How to add rational expressions with a common denominator

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GRE Quantitative Reasoning › How to add rational expressions with a common denominator

Questions 1 - 5

1

Add and simplify:

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

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

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

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

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

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

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.

2

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

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

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

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

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.

3

Simplify the following rational expression:

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

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

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

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

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

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

Explanation

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

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

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

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

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

4

Simplify the following expression:

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

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

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

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

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

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

Explanation

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

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

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

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

5

Simplify the following rational expression:

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

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

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

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

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

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

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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