Adding Mixed Numbers

Adding mixed numbers is quite easy.

We know that mixed number consists of integer part and fractional part.

To add mixed numbers three steps are needed:

Convert each mixed number to improper fraction.
Add improper fractions (using addition of fractions with unlike denominators)
Convert improper fraction to mixed number if needed (and if possible).

Example 1. Find ${1}\frac{{3}}{{5}}+{2}\frac{{4}}{{9}}$ .

Convert each mixed number into improper fraction: ${1}\frac{{3}}{{5}}=\frac{{8}}{{5}}$ and ${2}\frac{{4}}{{9}}=\frac{{22}}{{9}}$ .

Now add fractions: $\frac{{8}}{{5}}+\frac{{22}}{{9}}=\frac{{{8}\cdot{9}}}{{{5}\cdot{9}}}+\frac{{{22}\cdot{5}}}{{{9}\cdot{5}}}=\frac{{72}}{{45}}+\frac{{110}}{{45}}=\frac{{182}}{{45}}$ .

Convert fraction to mixed number: $\frac{{182}}{{45}}={4}\frac{{2}}{{45}}$ .

Answer: ${4}\frac{{2}}{{45}}$ .

Next example.

Example 2. Find $-{2}\frac{{1}}{{6}}+{3}\frac{{1}}{{2}}$ .

Convert each mixed number into improper fraction: $-{2}\frac{{1}}{{6}}=-\frac{{13}}{{6}}$ and ${3}\frac{{1}}{{2}}=\frac{{7}}{{2}}$ .

Now add fractions: $-\frac{{13}}{{6}}+\frac{{7}}{{2}}=-\frac{{13}}{{6}}+\frac{{{7}\cdot{3}}}{{{2}\cdot{3}}}=-\frac{{13}}{{6}}+\frac{{21}}{{6}}=\frac{{8}}{{6}}$ .

Reduce fraction: $\frac{{8}}{{6}}=\frac{{4}}{{3}}$ .

Convert fraction to mixed number: $\frac{{4}}{{3}}={1}\frac{{1}}{{3}}$ .

Answer: $\frac{{4}}{{3}}={1}\frac{{1}}{{3}}$ .

Next example.

Example 3. Find ${2}+{5}\frac{{8}}{{11}}$ .

Since first number has no fractional part, we can easier add numbers.

Add integer parts: ${2}+{5}={7}$ and fractional part leave the same.

Warning. This works only when we add either both positive numbers or both negative numbers.

Answer: ${7}\frac{{8}}{{11}}=\frac{{85}}{{11}}$ .

Now, take pen and paper and do following exercises.

Exercise 1. Find ${5}\frac{{1}}{{6}}+{3}\frac{{7}}{{8}}$ .

Answer: $\frac{{217}}{{24}}={9}\frac{{1}}{{24}}$ .

Next exercise.

Exercise 2. Find ${7}\frac{{4}}{{9}}+{\left(-{2}\frac{{5}}{{6}}\right)}$ .

Answer: $\frac{{83}}{{18}}={4}\frac{{11}}{{18}}$ .

Next exercise.

Exercise 3. Find $-{5}\frac{{1}}{{6}}+{\left(-{3}\frac{{7}}{{8}}\right)}$ .

Answer: $-\frac{{217}}{{24}}=-{9}\frac{{1}}{{24}}$ .

Next exercise.

Exercise 4. Find ${3}\frac{{5}}{{6}}+\frac{{5}}{{7}}$ .

${3}\frac{{5}}{{6}}+\frac{{5}}{{7}}=\frac{{23}}{{6}}+\frac{{5}}{{7}}=\frac{{161}}{{42}}+\frac{{30}}{{42}}=\frac{{191}}{{42}}={4}\frac{{23}}{{42}}$ .

Answer: $\frac{{191}}{{42}}={4}\frac{{23}}{{42}}$ .

Next exercise.

Exercise 5. Find $-{5}\frac{{1}}{{6}}+{7}$ .

Here we just can't add fractional parts to obtain ${2}\frac{{1}}{{6}}$ . This is not correct, because one of the addends is negative and another is positive.

We do it as always.

$-{5}\frac{{1}}{{6}}+{7}=-\frac{{31}}{{6}}+\frac{{42}}{{6}}=\frac{{11}}{{6}}={1}\frac{{5}}{{6}}$ .

Answer: $\frac{{11}}{{6}}={1}\frac{{5}}{{6}}$ .

If you are not sure whether it is possible to add integer parts, use the three-step method. It guarantees correct answer.