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ACT Math : Complex Fractions

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Example Question #1 : Complex Fractions

Which of the following is equal to $\frac{\left(7+\frac{1}{x}\right)}{\left(14+\frac{2}{x}\right)}$ ?

Possible Answers:

$\frac{1}{2}$

$\frac{x}{2}$

$\frac{1}{2x}$

$\frac{7x}{2}$

x^2+14x+2

Correct answer:

$\frac{1}{2}$

Explanation:

First we must take the numerator of our whole problem. There is a fraction in the numerator with as the denominator. Therefore, we multiply the numerator of our whole problem by $\frac{x}{x}$ , giving us $7+\frac{1}{x}=7x+1$ .

Now we look at the denominator of the whole problem, and we see that there is another fraction present with as a denominator. So now, we multiply the denominator by $\frac{x}{x}$ , giving us $14+\frac{2}{x}=14x+2$ .

Our fraction should now read $\frac{(7x+1)}{(14x+2)}$ . Now, we can factor our denominator, making the fraction $\frac{(7x+1)}{(2(7x+1))}$ .

Finally, we cancel out 7x+1 from the top and the bottom, giving us $\frac{1}{2}$ .

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Example Question #1 : Complex Fractions

Simplify: $\frac{3}{\frac{3}{2}}$

Possible Answers:

$\frac{9}{2}$

$\frac{3}{2}$

Correct answer:

Explanation:

Rewrite $\frac{3}{\frac{3}{2}}$ into the following form:

$3\div \frac{3}{2}$

Change the division sign to a multiplication sign by flipping the 2nd term and simplify.

$3\cdot \frac{2}{3}=2$

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Example Question #1 : Complex Fractions

Evaluate: $\frac{\frac{1}{x}}{x^2}$

Possible Answers:

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

$\frac{1}{x}$

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

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

Correct answer:

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

Explanation:

The expression $\frac{\frac{1}{x}}{x^2}$ can be rewritten as:

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

Change the division sign to a multiplication sign and take the reciprocal of the second term. Evaluate.

$\frac{1}{x}\cdot\frac{1}{x^2}=\frac{1}{x^3}$

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Example Question #4 : Complex Fractions

Simplify: $\frac{\frac{1}{6}}{3} + \frac{\frac{1}{3}}{6}$

Possible Answers:

$\frac{2}{9}$

$\frac{1}{4}$

$\frac{1}{9}$

$\frac{5}{2}$

Correct answer:

$\frac{1}{9}$

Explanation:

The expression $\frac{\frac{1}{6}}{3} + \frac{\frac{1}{3}}{6}$ can be simplified as follows:

We can simplify each fraction by multiplying the numerator by the reciprocal of the denominator.

$\frac{1}{6}\div3 \rightarrow \frac{1}{6}\cdot \frac{1}{3}=\frac{1}{9}$

$\frac{1}{3}\div 6 \rightarrow \frac{1}{3}\cdot \frac{1}{6}=\frac{1}{9}$

From here we add our two new fractions together and simplify.

$=\frac{1}{18}+\frac{1}{18}=\frac{2}{18}=\frac{1}{9}$

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Example Question #5 : Complex Fractions

Simplify the following:

$\frac{7+\frac{1}{3}}{\frac{3}{5}}$

Possible Answers:

$\frac{13}{7}$

$\frac{28}{5}$

$\frac{7}{10}$

$\frac{110}{9}$

$\frac{25}{4}$

Correct answer:

$\frac{110}{9}$

Explanation:

Begin by simplifying your numerator. Thus, find the common denominator:

$\frac{\frac{21}{3}+\frac{1}{3}}{\frac{3}{5}}$

Next, combine the fractions in the numerator:

$\frac{\frac{22}{3}}{\frac{3}{5}}$

Next, remember that to divide fractions, you multiply the numerator by the reciprocal of the denominator:

$\frac{22}{3}\times \frac{5}{3}$

Since nothing needs to be simplified, this is just:

$\frac{22}{3}\times \frac{5}{3}=\frac{110}{9}$

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Example Question #6 : Complex Fractions

Simplify,

$\frac{\frac{ab}{c} + \frac{c}{a}}{\frac{b}{c}} - a$

Possible Answers:

$\frac{b}{ac}$

$\frac{a^2bc + bc^2}{a}$

$\frac{c^2}{ab}$

$\frac{ba^2 + c}{ab}$

$\frac{ab + c - a}{c^2}$

Correct answer:

$\frac{c^2}{ab}$

Explanation:

Convert the numerators and denominators into single fractions, then simplify.

