Complex Fractions - ACT Math

Card 0 of 225

Question

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

Answer

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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Question

Steven purchased $1\frac{2}{3}:lbs$ of vegetables on Monday and $2\frac{3}{4}:lbs$ of vegetables on Tuesday. What was the total weight, in pounds, of vegetables purchased by Steven?

Answer

To solve this answer, we have to first make the mixed numbers improper fractions so that we can then find a common denominator. To make a mixed number into an improper fraction, you multiply the denominator by the whole number and add the result to the numerator. So, for the presented data:

$1\frac{2}{3}=\frac{(31)+2}{3}=\frac{3+2}{3}=\frac{5}{3}$

and

$2\frac{3}{4}=\frac{(42)+3}{4}=\frac{8+3}{4}=\frac{11}{4}$

Now, to find out how many total pounds of vegetables Steven purchased, we need to add these two improper fractions together:

$\frac{5}{3}+\frac{11}{4}$

To add these fractions, they need to have a common denominator. We can adjust each fraction to have a common denominator of by multiplying $\frac{5}{3}$ by $\frac{4}{4}$ and $\frac{11}{4}$ by $\frac{3}{3}$ :

$(\frac{5}{3}\frac{4}{4})+(\frac{11}{4}\frac{3}{3})$

To multiply fractions, just multiply across:

$\frac{20}{12}+\frac{33}{12}$

We can now add the numerators together; the denominator will stay the same:

$\frac{53}{12}$

Since all of the answer choices are mixed numbers, we now need to change our improper fraction answer into a mixed number answer. We can do this by dividing the numerator by the denominator and leaving the remainder as the numerator:

$53\div12=4: r.:5$

$\frac{53}{12}=4\frac{5}{12}$

This means that our final answer is $4\frac{5}{12}:lbs$ .

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Question

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

Answer

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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Question

Steven purchased $1\frac{2}{3}:lbs$ of vegetables on Monday and $2\frac{3}{4}:lbs$ of vegetables on Tuesday. What was the total weight, in pounds, of vegetables purchased by Steven?

Answer

To solve this answer, we have to first make the mixed numbers improper fractions so that we can then find a common denominator. To make a mixed number into an improper fraction, you multiply the denominator by the whole number and add the result to the numerator. So, for the presented data:

$1\frac{2}{3}=\frac{(31)+2}{3}=\frac{3+2}{3}=\frac{5}{3}$

and

$2\frac{3}{4}=\frac{(42)+3}{4}=\frac{8+3}{4}=\frac{11}{4}$

Now, to find out how many total pounds of vegetables Steven purchased, we need to add these two improper fractions together:

$\frac{5}{3}+\frac{11}{4}$

To add these fractions, they need to have a common denominator. We can adjust each fraction to have a common denominator of by multiplying $\frac{5}{3}$ by $\frac{4}{4}$ and $\frac{11}{4}$ by $\frac{3}{3}$ :

$(\frac{5}{3}\frac{4}{4})+(\frac{11}{4}\frac{3}{3})$

To multiply fractions, just multiply across:

$\frac{20}{12}+\frac{33}{12}$

We can now add the numerators together; the denominator will stay the same:

$\frac{53}{12}$

Since all of the answer choices are mixed numbers, we now need to change our improper fraction answer into a mixed number answer. We can do this by dividing the numerator by the denominator and leaving the remainder as the numerator:

$53\div12=4: r.:5$

$\frac{53}{12}=4\frac{5}{12}$

This means that our final answer is $4\frac{5}{12}:lbs$ .

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Question

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

Answer

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 from the top and the bottom, giving us $\frac{1}{2}$ .

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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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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Question

Simplify the following:

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

Answer

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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Question

Simplify,

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

Answer

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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Question

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

Answer

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.

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

$\frac{3(35)}{2(35)}-\frac{2(25)}{3(25)}-\frac{4(23)}{5(23)}$

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

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Question

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

Answer

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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Question

Simplify:

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

Answer

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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Question

Calculate

$\frac{\frac{1}{4}+\frac{2}{3}}{\frac{3}{8}+\frac{1}{6}} + \frac{4}{13}$

Answer

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.

