Converting Improper Fractions to Mixed Numbers

Converting improper fractions to mixed numbers is inverse of converting mixed number to improper fractions.

Suppose you want to convert ${3}\frac{{4}}{{5}}$ to improper fraction.

We already know that ${3}\frac{{4}}{{5}}=\frac{{3}}{{1}}+\frac{{4}}{{5}}=\frac{{{3}\cdot{5}}}{{5}}+\frac{{4}}{{5}}=\frac{{{3}\cdot{5}+{4}}}{{5}}=\frac{{19}}{{9}}$ .

Now, imagine that you want to convert $\frac{{19}}{{9}}$ to mixed number. We can look at above transformations from right to left to find that $\frac{{19}}{{9}}=\frac{{{3}\cdot{5}+{4}}}{{5}}=\frac{{{3}\cdot{5}}}{{5}}+\frac{{4}}{{5}}={3}+\frac{{4}}{{5}}={3}\frac{{4}}{{5}}$ .

From this we notice that we need only one thing to convert improper fraction to mixed number: division with remainder of numerator by denominator.

Indeed, when we talked about division with remainder we said that when 19 is divided by 5 result is 3 and something extra: ${19}={5}\cdot{3}+{4}$ . Now, we know that extra is nothing else than fraction $\frac{{4}}{{5}}$ .

If ${m}={n}\cdot{q}+{r}$ then ${\color{red}{{\frac{{m}}{{q}}={n}\frac{{r}}{{q}}}}}$ .

Now, let's go through a couple of examples.

Example 1. Convert $\frac{{23}}{{5}}$ to mixed number.

We can write 23 as ${23}={5}\cdot{4}+{3}$ (division with remainder is performed), so $\frac{{23}}{{5}}={4}\frac{{3}}{{5}}$ .

Answer: ${4}\frac{{3}}{{5}}$ .

Next example.

Example 2. Convert $\frac{{18}}{{14}}$ to mixed number.

First note that fraction is not irreducible. Reduce it: $\frac{{18}}{{14}}=\frac{{9}}{{7}}$ .

We can write 9 as ${9}={7}\cdot{1}+{2}$ (division with remainder is performed), so $\frac{{9}}{{7}}={1}\frac{{2}}{{5}}$ .

Answer: ${1}\frac{{2}}{{7}}$ .

Next example.

Example 3. Convert $-\frac{{45}}{{8}}$ to mixed number.

First ignore minus sign: work with $\frac{{45}}{{8}}$ .

We can write 45 as ${45}={8}\cdot{5}+{5}$ (division with remainder is performed), so $\frac{{45}}{{8}}={5}\frac{{5}}{{8}}$ .

Finally, don't forget about ignored minus sign.

Answer: $-{5}\frac{{5}}{{8}}$ .

Time to practice.

Exercise 1. Convert $\frac{{15}}{{7}}$ to mixed number.

Answer: ${2}\frac{{1}}{{7}}$ .

Next exercise.

Exercise 2. Convert $\frac{{24}}{{8}}$ to mixed number.

Answer: 3.

Next exercise.

Exercise 3. Convert $-\frac{{79}}{{10}}$ to mixed number.

Answer: $-{7}\frac{{9}}{{10}}$ .