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Algebra II : Properties of Fractions

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Example Questions

Example Question #11 : Rational Expressions

Add the fractions: $\frac{1}{3}+\frac{2}{4x^2}$

Possible Answers:

$\frac{2x+3}{12x^2}$

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

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

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

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

Correct answer:

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

Explanation:

To add these fractions, we must have a common denominator.

Multiply both denominators to obtain the least common denominator.

3(4x^2)=12x^2

Convert both fractions.

$\frac{1}{3}+\frac{2}{4x^2}=\frac{4x^2}{12x^2}+\frac{2(3)}{12x^2}$

Simplify the numerator and combine the fractions as a single fraction.

$\frac{4x^2+6}{12x^2}$

Pull out a common factor on the numerator.

$\frac{4x^2+6}{12x^2}=\frac{2(2x^2+3)}{12x^2}$

Reduce the fraction.

The answer is: $\frac{2x^2+3}{6x^2}$

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Example Question #12 : Rational Expressions

Which of the following is a misuse of fractional properties?

Possible Answers:

$\frac{a+b}{b+c}\div\frac{a+c}{b+d} = \frac{ab+ad+b^2+bd}{ab+ac+bc+c^2}$

$\frac{x}{y}\times \frac{w}{z} = \frac{xw}{yz}$

$\textup{All of the given scenarios are correct.}$

$\frac{b}{a}-\frac{d}{c} = \frac{bc}{ac}-\frac{ad}{ac}$

$\frac{a}{c}+\frac{b}{d}+\frac{c}{e} =\frac{ac+bd+ce}{cde}$

Correct answer:

$\frac{a}{c}+\frac{b}{d}+\frac{c}{e} =\frac{ac+bd+ce}{cde}$

Explanation:

For the additional and subtraction properties of fractions, we must determine the least common factor, change the fractions, and find the value of the numerators.

$\frac{a}{c}+\frac{b}{d}+\frac{c}{e} = \frac{a(de)}{c(de)}+\frac{b(ce)}{d(ce)}+\frac{c(cd)}{e(cd)}$

Combining all the terms, this fraction will become:

$\frac{a}{c}+\frac{b}{d}+\frac{c}{e} =\frac{ade+bce+c^2d}{cde}$

According to the multiplicative rule of fractions, we can simply multiply the numerators together, and denominators together.

For division, the division sign can be switched to a multiplication sign, but we must also take the reciprocal of what is being divided.

We can verify that:

$\frac{a+b}{b+c}\div\frac{a+c}{b+d} = \frac{ab+ad+b^2+bd}{ab+ac+bc+c^2}$

$\frac{a+b}{b+c}\div\frac{a+c}{b+d}=\frac{a+b}{b+c}\times \frac{b+d}{a+c} = \frac{(a+b)(b+d)}{(b+c)(a+c)}$

(a+b)(b+d) = ab+ad+b^2+bd

(b+c)(a+c) = ba+bc+ca+c^2

Both terms in the numerator and denominator are satisfied by FOIL method.

The answer is: $\frac{a}{c}+\frac{b}{d}+\frac{c}{e} =\frac{ac+bd+ce}{cde}$

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Example Question #13 : Rational Expressions

Simply the expression:

$\frac{3}{x+5}+ \frac{7}{x^{2}+3x -10}$

Possible Answers:

$\frac{5x-3}{x^{2}+2x-10}$

$\frac{7x+5}{x^{2}+4x-5}$

$\frac{3x+1}{x^{2}+3x-10}$

$\frac{7x-3}{x^{2}+3x-10}$

$\frac{4x-5}{x^{2}+5x-6}$

Correct answer:

$\frac{3x+1}{x^{2}+3x-10}$

Explanation:

In order to simplify the expression $\frac{3}{x+5}+ \frac{7}{x^{2}+3x -10}$ , first note that the denominators in both terms share a factor:

$\frac{3}{x+5}+ \frac{7}{(x+5)(x-2)}$

Find the Least Common Denominator (LCD) of both terms, and then simplify the expression:

$\frac{3(x-2)}{(x+5)(x-2)}+ \frac{7}{(x+5)(x-2)}$

$=\frac{3x-6}{(x+5)(x-2)}+ \frac{7}{(x+5)(x-2)}$

$=\frac{3x-6+7}{(x+5)(x-2)}$

Which equals:

$\frac{3x+1}{x^{2}+3x-10}$

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Algebra II : Properties of Fractions

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #11 : Rational Expressions

Example Question #12 : Rational Expressions

Example Question #13 : Rational Expressions

All Algebra II Resources

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