GRE Math : Rational Expressions

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Example Question #4 : Rational Expressions

Which of the following is equivalent to $\dpi{100} \frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ ? Assume that denominators are always nonzero.

Possible Answers:

$x-t$

$t-x$

$(xt)^{-1}$

$x^{2}-t^{2}$

$\frac{x}{t}$

Correct answer:

$(xt)^{-1}$

Explanation:

We will need to simplify the expression $\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ . We can think of this as a large fraction with a numerator of $\frac{1}{t}-\frac{1}{x}$ and a denominator of $\dpi{100} x-t$ .

In order to simplify the numerator, we will need to combine the two fractions. When adding or subtracting fractions, we must have a common denominator. $\frac{1}{t}$ has a denominator of $\dpi{100} t$ , and $\dpi{100} -\frac{1}{x}$ has a denominator of $\dpi{100} x$ . The least common denominator that these two fractions have in common is $\dpi{100} xt$ . Thus, we are going to write equivalent fractions with denominators of $\dpi{100} xt$ .

In order to convert the fraction $\dpi{100} \frac{1}{t}$ to a denominator with $\dpi{100} xt$ , we will need to multiply the top and bottom by $\dpi{100} x$ .

$\frac{1}{t}=\frac{1\cdot x}{t\cdot x}=\frac{x}{xt}$

Similarly, we will multiply the top and bottom of $\dpi{100} -\frac{1}{x}$ by $\dpi{100} t$ .

$\frac{1}{x}=\frac{1\cdot t}{x\cdot t}=\frac{t}{xt}$

We can now rewrite $\frac{1}{t}-\frac{1}{x}$ as follows:

$\frac{1}{t}-\frac{1}{x}$ = $\frac{x}{xt}-\frac{t}{xt}=\frac{x-t}{xt}$

Let's go back to the original fraction $\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ . We will now rewrite the numerator:

$\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ = $\frac{\frac{x-t}{xt}}{x-t}$

To simplify this further, we can think of $\frac{\frac{x-t}{xt}}{x-t}$ as the same as $\frac{x-t}{xt}\div (x-t)$ . When we divide a fraction by another quantity, this is the same as multiplying the fraction by the reciprocal of that quantity. In other words, $a\div b=a\cdot \frac{1}{b}$ .

$\frac{x-t}{xt}\div (x-t)$ = $\frac{x-t}{xt}\cdot \frac{1}{x-t}$ $=\frac{x-t}{xt(x-t)}$ $= \frac{1}{xt}$

Lastly, we will use the property of exponents which states that, in general, $\frac{1}{a}=a^{-1}$ .

$\frac{1}{xt}=(xt)^{-1}$

The answer is $(xt)^{-1}$ .

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Example Question #1 : Rational Expressions

Simplify:

$\frac{x^{2}+x-2}{3x^{2} + 9x + 6} \div (x-1)$

Possible Answers:

$\frac{1}{3(x-1)}$

$\frac{1}{3(x+2)}$

$\frac{1}{3(x^{2}+1)}$

$\frac{1}{3(x^{2}-2)}$

$\frac{1}{3(x+1)}$

Correct answer:

$\frac{1}{3(x+1)}$

Explanation:

Multiply by the reciprocal of $\frac{x-1}1{}$ .

$\frac{x^{2}+x-2}{3x^{2}+9x+6} \cdot \frac{1}{x-1}$

Factor $\frac{(x-1)(x+2)}{3(x+2)(x+1)} \cdot \frac{1}{x-1}$

Divide by common factors.

$= \frac{1}{3(x+1)}$

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Example Question #3 : Expressions

Solve for :

$\frac{\frac{2m^2 + 5}{4}}{ \frac{m}{2}} = 20$

Possible Answers:

m = 12

m = -10

m = -12,-8

m = 8

m=19.87,0.13

Correct answer:

m=19.87,0.13

Explanation:

To tackle this problem, you need to invert and multiply:

$\frac{\frac{2m^2 + 5}{4}} { \frac{m}{2}} = 20$

$\frac{2m^2 + 5}{4} \cdot \frac{2}{m} = 20$

$\frac{4m^2 + 10}{4m} = 20$

4m^2+10=80m

Here we see that we have created a quadratic equation. Therefore, we get all terms to one side, set it equal to zero and use the quadratic formula to solve.

4m^2-80m+10=0

The quadratic formula is:

$m=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where a=4, b=-80, c=10

Plugging these values in we get the following:

$\frac{80\pm\sqrt{80^2-4(4)(10)}}{2(4)}=\frac{80\pm\sqrt{6240}}{8}=\frac{80\pm78.99}{8}$

m=19.87,0.13

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Example Question #3 : Rational Expressions

Express the following as a single rational expression:

$\frac{\frac{2x^2 + 7}{12b} }{ \frac{y^2 + 2y}{3a^2 + 2}}$

Possible Answers:

$\frac{x^2a^2 +21a^2 +4x^2 +14}{12by^2 + 24by}$

$\frac{6x^2a^2 +84a^2x^2 +14}{12by^2 + 24by}$

$\frac{6x^2a^2 +21a^2 +4x^2 +14}{12by^2 + 24by}$

$\frac{6x^2a^2 +21a^2 +4x^2 +14}{36by}$

$\frac{6x^2a^2 +21a^2 +4x^2 +14}{12by^2 + by}$

Correct answer:

$\frac{6x^2a^2 +21a^2 +4x^2 +14}{12by^2 + 24by}$

Explanation:

To divide one rational expression by another, invert and multiply:

$\frac{\frac{2x^2 + 7}{12b}}{ \frac{y^2 + 2y}{3a^2 + 2}}$

$\frac{2x^2 + 7 }{12b} \cdot \frac{3a^2 + 2}{y^2 + 2y}$

Remember to foil the numerator meaning, multiply the first components of each binomial. Then multiply the outer components of each binomial. After that, multiply the inner components together, and lastly, multiply the components in the last position of the binomials together.

