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Adding Fractions — Definition, Formula & Examples

Adding fractions is the process of combining two or more fractions into a single fraction. When the denominators are the same, you add the numerators directly; when they differ, you first rewrite the fractions with a common denominator.

For fractions ab\frac{a}{b} and cd\frac{c}{d} where b≠0b \neq 0 and d≠0d \neq 0, their sum is defined as ab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}. In practice, using the least common denominator rather than the product bdbd keeps numbers smaller and the result in simplest form.

Key Formula

ab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
Where:
  • aa = Numerator of the first fraction
  • bb = Denominator of the first fraction (cannot be 0)
  • cc = Numerator of the second fraction
  • dd = Denominator of the second fraction (cannot be 0)

How It Works

If the fractions already share the same denominator, simply add the numerators and keep that denominator. If the denominators are different, find the least common denominator (LCD), convert each fraction so it has the LCD, then add the numerators. After adding, always check whether the result can be simplified by dividing the numerator and denominator by their greatest common factor. If the result is an improper fraction, you may convert it to a mixed number.

Worked Example

Problem: Add 2/5 + 1/4.
Find the LCD: The denominators are 5 and 4. The least common denominator is 20.
LCD(5,4)=20\text{LCD}(5, 4) = 20
Rewrite each fraction: Multiply the numerator and denominator of each fraction so both have a denominator of 20.
25=2×45×4=820,14=1×54×5=520\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}, \qquad \frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}
Add the numerators: Now that the denominators match, add the numerators and keep the denominator.
820+520=1320\frac{8}{20} + \frac{5}{20} = \frac{13}{20}
Simplify if possible: 13 and 20 share no common factor other than 1, so the fraction is already in simplest form.
1320\frac{13}{20}
Answer: 2/5 + 1/4 = 13/20

Another Example

Problem: Add 3/8 + 5/8.
Check the denominators: Both fractions already have the same denominator (8), so no conversion is needed.
Add the numerators: Add 3 and 5, keeping the denominator of 8.
38+58=3+58=88\frac{3}{8} + \frac{5}{8} = \frac{3 + 5}{8} = \frac{8}{8}
Simplify: 8 divided by 8 equals 1.
88=1\frac{8}{8} = 1
Answer: 3/8 + 5/8 = 1

Why It Matters

Adding fractions is a foundational skill you will use throughout elementary and middle school math, from working with mixed numbers to solving equations in pre-algebra. It also appears in everyday situations like combining partial quantities in recipes or splitting shared costs unevenly. Mastering this operation makes learning algebra, ratios, and proportions much smoother.

Common Mistakes

Mistake: Adding both the numerators and the denominators, e.g., writing 1/3 + 1/4 = 2/7.
Correction: Denominators are never added together. Convert to a common denominator first, then add only the numerators: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
Mistake: Forgetting to simplify the answer.
Correction: Always check if the numerator and denominator share a common factor. For example, 4/8 should be simplified to 1/2.

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