{2}-8A = 5A^{2}-8A$

The sign of $5A^{2}-8A$ can vary.

Case 1: A = 1

Then $5A^{2}-8A = 5 \cdot 1^{2}-8\cdot 1 = 5 - 8 = -3 < 0$

giving the equation two imaginary solutions.

Case 2: A = 2

Then $5A^{2}-8A = 5 \cdot 2^{2}-8\cdot 2 = 20 - 16 = 4 > 0$

giving the equation two real solutions.

Therefore, Statement 1, but not Statement 2, is enough to answer the question.

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Example Question #2 : Dsq: Solving Quadratic Equations

is a positive integer.

True or false:

$x^{2} - 6x + 8 = 0$

Statement 1: is an even integer

Statement 2: x < 6

Possible Answers:

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

EITHER 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.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

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

Explanation:

The quadratic expression can be factored as (x+? )(x+? ) , replacing the question marks with integers whose product is 8 and whose sum is . These integers are -2,-4 , so the equation becomes:

$x^{2} - 6x + 8 = 0$

(x-2 )(x-4 ) = 0

Set each linear binomial to 0 and solve:

$x - 2 = 0 \Rightarrow x = 2$

$x - 4 = 0 \Rightarrow x = 4$

Therefore, for the statement to be true, either x = 2 or x = 4 . Each of Statement 1 and Statement 2, taken alone, leaves other possible values of . Taken together, however, they are enough, since the only two positive even integers less than 6 are 2 and 4.

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #946 : Data Sufficiency Questions

Example Question #14 : Dsq: Solving Equations

Example Question #14 : Dsq: Solving Equations

Example Question #16 : Dsq: Solving Equations

Example Question #17 : Dsq: Solving Equations

Example Question #18 : Dsq: Solving Equations

Example Question #1 : Dsq: Solving Quadratic Equations

Example Question #2 : Dsq: Solving Quadratic Equations

Example Question #154 : Algebra

Example Question #2 : Dsq: Solving Quadratic Equations

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