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ACT Math : Fractions

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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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Example Question #3 : How To Add And Subtract Fractions

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?

Possible Answers:

$4\frac{3}{4}\:lbs$

$4\frac{1}{2}\:lbs$

$4\frac{5}{12}\:lbs$

$4\frac{7}{12}\:lbs$

$4\frac{1}{3}\:lbs$

Correct answer:

$4\frac{5}{12}\:lbs$

Explanation:

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:

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

and

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

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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Example Question #1 : How To Add Complex Fractions

Calculate

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

Possible Answers:

1.5

$\frac{88}{13}$

$\frac{11}{72}$

$\frac{22}{13}$

Correct answer:

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.

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

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

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

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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Example Question #2 : How To Add Complex Fractions

Calculate

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

Possible Answers:

$\frac{10}{63}$

$\frac{18}{35}$

$\frac{3}{5}$

$\frac{7}{15}$

$\frac{22}{35}$

Correct answer:

$\frac{3}{5}$

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.

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

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

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

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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Example Question #3 : How To Add Complex Fractions

Calculate

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

Possible Answers:

$\frac{27}{20}$

$\frac{108}{5}$

$\frac{87}{4}$

$\frac{39}{5}$

Correct answer:

$\frac{108}{5}$

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.

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

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

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

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.

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

Report an Error

Example Question #1 : How To Add Complex Fractions

Calculate

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

Possible Answers:

$\frac{1559}{1170}$

$\frac{91}{180}$

1.5

2.5

2.0

Correct answer:

1.5

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.

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

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

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

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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Example Question #5 : How To Add Complex Fractions

Calculate

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

Possible Answers:

$\frac{4}{3}$

$\frac{13}{10}$

$\frac{8}{15}$

$\frac{62}{15}$

$\frac{20}{6}$

Correct answer:

$\frac{4}{3}$

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.

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

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

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

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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Example Question #6 : How To Add Complex Fractions

Calculate

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

Possible Answers:

$\frac{43}{20}$

$\frac{17}{8}$

$\frac{7}{5}$

$\frac{31}{18}$

$\frac{35}{36}$

Correct answer:

$\frac{43}{20}$

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.

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

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

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

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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Example Question #7 : How To Add Complex Fractions

Simplify,

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

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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Example Question #8 : How To Add Complex Fractions

Simplify:

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

Possible Answers:

$\frac{3}{8}$

$\frac{1}{4}$

$\frac{5}{3}$

$\frac{2}{7}$

$\frac{15}{7}$

Correct answer:

$\frac{5}{3}$

Explanation:

Begin by simplifying the first fraction:

$\frac{\frac{1}{4+5}}{\frac{3-5}{6}}+\frac{3}{\frac{3}{2}}=\frac{\frac{1}{9}}{\frac{-2}{6}}+\frac{3}{\frac{3}{2}}$

Next, handle the division of each fraction by multiplying by the reciprocal in each case:

$\frac{\frac{1}{9}}{\frac{-2}{6}}+\frac{3}{\frac{3}{2}} = \frac{1}{9}*\frac{6}{-2}+\frac{3}{1}*\frac{2}{3}$

$\frac{1}{9}*\frac{6}{-2}+\frac{3}{1}*\frac{2}{3}=\frac{-1}{3}+\frac{2}{1}$

Now, with a common denominator, you are done!

$\frac{-1}{3}+\frac{6}{3}=\frac{5}{3}$

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ACT Math : Fractions

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #10 : Complex Fractions

Example Question #3 : How To Add And Subtract Fractions

Example Question #1 : How To Add Complex Fractions

Example Question #2 : How To Add Complex Fractions

Example Question #3 : How To Add Complex Fractions

Example Question #1 : How To Add Complex Fractions

Example Question #5 : How To Add Complex Fractions

Example Question #6 : How To Add Complex Fractions

Example Question #7 : How To Add Complex Fractions

Example Question #8 : How To Add Complex Fractions

All ACT Math Resources

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