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Dividing Mixed Numbers

To divide mixed numbers, simply follow these three steps:

1. Convert the mixed numbers to improper fractions
2. Divide the fractions
3. Convert back to a mixed number

Let's do an example.

Step One: Convert to Improper Fractions

$$8\frac{2}{3} \div 3\frac{5}{9} = \dfrac{26}{3} \div \dfrac{32}{9}$$

Step Two: Divide Fractions

$$\begin{align} \dfrac{26}{3} \div \dfrac{32}{9}&=\dfrac{26}{3} \cdot \dfrac{9}{32}\\ &=\dfrac{\cancel{26} 13}{\cancel{3} 1} \cdot \dfrac{\cancel{9}3}{\cancel{32} 16}\\ &=\dfrac{13}{1} \cdot \dfrac{3}{16}\\ &=\dfrac{13 \cdot 3}{1 \cdot 16}\\ &=\dfrac{39}{16}\\ \end{align}$$

Step Three: Convert Back to Mixed Number

$$\dfrac{39}{16}=2\frac{7}{16}$$

(Video) Multiplying and Dividing Mixed Numbers

by mathman1024

Here’s a video by mathman1024 that covers multiplying mixed numbers and dividing mixed numbers.

$\large{\text{Summary}}$

To multiply and divide mixed numbers, you first convert them into improper fractions. Then, you multiply or divide, and convert back into a mixed number.

For example let's look at $3\frac14\div1\frac38$

$$\begin{align}3\frac14 \div1\frac38&=\frac{3\cdot4 +1}{4} \div \frac{1\cdot8+3}{8}\\ &=\frac{13}{4} \div \frac{11}{8}\\ &=\frac{13}{\cancel{4}{\color{red}{1}}} \cdot \frac{\cancel{8}{\color{red}{2}}}{11}\\ &=\frac{26}{11}\\ &=2\frac{4}{11} \end{align}$$

By canceling out a factor of $4$, we avoid having to reduce the fraction to lowest terms at the end.

Convert them to improper fractions, then flip one upside-down!

To multiply mixed numbers, it is simplest to:

1. Convert them to improper fractions
2. Divide the fractions (this is where the flipping upside-down comes in).
3. Convert the result back to a mixed number (if necessary)

Let's try it out an actual problem. What is $5\frac{3}{4}$ divided by $6\frac{7}{10}$? Start with Step 1.

$$5\frac{3}{4} = \frac{23}{4}$$

We can find that by multiplying the denominator ($4$) by the whole number ($5$), then adding the result to the numerator ($3$). $$5 \times 4 = 20\\ 20 + 3 = 23$$ So, our answer is $\frac{23}{4}$.

$$6\frac{7}{10} = \frac{67}{10}$$

Now that we have both numbers as improper fractions, we can move on to Step 2!

$$\frac{23}{4} \div \frac{67}{10} = \frac{230}{268}$$

We can find that by first flipping the second fraction upside-down (so that its numerator and denominator switch), then multiplying the numerators by the numerators and the denominators by the denominators. $$\frac{23}{4} \div \frac{67}{10} = \frac{23}{4} \times \frac{10}{67}$$ $$23 \times 10 = 230$$ $$4 \times 67 = 268$$

So, our final fraction is $\frac{230}{268}$. But we're not quite done yet! Let's take a look at Step 3.

$\frac{230}{268}$ is not an improper fraction, as the numerator is less than the denominator. So, unless you need to simplify the fraction, you're done! If you do need to simplify the fraction, you can do that for this one by simply dividing the numerator and the denominator by $2$. $$\frac{230}{268} = \frac{230}{268} \div \frac{2}{2} = \frac{115}{134}$$