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ACT Math : How to add complex fractions

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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:

$\frac{11}{72}$

$\frac{88}{13}$

$\frac{22}{13}$

1.5

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 #1 : 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:

$\frac{18}{35}$

$\frac{10}{63}$

$\frac{7}{15}$

$\frac{22}{35}$

$\frac{3}{5}$

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 #2 : 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:

$\frac{39}{5}$

$\frac{87}{4}$

$\frac{27}{20}$

$\frac{108}{5}$

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 #3 : How To Add Complex Fractions

Calculate

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

Possible Answers:

2.0

2.5

$\frac{1559}{1170}$

$\frac{91}{180}$

1.5

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 #4 : How To Add Complex Fractions

Calculate

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

Possible Answers:

$\frac{20}{6}$

$\frac{13}{10}$

$\frac{8}{15}$

$\frac{4}{3}$

$\frac{62}{15}$

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 #3 : How To Add Complex Fractions

Calculate

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

Possible Answers:

$\frac{31}{18}$

$\frac{17}{8}$

$\frac{43}{20}$

$\frac{35}{36}$

$\frac{7}{5}$

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 #5 : How To Add Complex Fractions

Simplify,

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

Possible Answers:

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

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

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

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

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

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 #4 : How To Add Complex Fractions

Simplify:

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

Possible Answers:

$\frac{3}{8}$

$\frac{5}{3}$

$\frac{2}{7}$

$\frac{1}{4}$

$\frac{15}{7}$

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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Example Question #1 : How To Add Complex Fractions

Simplify:

$\frac{\frac{4-5}{12}+\frac{7+9}{10}}{\frac{5}{7}} + \frac{3}{4}$ .

Possible Answers:

$\frac{941}{100}$

$\frac{431}{150}$

$\frac{831}{11}$

$\frac{542}{177}$

$\frac{17}{91}$

Correct answer:

$\frac{431}{150}$

Explanation:

With a complex fraction like this, begin by simplifying the numerator of the first fraction:

$\frac{\frac{4-5}{12}+\frac{7+9}{10}}{\frac{5}{7}} + \frac{3}{4}=\frac{\frac{-1}{12}+\frac{16}{10}}{\frac{5}{7}} + \frac{3}{4}$

Next, find the common denominator of the numerator's fractions:

$\frac{\frac{-1}{12}+\frac{16}{10}}{\frac{5}{7}} + \frac{3}{4}=\frac{\frac{-5}{60}+\frac{96}{60}}{\frac{5}{7}} + \frac{3}{4}=\frac{\frac{91}{60}}{\frac{5}{7}} + \frac{3}{4}$

Next, simplify the left division by multiplying by the reciprocal:

$\frac{\frac{91}{60}}{\frac{5}{7}} + \frac{3}{4}=\frac{91}{60}*\frac{7}{5} + \frac{3}{4}$

$\frac{91}{60}*\frac{7}{5} + \frac{3}{4} = \frac{637}{300}+\frac{3}{4}$

Finally, combine the fractions:

$\frac{637}{300}+\frac{225}{300}=\frac{862}{300}$

Simplifying, this is:

$\frac{431}{150}$

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Example Question #1 : How To Add Complex Fractions

Susan is training to run a race. The week before the race she ran four times. The first time she ran $2\tfrac{1}{3}$ miles, her second run was $3\tfrac{1}{2}$ miles, her third run was $1\tfrac{5}{6}$ miles and her final run was $4\tfrac{1}{5}$ miles. How many miles did Susan run this week?

Possible Answers:

$11\tfrac{13}{15}$

$12\tfrac{7}{8}$

$10\tfrac{1}{15}$

$10\tfrac{1}{2}$

$14\tfrac{2}{5}$

Correct answer:

$11\tfrac{13}{15}$

Explanation:

In this problem we are adding complex fractions. The first step is to add the whole numbers preceding the fractions. 2+3+1+4=10 . Next we need to find a common denominator to add the fractions. This should be the smallest number that has all of the other denominators as a factor. The least common denominator in this case is 30. Now we need to multiply the top and bottom of each fraction by the number that will make the denominator 30. From here we can add and divide the top and bottom by two to simplify.

$\\\frac{1}{3}*\frac{10}{10}=\frac{10}{30};\\ \\\; \frac{1}{2}*\frac{15}{15}=\frac{15}{30};\\ \\ \; \frac{5}{6}*\frac{5}{5}=\frac{25}{30};\\ \\ \; \frac{1}{5}*\frac{6}{6}=\frac{6}{30}\\\frac{10}{30}+\frac{15}{30}+\frac{25}{30}+\frac{6}{30}=\frac{56}{30}=\frac{28}{15}$

From here we have an improper fraction so we must subtract the value of the denominator from the numerator to make a complex fraction. After subtracting once we get a proper fraction.

$\frac{28-15}{15}=\frac{13}{15}$ .

Since we subtracted once, that means we have a 1 attached to the fraction and can be added to the other 10 to make 11. Then to get the final answer we combine the whole numbers and the fraction to get $11\tfrac{13}{15}$ .

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ACT Math : How to add complex fractions

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Add Complex 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 #4 : How To Add Complex Fractions

Example Question #3 : How To Add Complex Fractions

Example Question #5 : How To Add Complex Fractions

Example Question #4 : How To Add Complex Fractions

Example Question #1 : How To Add Complex Fractions

Example Question #1 : How To Add Complex Fractions

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