ACT Math - Complex Fractions

Simplify,

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

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

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

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

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

$\frac{b}{ac}$

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.

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

Simplify:

$\frac{\frac{4}{5}}{\frac{8}{9}+\frac{8}{7}}$

$\frac{63}{160}$

$\frac{25}{140}$

$\frac{512}{315}$

$\frac{4}{5}$

$\frac{4}{21}$

Explanation

Begin by simplifying the denominator:

$\frac{\frac{4}{5}}{\frac{8}{9}+\frac{8}{7}}=\frac{\frac{4}{5}}{\frac{56}{63}+\frac{72}{63}}$

$\frac{\frac{4}{5}}{\frac{56}{63}+\frac{72}{63}}=\frac{\frac{4}{5}}{\frac{128}{63}}$

Then, you perform the division by multiplying the numerator by the reciprocal of the denominator:

$\frac{\frac{4}{5}}{\frac{128}{63}} = \frac{4}{5} * \frac{63}{128}$

Do your simplifying now:

$\frac{4}{5} * \frac{63}{128} = \frac{1}{5} * \frac{63}{32}$

Finally, multiply everything:

$\frac{1}{5} * \frac{63}{32} = \frac{63}{160}$

Simplify:

$\frac{\frac{4}{5}}{\frac{8}{9}+\frac{8}{7}}$

$\frac{63}{160}$

$\frac{25}{140}$

$\frac{512}{315}$

$\frac{4}{5}$

$\frac{4}{21}$

Explanation

Begin by simplifying the denominator:

$\frac{\frac{4}{5}}{\frac{8}{9}+\frac{8}{7}}=\frac{\frac{4}{5}}{\frac{56}{63}+\frac{72}{63}}$

$\frac{\frac{4}{5}}{\frac{56}{63}+\frac{72}{63}}=\frac{\frac{4}{5}}{\frac{128}{63}}$

Then, you perform the division by multiplying the numerator by the reciprocal of the denominator:

$\frac{\frac{4}{5}}{\frac{128}{63}} = \frac{4}{5} * \frac{63}{128}$

Do your simplifying now:

$\frac{4}{5} * \frac{63}{128} = \frac{1}{5} * \frac{63}{32}$

Finally, multiply everything:

$\frac{1}{5} * \frac{63}{32} = \frac{63}{160}$

Simplify,

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

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

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

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

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

$\frac{b}{ac}$

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.

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

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

$\frac{3}{2}$

$\frac{9}{2}$

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.

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

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

$\frac{3}{2}$

$\frac{9}{2}$

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.

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

Simplify the following:

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

$\frac{110}{9}$

$\frac{7}{10}$

$\frac{28}{5}$

$\frac{13}{7}$

$\frac{25}{4}$

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:

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

Simplify the following:

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

$\frac{110}{9}$

$\frac{7}{10}$

$\frac{28}{5}$

$\frac{13}{7}$

$\frac{25}{4}$

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:

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

Simplify,

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

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

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

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

$\frac{a\left ( c+b^2\right )}{bc}$

$\frac{a^2b}{c^2}$

Explanation

To add complex fractions, 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.

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

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

$\frac{\frac{ac}{bc} + \frac{ab^2}{bc}}{\frac{c+b^2}{b}} + \frac{b}{c^2}$

Add fractions with like denominators.

$\frac{\frac{ac + ab^2}{bc}}{\frac{c+b^2}{b}} + \frac{b}{c^2}$

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

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

Solve.

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

Simplify,

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

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

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

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

$\frac{a\left ( c+b^2\right )}{bc}$

$\frac{a^2b}{c^2}$

Explanation

To add complex fractions, 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.

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

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

$\frac{\frac{ac}{bc} + \frac{ab^2}{bc}}{\frac{c+b^2}{b}} + \frac{b}{c^2}$

Add fractions with like denominators.

$\frac{\frac{ac + ab^2}{bc}}{\frac{c+b^2}{b}} + \frac{b}{c^2}$

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

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

Solve.

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

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