Fractions - ACT Math

Card 0 of 20

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.

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

Compare your answer with the correct one above

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.

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

Compare your answer with the correct one above

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.

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

Compare your answer with the correct one above

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.

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

Compare your answer with the correct one above

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.

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

Compare your answer with the correct one above

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.

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

Compare your answer with the correct one above

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.

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

Compare your answer with the correct one above

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:

$\frac{431}{150}$

Compare your answer with the correct one above

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!

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

Compare your answer with the correct one above

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 $11\tfrac{13}{15}$ .

Compare your answer with the correct one above

Question

What is the result of adding of $\frac{2}{7}$ to $\frac{1}{4}$ ?

Answer

Let us first get our value for the percentage of the first fraction. 20% of 2/7 is found by multiplying 2/7 by 2/10 (or, simplified, 1/5): (2/7) * (1/5) = (2/35)

Our addition is therefore (2/35) + (1/4). There are no common factors, so the least common denominator will be 35 * 4 or 140. Multiply the numerator and denominator of 2/35 by 4/4 and the numerator of 1/4 by 35/35.

This yields:

(8/140) + (35/140) = 43/140, which cannot be reduced.

Compare your answer with the correct one above

Question

Add the following fractions

$\frac{2}{5}+\frac{3}{4}+\frac{1}{12}$

Answer

To add fractions you first must find the lowest common denominator. For these fractions it is 60. Then you must multiply the numerator and denominator by the number such that the denominator is equal to the LCD. For example $\frac{2}{5}$ gets multiplied by 12 (on both the numerator and denominator) because 5 times 12 is 60. When you do that you get the expression

$\frac{24}{60}+\frac{45}{60}+\frac{5}{60}$

then you just add the numerators and get

$\frac{74}{60}$

2 goes into both of those numbers so you get

$\frac{37}{30}$

Compare your answer with the correct one above

Question

Simplify the expression:

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

Answer

First you want to find the least common denominator (in this case it's 18):

$\frac{91}{92}-\frac{22}{29}+\frac{115}{118}$

$\frac{9}{18}-\frac{4}{18}+\frac{15}{18}$

$\frac{9-4+15}{18}$

$\frac{20}{18}=\frac{10}{9}$

Compare your answer with the correct one above

Question

Which of the following is equal to $\frac{1}{y(y+1)}$ ?

Answer

We have two options here. We can manipulate the answers to work backwards and find a common denominator. This involves simply subtracting or adding two fractions. We can also try to rewrite the numerator by adding and subtracting the value . This serves the purpose of creating a sum in the numerator than can be split into and . This gives us one of the factors in the denominator in each numerator. When we separate or decompose the fraction, we can divide out by the common factor to re-express this as the difference of two rational expressions.

$\frac{1}{y(y+1)}=\frac{1+y-y}{y(y+1)}=\frac{y+1}{y(y+1)}-\frac{y}{y(y+1)}=\frac{1}{y}-\frac{1}{y+1}$

Compare your answer with the correct one above

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 $\frac{1}{2}$ .

Compare your answer with the correct one above

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.

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

Compare your answer with the correct one above

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.

$\frac{1}{x}\cdot\frac{1}{x^2}=\frac{1}{x^3}$

Compare your answer with the correct one above

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.

$=\frac{1}{18}+\frac{1}{18}=\frac{2}{18}=\frac{1}{9}$

Compare your answer with the correct one above

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:

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

Compare your answer with the correct one above

Question

$\dpi{100} \small \frac{1}{3}\div \frac{3}{5} =$

Answer

Cross multiply or multiply using the reciprocal of the second fraction.

Compare your answer with the correct one above

Tap the card to reveal the answer