GMAT Math : Simplifying algebraic expressions

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Example Question #11 : Simplifying Algebraic Expressions

What is the coefficient of $x ^{5}$ in the expansion of $\left (x + 3 \right )^{8}$ ?

Possible Answers:

9,072

4,536

1,512

504

3,024

Correct answer:

1,512

Explanation:

By the Binomial Theorem, the $x^{5}$ term of $\left ( x+3\right ) ^{8}$ is:

$C(8,5) \cdot x^{5} \cdot 3 ^{8-5}$

The coefficient of $x ^{5}$ is therefore:

$C(8,5) \cdot 3 ^{8-5} = \frac{8!}{(8-5)!\; 5!}\cdot 3 ^{3}= \frac{8!}{3!\; 5!}\cdot 27= \frac{40,320}{6\cdot 120}\cdot 27 = 1,512$

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Example Question #11 : Simplifying Algebraic Expressions

Simplify the expression:

$(0.6x + 0.7) ^{3}$

Possible Answers:

$0.216x^{3} + 0.343$

$0.216x^{3} + 0.252x ^{2} + 0.294x + 0.343$

$0.216x^{3} + 2.52x ^{2} + 2.94x + 0.343$

$0.216x^{3} + 0.756x ^{2} + 0.882x + 0.343$

$0.216x^{3} + 7.56x ^{2} + 8.82x + 0.343$

Correct answer:

$0.216x^{3} + 0.756x ^{2} + 0.882x + 0.343$

Explanation:

You can use the pattern for cubing a binomial sum, setting A = 0.6x, B = 0.7 :

$\left (A + B \right )^{3} = A^{3} + 3A ^{2}B + 3AB^{2} + B ^{3}$

$(0.6x + 0.7) ^{3} =\left ( 0.6x \right )^{3} + 3 \cdot \left ( 0.6x \right ) ^{2} \cdot 0.7 + 3\cdot 0.6x \cdot 0.7 ^{2} + 0.7 ^{3}$

$=0.216x^{3} + 0.756x ^{2} + 0.882x + 0.343$

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Example Question #12 : Simplifying Algebraic Expressions

Simplify:

$\left ( 24x ^{3}-3x^{2}+ 12x - 18\right ) \div 6x$

Possible Answers:

$4 x ^{2} - \frac{ 1 }{2}x+ 2- \frac{3}{x}$

$4 x ^{2} - \frac{ 1 }{2}x+ 2- \frac{1}{3x}$

$4 x ^{2} - 2x+ 2- \frac{3}{x}$

$4 x ^{2} - 5x+ 2$

$4 x ^{2} - \frac{ 7 }{2}x+ 2$

Correct answer:

$4 x ^{2} - \frac{ 1 }{2}x+ 2- \frac{3}{x}$

Explanation:

Rewrite, distribute, and simplify where possible:

$\left ( 24x ^{3}-3x^{2}+ 12x - 18\right ) \div 6x$

$=\frac{ 24x ^{3}-3x^{2}+ 12x - 18 }{6x}$

$=\frac{ 24x ^{3}}{6x} - \frac{ 3x^{2} }{6x} + \frac{ 12x }{6x} - \frac{ 18 }{6x}$

$=\frac{ 24}{6} \cdot x ^{3-1} - \frac{ 3 }{6}\cdot x^{2-1} + \frac{ 12 }{6 } - \frac{ 18 }{6 } \cdot \frac{1}{x}$

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

$=4 x ^{2} - \frac{ 1 }{2}x+ 2- \frac{3}{x}$

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Example Question #11 : Simplifying Algebraic Expressions

What is the simplified result of following the steps?

(1) Add to .

(2) Multiply the result by .

(3) Subtract $\left ( 2x+y \right )$ from the result.

Possible Answers:

2x+9y

14x+7y

14x+9y

14x+y

21x

Correct answer:

14x+7y

Explanation:

From (1), we can easily get the result 4x+2y .

Then from (2), we need to multiply 4x+2y by . This gives us 16x+8y .

