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ACT Math : Complex Fractions

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Example Question #10 : 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:

$11\tfrac{13}{15}$

$10\tfrac{1}{15}$

$10\tfrac{1}{2}$

$14\tfrac{2}{5}$

$12\tfrac{7}{8}$

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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Example Question #1 : How To Multiply Complex Fractions

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

Possible Answers:

$\frac{3}{2}$

$\frac{4}{9}$

$\frac{16}{9}$

$\frac{27}{4}$

$\frac{9}{2}$

Correct answer:

$\frac{27}{4}$

Explanation:

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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Example Question #2 : How To Multiply Complex Fractions

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

Possible Answers:

$\frac{45}{64}$

$\frac{5}{3}$

$\frac{1}{5}$

$\frac{7}{15}$

$\frac{23}{45}$

Correct answer:

$\frac{45}{64}$

Explanation:

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 120 .

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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Example Question #3 : How To Multiply Complex Fractions

Simplify:

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

Possible Answers:

$\frac{4}{5}$

$\frac{4}{21}$

$\frac{25}{140}$

$\frac{63}{160}$

$\frac{512}{315}$

Correct answer:

$\frac{63}{160}$

Explanation:

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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Example Question #4 : How To Multiply Complex Fractions

Simplify:

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

Possible Answers:

$\frac{71}{22}$

$\frac{72}{343}$

$\frac{7}{8}$

$\frac{14}{9}$

$\frac{49}{3}$

Correct answer:

$\frac{14}{9}$

Explanation:

Generally, when you multiply fractions, it is a very easy affair. This does not change for complex fractions like this. You can begin by simply multiplying the numerators and denominators directly. Thus, you know:

$\frac{\frac{4}{5}}{\frac{3}{7}} * \frac{ \frac{5}{7}}{\frac{6}{7}}=\frac{\frac{4}{5}*\frac{5}{7}}{\frac{3}{7}*\frac{6}{7}}$

Now, simplify this to:

$\frac{\frac{4}{1}*\frac{1}{7}}{\frac{3}{7}*\frac{6}{7}}$ or $\frac{\frac{4}{7}}{\frac{18}{49}}$

Now, remember that when you divide fractions, you multiply the numerator by the reciprocal of the denominator:

$\frac{\frac{4}{7}}{\frac{18}{49}} = \frac{4}{7}*\frac{49}{18}$

Now, cancel your terms immediately:

$\frac{2}{1}*\frac{7}{9}$ , which is easy to finish:

$\frac{2}{1}*\frac{7}{9}=\frac{14}{9}$

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ACT Math : Complex Fractions

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #10 : How To Add Complex Fractions

Example Question #1 : How To Multiply Complex Fractions

Example Question #2 : How To Multiply Complex Fractions

Example Question #3 : How To Multiply Complex Fractions

Example Question #4 : How To Multiply Complex Fractions

All ACT Math Resources

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