Start by finding the lowest common denominator in both the numerator and denominator of the complex fraction.

$\frac{\frac{ab}{c} + \frac{c}{a}}{\frac{b}{c}} - a$

$\frac{\frac{ab}{c}\left ( \frac{a}{a} \right ) + \frac{c}{a}\left ( \frac{c}{c}\right )}{\frac{b}{c}} - a$

$\frac{\frac{a^2b}{ac} + \frac{c^2}{ac}}{\frac{b}{c}} - a$

Add fractions with like denominators.

$\frac{\frac{a^2b + c^2}{ac}}{\frac{b}{c}} - a$

Simplify. Divide complex fractions by multiplying the numerator by the reciprocal of the denominator.

$\frac{\frac{a^2b + c^2}{ac}}{\frac{b}{c}} - a = \frac{a^2b + c^2}{ac}\cdot \frac{c}{b} - a = \frac{a^2b + c^2}{ab} - a$

Solve.

$\frac{a^2b + c^2}{ab} - a\left ( \frac{ab}{ab} \right ) = \frac{a^2b + c^2}{ab} - \frac{a^2b}{ab} = \frac{c^2}{ab}$

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Example Question #7 : Complex Fractions

Subtract: $\frac{3}{2}-\frac{2}{3}-\frac{4}{5}$

Possible Answers:

$\frac{1}{6}$

$-\frac{4}{5}$

$-\frac{1}{2}$

$\frac{1}{30}$

Correct answer:

$\frac{1}{30}$

Explanation:

The least common multiple can be found by multiplying the denominators: 2, 3, and 5. The common denominator of these numbers is 30. Multiply the numerator with what was multiplied to the denominator of each term, and then solve.

$\frac{3}{2}-\frac{2}{3}-\frac{4}{5}$

$\frac{3(3*5)}{2(3*5)}-\frac{2(2*5)}{3(2*5)}-\frac{4(2*3)}{5(2*3)}$

$\frac{45}{30}-\frac{20}{30}-\frac{24}{30}=\frac{1}{30}$

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Example Question #1 : Complex Fractions

What is $\frac{\frac{4}{9}}{2} - \frac{\frac{5}{12}}3{}$ ?

Possible Answers:

$\frac{1}{4}$

$\frac{5}{36}$

$\frac{3}{18}$

$\frac{1}{12}$

$\frac{1}{36}$

Correct answer:

$\frac{1}{12}$

Explanation:

First, simplify both sides. $\frac{\frac{4}{9}}{2}$ becomes $\frac{4}{18}$ and $\frac{\frac{5}{12}}{3}$ becomes $\frac{5}{36}$ . The LCF between $\frac{4}{18}$ and $\frac{5}{36}$ is 36. Thus, $\frac{8}{36} - \frac{5}{36} = \frac{3}{36}.$ This simplifies to $\frac{1}{12}$ .

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Example Question #1 : How To Subtract Complex Fractions

Simplify:

$\frac{4}{\frac{4-12}{6}}-\frac{\frac{3}{4}}{5}$

Possible Answers:

$\frac{-15}{4}$

$\frac{-3}{7}$

$\frac{-63}{20}$

$\frac{40}{3}$

$\frac{3}{7}$

Correct answer:

$\frac{-63}{20}$

Explanation:

Begin by simplifying the denominator of the first fraction:

$\frac{4}{\frac{4-12}{6}}-\frac{\frac{3}{4}}{5}=\frac{4}{\frac{-8}{6}}-\frac{\frac{3}{4}}{5}$

Now, remember that division of fractions is done by multiplying the numerator by the reciprocal of the denominator. Thus:

$\frac{4}{\frac{-8}{6}}-\frac{\frac{3}{4}}{5}=4*\frac{-6}{8}-\frac{3}{4}*\frac{1}{5}$

Simplify a bit:

$\frac{-3}{1}-\frac{3}{20}=\frac{-60}{20}-\frac{3}{20}=\frac{-63}{20}$

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Example Question #10 : Complex Fractions

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Simplify the complex fraction by multiplying by the complex denominator:

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

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ACT Math : Complex Fractions

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Complex Fractions

Example Question #1 : Complex Fractions

Example Question #1 : Complex Fractions

Example Question #4 : Complex Fractions

Example Question #5 : Complex Fractions

Example Question #6 : Complex Fractions

Example Question #7 : Complex Fractions

Example Question #1 : Complex Fractions

Example Question #1 : How To Subtract Complex Fractions

Example Question #10 : Complex Fractions

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