Adding Fractions with Whole Numbers
Adding fractions with whole numbers is essentially the same as converting mixed number to improper fraction (just remember how to add integers correctly).
Indeed, suppose we want to add whole number $$${m}$$$ and fraction $$$\frac{{n}}{{q}}$$$.
If $$$\frac{{n}}{{q}}$$$ is proper fraction, then $$${m}\frac{{n}}{{q}}$$$ is mixed number and the task is to convert mixed number to improper fraction.
If $$$\frac{{n}}{{q}}$$$ is improper fraction, this doesn't change things much.
It is known that whole number $$${m}$$$ can be represented as fraction $$$\frac{{m}}{{1}}$$$.
Now, $$${m}+\frac{{n}}{{q}}=\frac{{m}}{{1}}+\frac{{n}}{{q}}=\frac{{{m}{q}}}{{q}}+\frac{{n}}{{q}}=\frac{{{m}{q}+{n}}}{{q}}$$$.
Formula for adding fractions with whole numbers: $$${\color{green}{{{m}+\frac{{n}}{{q}}=\frac{{{m}{q}+{n}}}{{q}}}}}$$$.
Example 1. Find $$${3}+\frac{{6}}{{7}}$$$.
Let's solve it step-by-step:
$$${3}+\frac{{6}}{{7}}=\frac{{3}}{{1}}+\frac{{6}}{{7}}=\frac{{{3}\cdot{\color{red}{{{7}}}}}}{{{1}\cdot{\color{red}{{{7}}}}}}+\frac{{6}}{{7}}=\frac{{21}}{{7}}+\frac{{6}}{{7}}=\frac{{27}}{{7}}$$$.
If you need mixed number, convert $$$\frac{{27}}{{7}}$$$ into the mixed number: $$$\frac{{27}}{{7}}={3}\frac{{6}}{{7}}$$$ (note that it is same as $$${3}+\frac{{6}}{{7}}$$$).
Answer: $$$\frac{{27}}{{7}}={3}\frac{{6}}{{7}}$$$.
Next example.
Example 2. Find $$$-{9}+\frac{{13}}{{8}}$$$.
Let's use direct formula:
$$$-{9}+\frac{{13}}{{8}}=\frac{{-{9}\cdot{8}+{13}}}{{8}}=\frac{{-{72}+{13}}}{{8}}=-\frac{{59}}{{8}}$$$.
Convert to mixed number if needed: $$$-\frac{{59}}{{8}}=-{7}\frac{{3}}{{8}}$$$
Answer: $$$-\frac{{59}}{{8}}=-{7}\frac{{3}}{{8}}$$$.
Next example.
Example 3. Find $$$-\frac{{9}}{{4}}+{\left(-{3}\right)}$$$.
$$$-\frac{{9}}{{4}}+{\left(-{3}\right)}=-\frac{{9}}{{4}}+{\left(-\frac{{{3}\cdot{\color{red}{{{4}}}}}}{{{1}\cdot{\color{red}{{{4}}}}}}\right)}=-\frac{{9}}{{4}}+{\left(-\frac{{12}}{{4}}\right)}=\frac{{-{9}+{\left(-{12}\right)}}}{{4}}=\frac{{-{9}-{12}}}{{4}}=-\frac{{21}}{{4}}$$$.
Convert to mixed fraction if needed: $$$-\frac{{21}}{{4}}=-{5}\frac{{1}}{{4}}$$$.
Answer: $$$-\frac{{21}}{{4}}=-{5}\frac{{1}}{{4}}$$$.
Now, it is time to practice.
Exercise 1. Find $$${2}+\frac{{6}}{{7}}$$$.
Answer: $$$\frac{{20}}{{7}}={2}\frac{{6}}{{7}}$$$.
Next exercise.
Exercise 2. Find $$${9}+{\left(-\frac{{29}}{{5}}\right)}$$$.
Answer: $$$\frac{{16}}{{5}}={3}\frac{{1}}{{5}}$$$.
Next exercise.
Exercise 3. Find $$$-{5}+\frac{{99}}{{8}}$$$.
Answer: $$$\frac{{59}}{{8}}={7}\frac{{3}}{{8}}$$$.