Adding and Subtracting Fractions with Unlike Denominators

When adding or subtracting fractions, the pieces must be the same size. This means the fractions need to have the same bottom number, called the denominator. If they have different (unlike) denominators, you cannot add or subtract them straight away.

To solve these problems, you first need to rewrite the fractions so they share a common denominator.

The 4-Step Process

Follow these four simple steps to add or subtract any fractions with unlike denominators:

1. Find a Common Denominator: Find a common multiple for both denominators. The easiest way is to find the Least Common Multiple (LCM), which becomes your Least Common Denominator (LCD).
2. Rewrite the Fractions: Multiply the top (numerator) and bottom (denominator) of each fraction by the same number to create equivalent fractions with the common denominator.
3. Add or Subtract the Numerators: Now that the denominators match, add or subtract the top numbers. Keep the denominator exactly the same!
4. Simplify: If possible, reduce your final fraction to its simplest form or convert improper fractions to mixed numbers.

Example 1: Adding Fractions

Problem: $\frac{2}{3} + \frac{3}{5}$

Step 1: Find a common denominator for $3$ and $5$. The multiples of $3$ are $3, 6, 9, 12, 15, \dots$ and the multiples of $5$ are $5, 10, 15, \dots$. The Least Common Denominator is $15$.

Step 2: Rewrite both fractions to have $15$ as the denominator.

• For $\frac{2}{3}$, multiply top and bottom by $5$: $\frac{2 \times 5}{3 \times 5} = \frac{10}{15}$
• For $\frac{3}{5}$, multiply top and bottom by $3$: $\frac{3 \times 3}{5 \times 3} = \frac{9}{15}$

Step 3: Add the numerators. $\frac{10}{15} + \frac{9}{15} = \frac{19}{15}$

Step 4: Simplify. Since $19$ is larger than $15$, this is an improper fraction. We can write it as a mixed number: $\frac{19}{15} = 1 \frac{4}{15}$

Example 2: Subtracting Fractions

Problem: $\frac{7}{8} - \frac{1}{3}$

Step 1: Find the LCD for $8$ and $3$. The smallest number both $8$ and $3$ divide into evenly is $24$.

Step 2: Rewrite the fractions.

• $\frac{7 \times 3}{8 \times 3} = \frac{21}{24}$
• $\frac{1 \times 8}{3 \times 8} = \frac{8}{24}$

Step 3: Subtract the numerators. $\frac{21}{24} - \frac{8}{24} = \frac{13}{24}$

Step 4: Simplify. The fraction $\frac{13}{24}$ cannot be reduced further, so this is our final answer.

Example 3: Using the Least Common Denominator

Problem: $\frac{5}{6} + \frac{2}{9}$

Step 1: Find the LCD for $6$ and $9$. You could multiply $6 \times 9 = 54$, but that's a big number! Let's find the LCM instead. Multiples of $9$: $9, 18, 27\dots$ Multiples of $6$: $6, 12, 18, 24\dots$ The LCD is $18$.

Step 2: Rewrite the fractions.

• $\frac{5 \times 3}{6 \times 3} = \frac{15}{18}$
• $\frac{2 \times 2}{9 \times 2} = \frac{4}{18}$

Step 3: Add the numerators. $\frac{15}{18} + \frac{4}{18} = \frac{19}{18}$

Step 4: Simplify to a mixed number. $\frac{19}{18} = 1 \frac{1}{18}$