$\frac{\frac{1}{4}+\frac{2}{3}}{\frac{3}{8}+\frac{1}{6}} + \frac{4}{13}$

$\frac{\frac{1}{4}\left (\frac{ 3}{3}\right )+\frac{2}{3}\left (\frac{4}{4}\right )}{\frac{3}{8}\left (\frac{ 3}{3}\right )+\frac{1}{6}\left (\frac{4}{4}\right )} + \frac{4}{13}$

$\frac{\frac{3}{12}+\frac{8}{12}}{\frac{9}{24}+\frac{4}{24}} + \frac{4}{13}$

Add fractions with like denominators.

$\frac{\frac{11}{12}}{\frac{13}{24}} + \frac{4}{13}$

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

$\frac{\frac{11}{12}}{\frac{13}{24}} + \frac{4}{13} = \frac{11}{12}\cdot \frac{24}{13}+ \frac{4}{13} = \frac{11\cdot 2}{13} + \frac{4}{13} =\frac{22}{13} + \frac{4}{13}$

Solve.

$\frac{22}{13} + \frac{4}{13} = \frac{26}{13} = 2$

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Question

Calculate

$\frac{\frac{2}{7}}{\frac{1}{3} + \frac{2}{9}} + \frac{\frac{1}{5}}{\frac{1}{3} + 2}$

Answer

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.

$\frac{\frac{2}{7}}{\frac{1}{3} + \frac{2}{9}} + \frac{\frac{1}{5}}{\frac{1}{3} + 2}$

$\frac{\frac{2}{7}}{\frac{1}{3}\left ( \frac{3}{3} \right ) + \frac{2}{9}} + \frac{\frac{1}{5}}{\frac{1}{3} + 2\left ( \frac{3}{3} \right )}$

$\frac{\frac{2}{7}}{\frac{3}{9} + \frac{2}{9}} + \frac{\frac{1}{5}}{\frac{1}{3} + \frac{6}{3}}$

Add fractions with like denominators.

$\frac{\frac{2}{7}}{\frac{5}{9} } + \frac{\frac{1}{5}}{\frac{7}{3}}$

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

$\frac{\frac{2}{7}}{\frac{5}{9} } + \frac{\frac{1}{5}}{\frac{7}{3}} =\frac{2}{7}\cdot \frac{9}{5} + \frac{1}{5}\cdot \frac{3}{7} =\frac{18}{35} + \frac{3}{35}$

Solve.

$\frac{18}{35} + \frac{3}{35} = \frac{21}{35} = \frac{3}{5}$

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Question

Calculate

$\frac{\frac{4}{5}}{\frac{3}{12}} + \frac{\frac{3}{5} + 4}{\frac{1}{9} + \frac{5}{36}}$

Answer

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.

$\frac{\frac{4}{5}}{\frac{3}{12}} + \frac{\frac{3}{5} + 4}{\frac{1}{9} + \frac{5}{36}}$

$\frac{\frac{4}{5}}{\frac{3}{12}} + \frac{\frac{3}{5} + 4\left ( \frac{5}{5} \right )}{\frac{1}{9}\left ( \frac{4}{4} \right ) + \frac{5}{36}}$

$\frac{\frac{4}{5}}{\frac{3}{12}} + \frac{\frac{3}{5} + \frac{20}{5}}{\frac{4}{36} + \frac{5}{36}}$

Add fractions with like denominators.

$\frac{\frac{4}{5}}{\frac{3}{12}} + \frac{\frac{3}{5} + \frac{20}{5}}{\frac{4}{36} + \frac{5}{36}} =\frac{\frac{4}{5}}{\frac{3}{12}} + \frac{\frac{23}{5}}{\frac{9}{36}}$

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

$\frac{\frac{4}{5}}{\frac{3}{12}} + \frac{\frac{23}{5}}{\frac{9}{36}} = \frac{4}{5}\cdot \frac{12}{3} + \frac{23}{5}\cdot \frac{36}{9} = \frac{4\cdot 4}{5} + \frac{23\cdot 4}{5} = \frac{16}{5} + \frac{92}{5}$

Solve.

