How to multiply complex fractions - ACT Math

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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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Question

Simplify:

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

Answer

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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