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

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Example Questions

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

Compute the following: $\frac{x+2}{2-x}+\frac{x-2}{2-x}$

Possible Answers:

$\frac{2x}{1-2x}$

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

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

Correct answer:

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

Explanation:

Notice that the denominator are the same for both terms. Since they are both the same, the fractions can be added. The denominator will not change in this problem.

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

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

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

Possible Answers:

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

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

3x-10

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

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 #3 : 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:

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

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

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

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

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

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 #4 : How To Add Rational Expressions With A Common Denominator

Simplify the following expression:

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

Possible Answers:

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

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

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

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

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

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 #5 : How To Add Rational Expressions With A Common Denominator

Simplify the following rational expression:

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

Possible Answers:

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

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

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

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

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

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 #5 : How To Add Rational Expressions With A Common Denominator

Combine the following rational expressions:

$\frac{3x^2-4x+5}{x+1}+\frac{4x^2-5x+9}{x+1}$

Possible Answers:

$\frac{7x^2 -9x+7}{x-1}$

$\frac{7x^2 +x+14}{x+1}$

$\frac{12x^2 +6x+15}{x+1}$

$\frac{-x^2 -9x+14}{x+1}$

$\frac{7x^2 -9x+14}{x+1}$

Correct answer:

$\frac{7x^2 -9x+14}{x+1}$

Explanation:

When working with complex fractions, it is important not to let them intimidate you. They follow the same rules as regular fractions!

$\frac{3x^2-4x+5}{x+1}+\frac{4x^2-5x+9}{x+1}$

In this case, our problem is made easier by the fact that we already have a common denominator. Nothing fancy is required to start. Simply add the numerators:

$\frac{(3x^2-4x+5)+(4x^2-5x+9)}{x+1}$

For our next step, we need to combine like terms. This is easier to see if we group them together.

$\frac{({\color{Blue} 3x^2+4x^2})+({\color{Red} -4x+-5x})+({\color{DarkOrange} 5+9})}{x+1}$

$\frac{({\color{Blue}7x^2})+({\color{Red} -9x})+({\color{DarkOrange} 14})}{x+1}$

Thus, our final answer is:

$\frac{7x^2 -9x+14}{x+1}$

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

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

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

Example Question #3 : How To Add Rational Expressions With A Common Denominator

Example Question #4 : How To Add Rational Expressions With A Common Denominator

Example Question #5 : How To Add Rational Expressions With A Common Denominator

Example Question #5 : How To Add Rational Expressions With A Common Denominator

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