Complex Fractions - ACT Math

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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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Question

Simplify:

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

Answer

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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Question

Simplify:

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

Answer

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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Question

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?

Answer

In this problem we are adding complex fractions. The first step is to add the whole numbers preceding the fractions. . 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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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 $\frac{x + \frac{1}{x}}{x}$

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

What is $\frac{3}{\frac{1}{2}} \times \frac{\frac{1}{4}}{\frac{2}{9}}$ ?

Answer

Simplify both sides first. $\frac{3}{\frac{1}{2}}$ simplifies to 6. $\frac{\frac{1}{4}}{\frac{2}{9}}$ simplifies to $\frac{9}{8}$ . Finally 6 $\times$ $\frac{9}{8}$ = $\frac{27}{4}$ .

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Question

What is $\frac{\frac{1}{2}}{\frac{2}{3}}\cdot \frac{\frac{3}{4}}{\frac{4}{5}}$ equal to?

Answer

When multiplying fractions, we can simply multiply the numerators and then multiply the denominators. Therefore, $\frac{\frac{1}{2}}{\frac{2}{3}}\cdot \frac{\frac{3}{4}}{\frac{4}{5}}$ is equal to $\frac{ \frac{1}{2}\cdot\frac{3}{4}}{\frac{2}{3}\cdot \frac{4}{5}}$

We then do the same thing again, giving us $\frac{\frac{3}{8}}{\frac{8}{15}}$ .

Now we must find the least common denominator, which is .

We multiply the top by $\frac{15}{15}$ and the bottom by $\frac{8}{8}$ . After we do this we can multiply our numerator by the reciprocal of the denominator.

Therefore our answer becomes,

$\frac{\frac{3\cdot 15}{8\cdot 15}}{\frac{8 \cdot 8}{15 \cdot 8}}=\frac{\frac{45}{120}}{\frac{64}{120}}=\frac{45}{120}\cdot \frac{120}{64}=\frac{45}{64}$ .

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Question

Simplify:

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

Answer

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:

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

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