GMAT Math : Problem-Solving Questions

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Example Questions

Example Question #64 : Understanding Exponents

Solve for :

$2^{x+1} \cdot 4 ^{x} = 8^{3x}$

Possible Answers:

x = 1

x= 6

$x= \frac{3}{2}$

The equation has no solution.

$x = \frac{1}{6}$

Correct answer:

$x = \frac{1}{6}$

Explanation:

Rewrite all expressions as powers of 2, and use the exponent rules as follows:

$2^{x+1} \cdot \left ( 2^{2}\right ) ^{x} = \left ( 2^{3}\right ) ^{3x}$

$2^{x+1} \cdot 2^{2 \cdot x} = 2^{3 \cdot 3x }$

$2^{x+1} \cdot 2^{2x} = 2^{9x }$

$2^{x+1 + 2x} = 2^{9x }$

$2^{3x+1 } = 2^{9x }$

Since powers of the same base are equal, set the exponents equal to each other:

3x+1=9x

3x+1 - 3x=9x - 3x

1 = 6x

$x = \frac{1}{6}$

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Example Question #1171 : Problem Solving Questions

Solve for :

$2 ^{x} + 2 ^{x-1} = 24$

Possible Answers:

x= 4

x= 3

$x= \frac{7}{2}$

The equation has no solution.

$x = \frac{9}{2}$

Correct answer:

x= 4

Explanation:

Rewrite the first expression to get:

$2 ^{x} + 2 ^{x-1} = 24$

$2 \cdot 2 ^{x-1} + 2 ^{x-1} = 24$

$2 \cdot 2 ^{x-1} + 1 \cdot 2 ^{x-1} = 24$

$3 \cdot 2 ^{x-1} = 24$

$\frac{3 \cdot 2 ^{x-1} }{3}= \frac{24}{3}$

$2 ^{x-1} = 8$

$2 ^{x-1} = 2^{3}$

x-1 = 3

x=4

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Example Question #65 : Understanding Exponents

Solve for :

$2^{3t} = 8^{t - 4}$

Possible Answers:

t = 0

t = 3

t = -4

t = 1

The equation has no solution.

Correct answer:

The equation has no solution.

Explanation:

Rewrite both sides as powers of 2 using the exponent rules as follows:

$2^{3t} = 8^{t - 4}$

$2^{3t} = (2^{3}) ^{t - 4}$

$2^{3t} = 2^{3\left ( t - 4 \right )}$

Since powers of the same base are equal, set the exponents equal to each other:

3t = 3t-12

3t - 3t= 3t-12 - 3t

0 = -12

This is identically false, so the equation has no solution.

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Example Question #1171 : Problem Solving Questions

Simplify:

$\frac{(x^3y^4z)^2}{x^{\frac{1}{2}}y^9}$

Possible Answers:

$\frac{x^3z^2}{y}$

$\frac{x^{\frac{11}{2}}z^2}{y}$

$\frac{x^3z}{y^5}$

$\frac{x^{\frac{5}{2}}z^2}{y^5}$

$\frac{x^5z^2}{y^{10}}$

Correct answer:

$\frac{x^{\frac{11}{2}}z^2}{y}$

Explanation:

The first step is two distribute the squared exponent across the numerator:

$\frac{(x^3y^4z)^2}{x^{\frac{1}{2}}y^9} = \frac{x^6y^8z^2}{x^{\frac{1}{2}}y^9}$

We can then subtract the denominator's exponents from the numerator's leaving us with the answer:

$\frac{x^6y^8z^2}{x^{\frac{1}{2}}y^9} = \frac{x^{\frac{11}{2}} z^2}{y}$

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Example Question #61 : Exponents

The first two terms of a geometric sequence are and $C ^{3}$ , in that order. Give the tenth term.

(Assume is positive.)

Possible Answers:

$C^{19}$

$C^{30}$

$C^{21}$

$C^{27}$

$C^{11}$

Correct answer:

$C^{19}$

Explanation:

The common ratio of the geometric sequence can be found by dividing the second term by the first:

$\frac{C^{3}}{C} = C^{2}$

The tenth term of the sequence is therefore

$a_{10} = a_{1}r^{10-1}=C \cdot \left ( C^{2}\right ) ^{9} = C \cdot C^{18} = C^{19}$

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Example Question #1172 : Problem Solving Questions

$7^{3x+2} = 49 ^{y - 12 }$

Express in terms of .

