How do you add two or more fractions together?
Understanding the basics of fractions is essential to adding them successfully. In a fraction, the top number is called the numerator, and the bottom number is called the denominator.
$$ \frac{\text{numerator}}{\text{denominator}} $$
When adding fractions, it’s important to ensure that the denominators are the same.
If they are not, you will need to find a common denominator by changing the fractions into their equivalent form.
Once you have the same denominator, you can add the numerators together and place the sum over the common denominator. Finally, simplify the fraction if possible.
With a little practice, adding fractions will become second nature to you.
Understanding Fractions
What are Fractions?
Fractions represent parts of a whole, composed of:
- Numerator: how many parts you have
- Denominator: how many parts the whole is divided into
Example: $\frac{3}{4}$ → 3 parts out of 4
Simplifying Fractions
Fractions can be simplified by dividing both numerator and denominator by their GCF:
$$ \frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2} $$
Fractions can be converted to decimals by division:
$$ \frac{3}{5} = 3 \div 5 = 0.6 $$
Adding Fractions
Fractions With Same Denominator
Just add the numerators:
$$ \frac{1}{7} + \frac{3}{7} = \frac{1 + 3}{7} = \frac{4}{7} $$
Fractions With Different Denominators
Step-by-step:
Find the Lowest Common Denominator (LCD)
Example: $\frac{1}{3} + \frac{1}{4}$
Multiples: $$ {3, 6, 9, \fcolorbox{red}{yellow}{12}, …}, \quad {4, 8, \fcolorbox{red}{yellow}{12}, …} $$Convert to Equivalent Fractions
$$ \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12} $$Add
$$ \frac{4}{12} + \frac{3}{12} = \frac{7}{12} $$
Adding Mixed Numbers
Method 1: Add Fractions Then Whole Numbers
Example 1: $$ 1\frac{1}{2} + 2\frac{1}{3} = 1 + \frac{1}{2} + 2 + \frac{1}{3} $$
Add fractions: $$ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} $$
Add wholes: $$ 1 + 2 = 3 $$
Final answer: $$ 3\frac{5}{6} $$
Example 2 (Improper Fraction Result): $$ 3\frac{3}{4} + 2\frac{4}{5} $$
Convert: $$ \frac{3}{4} = \frac{15}{20}, \quad \frac{4}{5} = \frac{16}{20} $$
Add: $$ \frac{15}{20} + \frac{16}{20} = \frac{31}{20} = 1\frac{11}{20} $$
Then: $$ 3 + 2 + 1 = 6 \Rightarrow 6\frac{11}{20} $$
Summary
- If denominators differ, find the LCM.
- Convert each fraction to an equivalent with the same denominator.
- Add numerators, keep the denominator.
- Simplify the final fraction if possible.
- For mixed numbers, add whole parts and fractions separately or convert to improper fractions.