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{m} can be expressed as fraction m1\frac{{m}}{{1}}.

Mixed number, as known, is a sum of integer and fraction, so mixed number mnq{m}\frac{{n}}{{q}} is m+nq=m1+nq{m}+\frac{{n}}{{q}}=\frac{{m}}{{1}}+\frac{{n}}{{q}}.

Now, we perform addition of fractions with unlike denominators: m1+nq=mqq+nq=mq+nq\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: mnq=mq+nq{\color{green}{{{m}\frac{{n}}{{q}}=\frac{{{m}{q}+{n}}}{{q}}}}}.

Example 1. Convert 537{5}\frac{{3}}{{7}} to improper fraction.

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

537=5+37=51+37=5717+37=357+37=387{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: 387\frac{{38}}{{7}}.

Next example.

Example 2. Convert 925{9}\frac{{2}}{{5}} to improper fraction.

Now, we use direct formula:

925=95+25=475{9}\frac{{2}}{{5}}=\frac{{{9}\cdot{5}+{2}}}{{5}}=\frac{{47}}{{5}}.

Answer: 475\frac{{47}}{{5}}.

Next example.

Example 3. Convert 2811-{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: 2811=211+811=3011{2}\frac{{8}}{{11}}=\frac{{{2}\cdot{11}+{8}}}{{11}}=\frac{{30}}{{11}}.

Now, add ignored minus sign: 3011-\frac{{30}}{{11}}.

Answer: 3011-\frac{{30}}{{11}}.

Now, it is time to practice.

Exercise 1. Convert 178{1}\frac{{7}}{{8}} to improper fraction.

Answer: 158\frac{{15}}{{8}}.

Next exercise.

Exercise 2. Convert 923{9}\frac{{2}}{{3}} to improper fraction.

Answer: 293\frac{{29}}{{3}}.

Next exercise.

Exercise 3. Convert 579-{5}\frac{{7}}{{9}} to improper fraction.

Answer: 529-\frac{{52}}{{9}}.