$\frac{16}{4} + \frac{92}{4} = \frac{108}{5}$

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Question

Calculate

$\frac{\frac{1}{3} + \frac{1}{4}}{\frac{1}{5} + \frac{2}{3}} + \frac{43}{52}$

Answer

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.

$\frac{\frac{1}{3} + \frac{1}{4}}{\frac{1}{5} + \frac{2}{3}} + \frac{43}{52}$

$\frac{\frac{1}{3}\left ( \frac{4}{4} \right ) + \frac{1}{4}\left ( \frac{3}{3} \right )}{\frac{1}{5}\left ( \frac{3}{3} \right ) + \frac{2}{3}\left ( \frac{5}{5} \right )} + \frac{43}{52}$

$\frac{\frac{4}{12} + \frac{3}{12}}{\frac{3}{15} + \frac{10}{15}} + \frac{43}{52}$

Add fractions with like denominators.

$\frac{\frac{7}{12}}{\frac{13}{15} } + \frac{43}{52}$

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

$\frac{\frac{7}{12}}{\frac{13}{15} } + \frac{43}{52} = \frac{7}{12}\cdot \frac{15}{13} + \frac{43}{52} = \frac{7\cdot 5}{4\cdot 13} + \frac{43}{52} = \frac{35}{52} + \frac{43}{52}$

Solve.

$\frac{35}{52} + \frac{43}{52} = \frac{78}{52} = \frac{3}{2} = 1.5$

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Question

Calculate

$\frac{\frac{5}{7} + \frac{13}{21}}{1 + \frac{3}{2}} + \frac{4}{5}$

Answer

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.

$\frac{\frac{5}{7} + \frac{13}{21}}{1 + \frac{3}{2}} + \frac{4}{5}$

$\frac{\frac{5}{7}\left ( \frac{3}{3} \right ) + \frac{13}{21}}{1\left ( \frac{2}{2} \right ) + \frac{3}{2}} + \frac{4}{5}$

$\frac{\frac{15}{21} + \frac{13}{21}}{\frac{2}{2} + \frac{3}{2}} + \frac{4}{5}$

Add fractions with like denominators.

$\frac{\frac{28}{21}}{\frac{5}{2} } + \frac{4}{5}$

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

$\frac{\frac{28}{21}}{\frac{5}{2} } + \frac{4}{5} = \frac{28}{21}\cdot \frac{2}{5} + \frac{4}{5} = \frac{4\cdot 2}{3\cdot 5} + \frac{4}{5} = \frac{8}{15} + \frac{4}{5}$

Solve.

$\frac{8}{15} + \frac{4}{5}\left ( \frac{3}{3} \right ) = \frac{8}{15} + \frac{12}{15} = \frac{20}{15} = \frac{4}{3}$

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Question

Calculate

$\frac{\frac{1}{2} + \frac{2}{3}}{\frac{5}{6}} + \frac{3}{4}$

Answer

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.

$\frac{\frac{1}{2} + \frac{2}{3}}{\frac{5}{6}} + \frac{3}{4}$

$\frac{\frac{1}{2}\left ( \frac{3}{3} \right ) + \frac{2}{3}\left ( \frac{2}{2} \right )}{\frac{5}{6}} + \frac{3}{4}$

$\frac{\frac{3}{6} + \frac{4}{6}}{\frac{5}{6}} + \frac{3}{4}$

Add fractions with like denominators.

$\frac{\frac{7}{6}}{\frac{5}{6} } + \frac{3}{4}$

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

$\frac{\frac{7}{6}}{\frac{5}{6} } + \frac{3}{4} = \frac{7}{6}\cdot \frac{6}{5} + \frac{3}{4} =\frac{7}{5} + \frac{3}{4}$

Solve.

$\frac{7}{5}\left ( \frac{4}{4} \right ) + \frac{3}{4}\left ( \frac{5}{5} \right ) = \frac{28}{20} + \frac{15}{20} = \frac{43}{20}$

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Question

Simplify,

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

Answer

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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