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.