This arrives at the following:

$\frac{6x^2a^2 +21a^2 +4x^2 +14}{12by^2 + 24by}$

You can't factor anything out, so that's your final answer.

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Example Question #3 : Rational Expressions

Simplify the following rational expression: (9x - 2)/(x²) MINUS (6x - 8)/(x²)

Possible Answers:

$\frac{12x-6}{x}$

$\frac{15x+6}{x}$

$\frac{3x+6}{x^{2}}$

3x-10

Correct answer:

$\frac{3x+6}{x^{2}}$

Explanation:

Since both expressions have a common denominator, x², we can just recopy the denominator and focus on the numerators. We get (9x - 2) - (6x - 8). We must distribute the negative sign over the 6x - 8 expression which gives us 9x - 2 - 6x + 8 ( -2 minus a -8 gives a +6 since a negative and negative make a positive). The numerator is therefore 3x + 6.

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Example Question #1 : How To Add Rational Expressions With A Common Denominator

Simplify the following rational expression:

$\frac{7x-18}{x^{2}}+\frac{6x-14}{x^{2}}$

Possible Answers:

$\frac{13x-28}{x^{2}}$

$\frac{13x-4}{x^{2}}$

$\frac{x-32}{x^{2}}$

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

$\frac{13x-32}{x^{2}}$

Correct answer:

$\frac{13x-32}{x^{2}}$

Explanation:

Since both fractions in the expression have a common denominator of $x^{2}$ , we can combine like terms into a single numerator over the denominator:

$\frac{7x-18}{x^{2}}+\frac{6x-14}{x^{2}}$

$=\frac{(7x-18)+(6x-14)}{x^{2}}$

$=\frac{13x-32}{x^{2}}$

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Example Question #2 : Expressions

Simplify the following expression:

$\frac{10x-9}{x^{3}}+\frac{11x+12}{x^{3}}$

Possible Answers:

$\frac{21x-3}{x^3}$

$\frac{3-21x}{x^{3}}$

$\frac{x+3}{x^{3}}$

$\frac{3-x}{x^{3}}$

$\frac{21x+3}{x^{3}}$

Correct answer:

$\frac{21x+3}{x^{3}}$

Explanation:

Since both terms in the expression have the common denominator $x^{3}$ , combine the fractions and simplify the numerators:

$\frac{10x-9}{x^{3}}+\frac{11x+12}{x^{3}}$

$=\frac{(10x-9)+(11x+12)}{x^{3}}$

$=\frac{21x+3}{x^{3}}$

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Example Question #4 : Rational Expressions

Simplify the following rational expression:

$\frac{5x-5}{2x^{2}} + \frac{7x+9}{2x^{2}}$

Possible Answers:

$\frac{12x+8}{2x^{2}}$

$\frac{8x+4}{2x^{2}}$

$\frac{6x+4}{2x^{2}}$

$\frac{6x+2}{x^{2}}$

$\frac{6x+8}{2x^{2}}$

Correct answer:

$\frac{6x+2}{x^{2}}$

Explanation:

Since both rational terms in the expression have the common denominator $2x^{2}$ , combine the numerators and simplify like terms:

$\frac{5x-5}{2x^{2}} + \frac{7x+9}{2x^{2}}$

$=\frac{(5x-5)+(7x+9)}{2x^{2}}$

$=\frac{12x+4}{2x^{2}}$

$=\frac{6x+2}{x^2}$

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Example Question #281 : Algebra

Add and simplify:

$\frac{3x^2 + 3x + 3}{2y} + \frac{10x^2 + 3x}{2y}$

Possible Answers:

$\frac{13x^2 + 6x + 3}{2y}$

$\frac{6x^2 + 3x +1}{2y}$

$\frac{13x^2 + 6x + 3}{4y}$

$\frac{10x^2 + 3x^2 +3x + 3x + 3}{2y}$

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

Correct answer:

$\frac{13x^2 + 6x + 3}{2y}$

Explanation:

When adding rational expressions with common denominators, you simply need to add the like terms in the numerator.

Therefore, $\frac{13x^2 + 6x + 3}{2y}$ is the best answer.

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Example Question #281 : Algebra

Choose the answer which best simplifies the following expression:

$\frac{2p^2 + 3p}{2a} - \frac{5p}{3}$

Possible Answers:

$\frac{6p^2 + 9p -10p}{6}$

$\frac{6p + 9p -10pa}{6a}$

$\frac{6p^2 + 9p -10pa}{6a}$

$\frac{6p^2 + 9p +10pa}{6a}$

$\frac{15p -10pa}{6a}$

Correct answer:

$\frac{6p^2 + 9p -10pa}{6a}$

Explanation:

To simplify this expression, first multiply both terms by the denominator of the other over itself:

$\frac{2p^2 + 3p}{2a}\left(\frac{3}{3}\right) - \frac{5p}{3}\left(\frac{2a}{2a}\right)$

$\frac{6p^2 + 9p}{6a} - \frac{10pa}{6a}$

Now that you have a common denominator, you may subtract:

$\frac{6p^2 +9p - 10pa}{6a}$

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GRE Math : Rational Expressions

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #4 : Rational Expressions

Example Question #1 : Rational Expressions

Example Question #3 : Expressions

Example Question #3 : Rational Expressions

Example Question #3 : Rational Expressions

Example Question #1 : How To Add Rational Expressions With A Common Denominator

Example Question #2 : Expressions

Example Question #4 : Rational Expressions

Example Question #281 : Algebra

Example Question #281 : Algebra

All GRE Math Resources

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