Multiplying and Dividing Fractions - Algebra II

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Question

Simplify $\frac{3}{6}\cdot\frac{12}{4}$ .

Answer

The problem can be made easier by first simplifying each fraction: $\frac{3}{6}=\frac{1}{2}$ and $\frac{12}{4}=\frac{3}{1}$ .

This brings our new problem to $\frac{1}{2}\cdot\frac{3}{1}$ .

Now, the numerators are multiplied by each other then the denomenators are multiplied by each other: $\frac{1\cdot3}{2\cdot1}=\frac{3}{2}$ .

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Question

Simplify $\frac{5}{3}\div\frac{7}{6}$ .

Answer

To solve, we must turn the division problem into a multiplication problem by "flipping" the second fraction (dividing by a fraction is the same as multiplying by its reciprocal):

$\frac{5}{3}\div\frac{7}{6}=\frac{5}{3}\cdot\frac{6}{7}$ .

Then, we multiply the numerators followed by the denomenators:

$\frac{5\cdot6}{3\cdot7}=\frac{30}{21}$ .

Lastly, the fraction must be simplified by a factor of 3:

$\frac{30}{21}=\frac{10}{7}$ , which gives us our final answer.

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Question

Solve the following equation to find .

$\frac{2}{3}x=\frac{4}{15}+\frac{3}{15}$

Answer

The first step in solving this equation is to add the fractions, giving us:

$\frac{2}{3}x=\frac{7}{15}$

To solve for , we need to divide both sides by $\frac{2}{3}$ .

Remember: When we divide a number by a fraction, we "switch" (find the reciprocal) of the fraction and mulitply it to the number.

$\bigg(\frac{3}{2}\bigg)\bigg(\frac{2}{3}\bigg)x=\bigg(\frac{7}{15}\bigg) \bigg(\frac{3}{2}\bigg)$

The right side of the equation cancels out leaving alone:

$x=\frac{7\times 3}{15\times 2}=\frac{21}{30}$

Notice: Both the numerator and denominator are divisble by so we can simplify this further.

$x=\frac{7}{10}$

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Question

Multiply:

$\left(\frac{x^2}{z}\right)\cdot \left(\frac{x^2z^2}{y}\right)$

Answer

To multiply fractions, just multiply the numerators, then the denominators, and then simplify.

$(\frac{x^2}{z})(\frac{x^2z^2}{y})=\frac{x^4z^2}{zy}=\frac{x^4z}{y}$

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Question

Multiply:

$\left(\frac{ab^2c}{a^{\frac{1}{2}}b^4c^3}\right)\cdot \left(\frac{abc^2}{c}\right)$

Answer

To multiply fractions, multiply the numerators and denominators together, then simplify.

$(\frac{ab^2c}{a^{\frac{1}{2}}b^4c^3})(\frac{abc^2}{c})=\frac{a^2b^3c^3}{a^{\frac{1}{2}}b^4c^4}=\frac{a^{\frac{3}{2}}}{bc}$

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Question

Multiply:

$\left(\frac{4}{5}x\right)\cdot \left(\frac{6}{13}y\right)$

Answer

Multiply the numerators and denominators. Then, simplify.

$(\frac{4}{5}x)(\frac{6}{13}y)=(\frac{4x}{5})(\frac{6y}{13})=\frac{24xy}{65}$

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Question

Multiply:

$\left(\frac{15}{7x}\right)\cdot \left(\frac{12x^2}{5}\right)$

Answer

Multiply the numerators and denominators, then simplify.

$(\frac{15}{7x})(\frac{12x^2}{5})=\frac{180x^2}{35x}=\frac{36}{7}x$

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Question

Simplify:

$\frac{36x^2y}{5x^5 y^{-2}}\div\frac{9x^{-3}}{25}$

Answer

In order to divide fractions, you need to multiply the first fraction by the reciprocal of the second one.

$\frac{36x^2y}{5x^5y^{-2}}\div\frac{9x^{-3}}{25}=\frac{36x^2y}{5x^5y^{-2}}\times\frac{25}{9x^{-3}}$

Now, multiply the numerators and denominators together, then simplify.

$\frac{36x^2y}{5x^5y^{-2}}\times\frac{25}{9x^{-3}}=\frac{900x^2y}{45x^2y^{-2}}=20y^3$

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Question

Simplify:

$\frac{25st^2}{6s^3t^5}\div\frac{30s^{-9}}{45s^2t^7}$

Answer

To divid fractions, you need to multiply the first fraction by the reciprocal of the second.

$(\frac{25st^2}{6s^3t^5})\div(\frac{30s^{-9}}{45s^2t^7})=(\frac{25st^2}{6s^3t^5})\times(\frac{45s^2t^7}{30s^{-9}})$

Now, multiply the numerators and denominators together, then simplify.