Possible Answers:

$y = \frac{3}{2} x+13$

y = 6x+16

y = 6x - 8

$y = \frac{3}{2} x+25$

$y = \frac{3}{2} x-11$

Correct answer:

$y = \frac{3}{2} x+13$

Explanation:

$7^{3x+2} = 49 ^{y - 12 }$

$7^{3x+2} = \left (7^{2} \right ) ^{y - 12 }$

$7^{3x+2} = 7^{2 \cdot (y - 12 )}$

$7^{3x+2} = 7^{2 y - 24}$

2y-24 = 3x+2

2y-24 + 24= 3x+2 + 24

2y = 3x+2 6

$\frac{1}{2}\cdot 2y = \frac{1}{2}\cdot (3x+2 6)$

$y = \frac{3}{2} x+13$

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Example Question #1177 : Gmat Quantitative Reasoning

$4^{t} = \frac{2^{x+3}}{8^{y - 5}}$

Which of the following is a true statement?

Possible Answers:

$t= \frac{1}{2} x - \frac{3}{2} y-6$

t = 2x-6y- 24

t = 2x-6y+ 36

t = 2x-6y+12

$t= \frac{1}{2} x - \frac{3}{2} y+9$

Correct answer:

$t= \frac{1}{2} x - \frac{3}{2} y+9$

Explanation:

$4^{t} = \frac{2^{x+3}}{8^{y - 5}}$

$(2^{2}) ^{t} = \frac{2^{x+3}}{(2^{3}) ^{y - 5}}$

$2^{2t} = \frac{2^{x+3}}{2^{3(y-5)}}$

$2^{2t} = \frac{2^{x+3}}{2^{3y-15}}$

$2^{2t} = 2^{(x+3)- (3y-15)}$

2t=(x+3)- (3y-15)

2t=x+3- 3y+15

2t=x - 3y+18

$\frac{1}{2} \cdot 2t= \frac{1}{2} \cdot ( x - 3y+18 )$

$t= \frac{1}{2} x - \frac{3}{2} y+9$

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Example Question #1178 : Gmat Quantitative Reasoning

$0.25 ^{t} = 2 ^{3y}$

Express in terms of .

Possible Answers:

$t = -\frac{2}{3}y$

$t = -\frac{3}{2}y$

$t = \sqrt{3y}$

t = 3y+2

$t = \sqrt[3]{2y}$

Correct answer:

$t = -\frac{3}{2}y$

Explanation:

$0.25 ^{t} = 2 ^{3y}$

$\left ( \frac{1}{4} \right )^{t} = 2 ^{3y}$

$\left ( 2 ^{-2 }\right )^{t} = 2 ^{3y}$

$2 ^{-2t} = 2 ^{3y}$

-2t = 3y

$\frac{-2t }{-2 }= \frac{3y}{-2}$

$t = -\frac{3}{2}y$

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Example Question #71 : Exponents

$0.04 ^{t} = 25 ^{x+2}$

Express in terms of .

Possible Answers:

t = x-2

t = -x-2

$t = \frac{-1}{x+2}$

t = -x+2

$t = \frac{1}{x+2}$

Correct answer:

t = -x-2

Explanation:

$0.04 ^{t} = 25 ^{x+2}$

$\left ( \frac{1}{25} \right )^{t} = 25 ^{x+2}$

$\left ( 25 ^{-1} \right )^{t} = 25 ^{x+2}$

$25 ^{- t} = 25 ^{x+2}$

-t = x+2

t = -x-2

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Example Question #91 : Algebra

The first and third terms of a geometric sequence are and $K ^{4}$ , in that order. Give the eighth term.

(Assume is positive.)

Possible Answers:

$K^{11 } \sqrt{K}$

$K^{25}$

$K^{10} \sqrt{K}$

$K^{22}$

$K^{14}$

Correct answer:

$K^{11 } \sqrt{K}$

Explanation:

Let be the common ratio of the geometric sequence. Then the third term is $r^{2}$ times the first, so

$r^{2} = \frac{K^{4}}{K} = K^{3}$

and

$r = \sqrt{K^{3}} = K^ { \frac{3}{2}}$ .

The eighth term of the sequence is

$a_{8} = a_{1} \cdot r ^{7}$

$= K \cdot (K^{\frac{3}{2}} )^{7}$

$= K \cdot K^{\frac{21}{2}}$

$= K^{\frac{23}{2}}$

$= K^{11 \frac{1}{2}}$

$= K^{11} \cdot K^{ \frac{1}{2}}$

$= K^{11 } \sqrt{K}$

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All GMAT Math Resources

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GMAT Math : Problem-Solving Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #64 : Understanding Exponents

Example Question #1171 : Problem Solving Questions

Example Question #65 : Understanding Exponents

Example Question #1171 : Problem Solving Questions

Example Question #61 : Exponents

Example Question #1172 : Problem Solving Questions

Example Question #1177 : Gmat Quantitative Reasoning

Example Question #1178 : Gmat Quantitative Reasoning

Example Question #71 : Exponents

Example Question #91 : Algebra

All GMAT Math Resources

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