The last step is to subtract $\left ( 2x+y \right )$ from 16x+8y :

$\left ( 16x+8y \right )-\left ( 2x+y \right )=16x+8y-2x-y=14x+7y$

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Example Question #12 : Simplifying Algebraic Expressions

If positive integer N is divided by 24, the remainder is 6. What is the remainder when N is divided by 10?

Possible Answers:

Correct answer:

Explanation:

The simplest way to solve this problem is to start by picking the smallest positive integer that can be divided by 24. That would be 24, since $24\ast1=24$ . Then, since we need a number that when divided by 24, leaves a remainder of 6, simply add 6 to 24. That gives us 30.

30 is the smallest positive integer that leaves a remainder of 6 when divided by 24.

Finally, divide 30 by 10.

The remainder is 0.

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Example Question #15 : Simplifying Algebraic Expressions

X and Y are positive integers, such that x/y=37.14 . Which of following numbers could be the remainder of x/y ?

Possible Answers:

Correct answer:

Explanation:

Let's set up the problem using algebraic symbols.

x/y=Q+r/y ,

where Q is the quotient of the answer, and r is the remainder.

$x/y=37.14=37+\frac{14}{100}=37+\frac{7}{50}$ ,

which means that $r/y=\frac{7}{50}$ .

Hence, the remainder MUST be a multiple of 7. The only multiple of 7 in the answer choices is 21, so that is our answer.

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Example Question #1322 : Gmat Quantitative Reasoning

A positive integer, X, leaves a remainder of 3 when divided by 5, and a remainder of 4 when divided by 8. What is the remainder when X is divided by 9?

Possible Answers:

Correct answer:

Explanation:

We want to find the smallest number that fits initial conditions, so we need to write two equations to represent the two conditions. If X leaves a remainder of 3 when divided by 5, that is equivalent to:

X=5p+3 , where is some integer greater than or equal to 0.

Likewise, the second condition can be expressed by:

X=8q+4 , where is some integer greater than or equal to 0.

For the first equation, X could be 3, 8, 13, 18, 23, 28, 33, 38...

For the second equation, X could be 4, 12, 20, 28, 36, 42...

The first number that fits our conditions is 28.

So, finally, what's the remainder when 28 is divided by 9?

The answer is 1.

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Example Question #13 : Simplifying Algebraic Expressions

Simplify the following fraction:

$\frac{(a^{3}b^{2}+3a^{2}b^{3}+3ab^{4}+b^{5})}{a^{2}b+2ab^{2}+b^{3}}$

Possible Answers:

$b^{2}(a^{3}+\frac{11}{2})$

b(a+b)

$\frac{b^{2}(a^{3}+3a^{2}b+3ab^{2}+b^3)}{b(a^{2}+2ab+b^{2})}$

$\frac{(ab^{2}+b^{3})^{3}}{(ab+b^{2})}$

$b^{2}(a+b)^{3}$

Correct answer:

b(a+b)

Explanation:

We start by factoring out the $b^{2}$ in the numerator and b in the denominator:

$\frac{(a^{3}b^{2}+3a^{2}b^{3}+3ab^{4}+b^{5})}{a^{2}b+2ab^{2}+b^{3}}$ $\Leftrightarrow$ $\frac{b^{2}(a^{3}+3a^{2}b+3ab^{2}+b^3)}{b(a^{2}+2ab+b^{2})}$ $\Leftrightarrow$

$\frac{b(a^{3}+3a^{2}b+3ab^{2}+b^3)}{(a^{2}+2ab+b^{2})}$

Note that $a^{3}+3a^{2}b+3ab^{2}+b^{3}=(a+b)^{3}$ and $a^{2}+2ab+b^{2}=(a+b)^{2}$

So the previous expression can be further simplified as follow:

$\frac{b(a+b)^{3}}{(a+b)^{2}}$

This can be simplified even further to get b (a+b) which is the correct answer.

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Example Question #10 : How To Factor An Equation

Which of the following equations is NOT equivalent to the following equation?