$(\frac{25st^2}{6s^3t^5})\times(\frac{45s^2t^7}{30s^{-9}})=\frac{1125s^3t^9}{180s^{-6}t^5}=\frac{25s^9t^4}{4}$

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Question

Simplify:

$\frac{12m^2n}{16m^3n^4}\div\frac{8m^{-4}n}{18m^4n^2}$

Answer

To divide fractions, you need to multiply the first fraction by the reciprocal of the second.

$(\frac{12m^2n}{16m^3n^4})\div(\frac{8m^{-4}n}{18m^4n^2})=(\frac{12m^2n}{16m^3n^4})\times(\frac{18m^4n^2}{8m^{-4}n})$

Now, multiply the numerators and denominators, then simplify.

$(\frac{12m^2n}{16m^3n^4})\times(\frac{18m^4n^2}{8m^{-4}n})=\frac{216m^6n^3}{128m^{-1}n^5}=\frac{27m^7}{16n^2}$

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Question

Evaluate: $\frac{7}{6}\div 2\frac{3}{8}$

Answer

Change all the fractions to improper fractions.

$\frac{7}{6}\div 2\frac{3}{8}=\frac{7}{6}\div \frac{19}{8}$

Convert the division sign to a multiplication sign, but flip the second term. Simplify.

$\frac{7}{6}\times \frac{8}{19}= \frac{7}{3}\times \frac{4}{19}= \frac{28}{57}$

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Question

Divide and simplify:

$\frac{2}{11}\div \frac{9}{22}$ .

Answer

Multiplying fractions is a two-step process. First, you must flip the second fraction (make it its reciprocal) and then set up the equation as a multiplication problem:

$\frac{2}{11}\cdot \frac{22}{}9$ .

Then, cross-reduce: 11 goes into both 11 and 22, so you can take that out so it looks like:

$\frac{2}{1}\cdot \frac{2}{9}$ .

Then, multiply straight across so that you get $\frac{4}{}9$ .

You can't simplify any further, so that's your answer!

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Question

Multiply and simplify:

$\frac{6y^2}{14x^9}\cdot \frac{2x^3}{4y}$

Answer

Before multiplying straight across, I would start with some cross-reducing. Look diagonally and see what can be simplified.

Between and , you can take out a

Between and , you can take out a .

The simplified version looks like:

$\frac{3y}{7x^6}\cdot \frac{1}{2}=\frac{3y}{14x^6}$

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Question

Multiply these fractions:

$\frac{5}{12}*\frac{3}{7}$

Answer

When multiplying fractions, all we have to do is multiply the numerators together and multiply the denominators together:

$\frac{5}{12}\frac{3}{7}=\frac{53}{12*7}=\frac{15}{84}$

Simply the fraction to get the final answer:

$\frac{15}{84}\div \frac{3}{3}=\frac{5}{28}$

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Question

Multiply these fractions:

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

Answer

When multiplying fractions, all we have to do is multiply the numerators together and multiply the denominators together:

$\frac{7}{9}\frac{1}{14}=\frac{71}{9*14}=\frac{7}{126}$

Simply the fraction to get the final answer:

$\frac{7}{126}\div \frac{7}{7}=\frac{1}{18}$

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Question

Multiply these fractions:

$\frac{2}{3}*\frac{5}{6}$

Answer

When multiplying fractions, all we have to do is multiply the numerators together and multiply the denominators together:

$\frac{2}{3}\frac{5}{6}=\frac{25}{3*6}=\frac{10}{18}$

Simply the fraction to get the final answer:

$\frac{10}{18}\div \frac{2}{2}=\frac{5}{9}$

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Question

Multiply these fractions:

$\frac{1}{2}*\frac{2}{3}$

Answer

When multiplying fractions, all we have to do is multiply the numerators together and multiply the denominators together:

$\frac{1}{2}\frac{2}{3}=\frac{12}{2*3}=\frac{2}{6}$

Simply the fraction to get the final answer:

$\frac{2}{6}\div \frac{2}{2}=\frac{1}{3}$

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Question

Multiply these fractions:

$\frac{5}{7}*\frac{1}{3}$

Answer

When multiplying fractions, all we have to do is multiply the numerators together and multiply the denominators together:

$\frac{5}{7}\frac{1}{3}=\frac{51}{7*3}=\frac{5}{21}$

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Question

Multiply these fractions:

$\frac{4}{9}*\frac{3}{7}$

Answer

When multiplying fractions, all we have to do is multiply the numerators together and multiply the denominators together:

$\frac{4}{9}\frac{3}{7}=\frac{43}{9*7}=\frac{12}{63}$

Simply the fraction to get the final answer:

$\frac{12}{63}\div \frac{3}{3}=\frac{4}{21}$

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Question

Multiply these fractions:

$\frac{8}{9}*\frac{3}{4}$

Answer

When multiplying fractions, all we have to do is multiply the numerators together and multiply the denominators together:

$\frac{8}{9}\frac{3}{4}=\frac{83}{9*4}=\frac{24}{36}$

Simplify the fraction to get the final answer:

$\frac{24}{36}\div \frac{6}{6}=\frac{4}{6}\div \frac{2}{2}=\frac{2}{3}$

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