Adding Fractions

Adding fractions is one of the most practical math skills you can have. Whether you are doubling a recipe or adding up measurements on a job site, you need to know how to combine fractions accurately.

The key rule: you can only add fractions when they share the same denominator (bottom number). If they don’t, you need to find a common denominator first.

Adding Fractions with Like Denominators

When the denominators are already the same, simply add the numerators and keep the denominator.

Formula:

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$

Example 1: Add $\frac{2}{7} + \frac{3}{7}$

Both fractions have a denominator of 7, so add the numerators:

$\frac{2}{7} + \frac{3}{7} = \frac{2 + 3}{7} = \frac{5}{7}$

Answer: $\frac{5}{7}$

Example 2: Add $\frac{5}{8} + \frac{7}{8}$

$\frac{5}{8} + \frac{7}{8} = \frac{5 + 7}{8} = \frac{12}{8}$

Since $\frac{12}{8}$ is an improper fraction (numerator larger than denominator), simplify:

$\frac{12}{8} = \frac{3}{2} = 1\frac{1}{2}$

Answer: $1\frac{1}{2}$

Adding Fractions with Unlike Denominators

When the denominators are different, you must find the Least Common Denominator (LCD) before adding. The LCD is the smallest number that both denominators divide into evenly.

Steps:

1. Find the LCD of the two denominators
2. Rewrite each fraction as an equivalent fraction with the LCD
3. Add the numerators
4. Simplify if needed

Finding the LCD

The LCD is the Least Common Multiple (LCM) of the denominators. For small numbers, you can list multiples:

• Multiples of 4: 4, 8, 12, 16, 20…
• Multiples of 6: 6, 12, 18, 24…
• LCD of 4 and 6 = 12

Example 3: Add $\frac{1}{4} + \frac{2}{6}$

Step 1: Find the LCD of 4 and 6. The LCD is 12.

Step 2: Convert each fraction:

$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$

$\frac{2}{6} = \frac{2 \times 2}{6 \times 2} = \frac{4}{12}$

Step 3: Add the numerators:

$\frac{3}{12} + \frac{4}{12} = \frac{7}{12}$

Answer: $\frac{7}{12}$

Example 4: Add $\frac{2}{3} + \frac{3}{5}$

Step 1: Find the LCD of 3 and 5. Since 3 and 5 share no common factors, the LCD is $3 \times 5 = 15$.

Step 2: Convert each fraction:

$\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$

$\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$

Step 3: Add:

$\frac{10}{15} + \frac{9}{15} = \frac{19}{15} = 1\frac{4}{15}$

Answer: $1\frac{4}{15}$

Adding Mixed Numbers

A mixed number has a whole number part and a fraction part, like $2\frac{3}{4}$. To add mixed numbers:

1. Add the whole number parts together
2. Add the fraction parts (using LCD if needed)
3. If the fraction sum is improper, convert and carry over to the whole number

Example 5: Add $3\frac{1}{4} + 2\frac{2}{3}$

Step 1: Add whole numbers: $3 + 2 = 5$

Step 2: Add fractions. The LCD of 4 and 3 is 12:

$\frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12}$

Step 3: Combine: $5 + \frac{11}{12} = 5\frac{11}{12}$

Answer: $5\frac{11}{12}$

Example 6: Add $4\frac{5}{6} + 1\frac{3}{4}$

Step 1: Add whole numbers: $4 + 1 = 5$

Step 2: Add fractions. The LCD of 6 and 4 is 12:

$\frac{5}{6} + \frac{3}{4} = \frac{10}{12} + \frac{9}{12} = \frac{19}{12}$

Step 3: Convert the improper fraction: $\frac{19}{12} = 1\frac{7}{12}$

Step 4: Add the extra whole number: $5 + 1\frac{7}{12} = 6\frac{7}{12}$

Answer: $6\frac{7}{12}$

Common LCD Reference Table

This table shows the LCD for denominators you will encounter frequently:

Real-World Application: Cooking

You are making banana bread. The recipe calls for $\frac{2}{3}$ cup of sugar and $\frac{3}{4}$ cup of flour for the topping. You want to know the total amount of dry ingredients for the topping (sugar + flour) so you can check if your mixing bowl is big enough.

Step 1: Find the LCD of 3 and 4, which is 12.

Step 2: Convert and add:

$\frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12} = 1\frac{5}{12} \text{ cups}$

You need a bowl that holds at least $1\frac{5}{12}$ cups just for the topping dry ingredients. In practice, you would round up and grab a 2-cup bowl to give yourself room to mix.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Like denominators: add the numerators and keep the denominator
• Unlike denominators: find the LCD, convert both fractions, then add
• Mixed numbers: add whole parts and fraction parts separately, then combine
• Always simplify your final answer when possible
• When in doubt, list the multiples of each denominator to find the LCD

Return to Arithmetic for more foundational math topics.