Converting Mixed Numbers to Improper Fractions

Let's see how to convert mixed numbers into improper fractions.

Actually, we convert them almost in the same way as we add fractions with unlike denominators.

Recall that every integer ${m}$ can be expressed as fraction $\frac{{m}}{{1}}$ .

Mixed number, as known, is a sum of integer and fraction, so mixed number ${m}\frac{{n}}{{q}}$ is ${m}+\frac{{n}}{{q}}=\frac{{m}}{{1}}+\frac{{n}}{{q}}$ .

Now, we perform addition of fractions with unlike denominators: $\frac{{m}}{{1}}+\frac{{n}}{{q}}=\frac{{{m}{q}}}{{q}}+\frac{{n}}{{q}}=\frac{{{m}{q}+{n}}}{{q}}$ .

To convert mixed number into improper fraction following formula is used: ${\color{green}{{{m}\frac{{n}}{{q}}=\frac{{{m}{q}+{n}}}{{q}}}}}$ .

Example 1. Convert ${5}\frac{{3}}{{7}}$ to improper fraction.

Let's first do converting step-by-step:

${5}\frac{{3}}{{7}}={5}+\frac{{3}}{{7}}=\frac{{5}}{{1}}+\frac{{3}}{{7}}=\frac{{{5}\cdot{\color{red}{{{7}}}}}}{{{1}\cdot{\color{red}{{{7}}}}}}+\frac{{3}}{{7}}=\frac{{35}}{{7}}+\frac{{3}}{{7}}=\frac{{38}}{{7}}$ .

Answer: $\frac{{38}}{{7}}$ .

Next example.

Example 2. Convert ${9}\frac{{2}}{{5}}$ to improper fraction.

Now, we use direct formula:

${9}\frac{{2}}{{5}}=\frac{{{9}\cdot{5}+{2}}}{{5}}=\frac{{47}}{{5}}$ .

Answer: $\frac{{47}}{{5}}$ .

Next example.

Example 3. Convert $-{2}\frac{{8}}{{11}}$ to improper fraction.

In case of negative mixed number way of converting isn't changed.

First we just ignore minus sign: ${2}\frac{{8}}{{11}}=\frac{{{2}\cdot{11}+{8}}}{{11}}=\frac{{30}}{{11}}$ .

Now, add ignored minus sign: $-\frac{{30}}{{11}}$ .

Answer: $-\frac{{30}}{{11}}$ .

Now, it is time to practice.

Exercise 1. Convert ${1}\frac{{7}}{{8}}$ to improper fraction.

Answer: $\frac{{15}}{{8}}$ .

Next exercise.

Exercise 2. Convert ${9}\frac{{2}}{{3}}$ to improper fraction.

Answer: $\frac{{29}}{{3}}$ .

Next exercise.

Exercise 3. Convert $-{5}\frac{{7}}{{9}}$ to improper fraction.

Answer: $-\frac{{52}}{{9}}$ .