How to Add and Subtract Fractions with Different Denominators

Adding and subtracting fractions with different denominators is a skill you'll need often in school and in everyday life. After this lesson, you'll be able to confidently solve problems like $\frac{2}{3} + \frac{1}{4}$ or $\frac{5}{6} - \frac{1}{2}$, step by step.

The Method: Finding a Common Denominator

When the denominators (the bottom numbers) are different, you can't add or subtract the fractions directly. You first need to rewrite them with a common denominator. Usually, we use the least common denominator (LCD), which is the smallest number both denominators divide into.

Steps:

1. Find the least common denominator (LCD) of the fractions.
2. Rewrite each fraction as an equivalent fraction with the LCD.
3. Add or subtract the numerators (top numbers), keeping the denominator the same.
4. Simplify the answer if possible.

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

Step 1: The denominators are 3 and 4. The LCD is 12.

Step 2: Rewrite each fraction with denominator 12:

• $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
• $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$

Step 3: Add the numerators:

• $\frac{8}{12} + \frac{3}{12} = \frac{8+3}{12} = \frac{11}{12}$

Step 4: $\frac{11}{12}$ is already simplified.

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

Worked Example 2: $\frac{5}{6} - \frac{1}{2}$

Step 1: The denominators are 6 and 2. The LCD is 6.

Step 2: Rewrite each fraction with denominator 6:

• $\frac{5}{6}$ already has denominator 6.
• $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

Step 3: Subtract the numerators:

• $\frac{5}{6} - \frac{3}{6} = \frac{5-3}{6} = \frac{2}{6}$

Step 4: Simplify $\frac{2}{6}$ by dividing top and bottom by 2:

• $\frac{2}{6} = \frac{1}{3}$

Final Answer: $\frac{1}{3}$

Common Mistakes and How to Avoid Them

• Forgetting to find a common denominator: You can't add or subtract the numerators unless the denominators match.
• Not multiplying both top and bottom: When changing denominators, multiply both the numerator and denominator by the same number to keep the value the same.
• Forgetting to simplify the final answer: Always check if your answer can be reduced.

Real-Life Application

Adding and subtracting fractions is useful when following recipes, splitting bills, or measuring materials (like wood or fabric) that come in parts.

Try It Yourself

Calculate $\frac{3}{5} + \frac{2}{3}$.

Takeaway

Whenever you add or subtract fractions with different denominators, always find a common denominator first. Practice a few more examples, and you'll master this skill quickly. If you want more step-by-step practice, you can sign up for free and try the Solve feature at https://gettutor.app/signup.