$4y^{2}=169x^{2}-81$

Possible Answers:

$y^{2}=(\frac{13x-9}{2})^{2}$

$(2y)^{2}=(13x+9)(13x-9)$

$\frac{(2y)^{2}}{13x+9}=27x-9-14x$

$6y^{2}=\frac{3\times (13x+9)\times (13x-9)}{2}$

$y=\sqrt{\frac{338x^{2}-162}{8}}$

Correct answer:

$y^{2}=(\frac{13x-9}{2})^{2}$

Explanation:

The equation presented in the problem is:

$4y^{2}=169x^{2}-81$

We know that:

$169x^{2}-81=(13x)^{2}-9^{2}=(13x+9)(13x-9)$

Therefore we can see that the answer choice $(2y)^{2}=(13x+9)(13x-9)$ is equivalent to $4y^{2}=169x^{2}-81$ .

$\frac{(2y)^{2}}{13x+9}=27x-9-14x$ is equivalent to $4y^{2}=169x^{2}-81$ . You can see this by first combining like terms on the right side of the equation:

$\frac{(2y)^{2}}{13x+9}=13x-9$

Multiplying everything by 13x+9 , we get back to:

$(2y)^{2}=(13x+9)(13x-9)$

We know from our previous work that this is equivalent to $4y^{2}=169x^{2}-81$ .

$6y^{2}=\frac{3\times (13x+9)\times (13x-9)}{2}$ is also equivalent $4y^{2}=169x^{2}-81$ since both sides were just multiplied by $\frac{3}{2}$ . Dividing both sides by $\frac{3}{2}$ , we also get back to:

$(2y)^{2}=(13x+9)(13x-9)$ .

We know from our previous work that this is equivalent to $4y^{2}=169x^{2}-81$ .

$y=\sqrt{\frac{338x^{2}-162}{8}}$ is also equivalent to $4y^{2}=169x^{2}-81$ since

$y^{2}=(\sqrt{\frac{338x^{2}-162}{8}})^{2}=\frac{338x^{2}-162}{8}$

$4y^{2}=\frac{338x^{2}-162}{2}=169x^{2}-81$

Only $y^{2}=(\frac{13x-9}{2})^{2}$ is NOT equivalent to $4y^{2}=169x^{2}-81$

because

$y^{2}=(\frac{13x-9}{2})^{2}=\frac{169x^{2}-234x-81}{4}$

$4y^{2}=169x^{2}-81-234x \neq 4y^{2}=169x^{2}-81$

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Example Question #13 : Simplifying Algebraic Expressions

Which of the following is equal to the expresion $\sqrt{6x} \cdot \sqrt{15x} \cdot \sqrt{35x}$ ?

Possible Answers:

$14x \sqrt{15 x }$

$15x \sqrt{14 x }$

$10x \sqrt{21 x }$

$21x \sqrt{10 x }$

$35x \sqrt{6 x }$

Correct answer:

$15x \sqrt{14 x }$

Explanation:

$\sqrt{6x} \cdot \sqrt{15x} \cdot \sqrt{35x}$

$=\sqrt{2 \cdot 3 \cdot x} \cdot \sqrt{3 \cdot 5 \cdot x} \cdot \sqrt{5 \cdot 7 \cdot x}$

$=\sqrt{2 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdot x \cdot x \cdot x}$

$=\sqrt{ 3 \cdot 3 } \cdot \sqrt{ 5 \cdot 5 } \cdot \sqrt{ x \cdot x} \cdot \sqrt{2 \cdot 7 \cdot x }$

$=3 \cdot 5 \cdot x \cdot \sqrt{14 x }$

$=15x \sqrt{14 x }$

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GMAT Math : Simplifying algebraic expressions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Simplifying Algebraic Expressions

Example Question #11 : Simplifying Algebraic Expressions

Example Question #12 : Simplifying Algebraic Expressions

Example Question #11 : Simplifying Algebraic Expressions

Example Question #12 : Simplifying Algebraic Expressions

Example Question #15 : Simplifying Algebraic Expressions

Example Question #1322 : Gmat Quantitative Reasoning

Example Question #13 : Simplifying Algebraic Expressions

Example Question #10 : How To Factor An Equation

Example Question #13 : Simplifying Algebraic Expressions

All GMAT Math Resources

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