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SAT Math : How to add rational expressions with a common denominator

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Example Questions

Example Question #51 : Expressions

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

Possible Answers:

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

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

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

3x-10 $\displaystyle 3x-10$

Correct answer:

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

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 #11 : Expressions

Simplify the following rational expression:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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Example Question #2 : How To Add Rational Expressions With A Common Denominator

Simplify the following rational expression:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

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Example Question #702 : Algebra

Simplify the following expression:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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SAT Math : How to add rational expressions with a common denominator

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #51 : Expressions

Example Question #11 : Expressions

Example Question #2 : How To Add Rational Expressions With A Common Denominator

Example Question #702 : Algebra

All SAT Math Resources

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