Lesson 4.5: Add and Subtract Fractions with Different Denominators

New Concept

To add or subtract fractions with unlike denominators, you must first find a common ground. This involves finding the Least Common Denominator (LCD) and rewriting each fraction into an equivalent form before combining them.

What’s next

Next, you'll see how to find the LCD with interactive examples. Then, you'll apply this skill in a series of practice challenges.

Property

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

Find the least common denominator (LCD) of two fractions.
Step 1. Factor each denominator into its primes.
Step 2. List the primes, matching primes in columns when possible.
Step 3. Bring down the columns.
Step 4. Multiply the factors. The product is the LCM of the denominators.
Step 5. The LCM of the denominators is the LCD of the fractions.

Examples

• To find the LCD for $\frac{5}{8}$ and $\frac{7}{12}$, find the LCM of the denominators 8 and 12. Factoring gives $8 = 2 \cdot 2 \cdot 2$ and $12 = 2 \cdot 2 \cdot 3$. The LCM is $2 \cdot 2 \cdot 2 \cdot 3 = 24$. The LCD is 24.

Property

Convert two fractions to equivalent fractions with their LCD as the common denominator.
Step 1. Find the LCD.
Step 2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
Step 3. Use the Equivalent Fractions Property to multiply both the numerator and denominator by the number you found in Step 2.
Step 4. Simplify the numerator and denominator.

Examples

• Convert $\frac{1}{4}$ and $\frac{1}{6}$ to use their LCD, 12. For $\frac{1}{4}$, multiply by $\frac{3}{3}$ to get $\frac{3}{12}$. For $\frac{1}{6}$, multiply by $\frac{2}{2}$ to get $\frac{2}{12}$.

• Convert $\frac{3}{10}$ and $\frac{5}{18}$ to use their LCD, 90. For $\frac{3}{10}$, multiply by $\frac{9}{9}$ to get $\frac{27}{90}$. For $\frac{5}{18}$, multiply by $\frac{5}{5}$ to get $\frac{25}{90}$.

Property

Add or subtract fractions with different denominators.
Step 1. Find the LCD.
Step 2. Convert each fraction to an equivalent form with the LCD as the denominator.
Step 3. Add or subtract the fractions.
Step 4. Write the result in simplified form.

Examples

• To add $\frac{2}{5} + \frac{3}{4}$, the LCD is 20. The problem becomes $\frac{2 \cdot 4}{5 \cdot 4} + \frac{3 \cdot 5}{4 \cdot 5} = \frac{8}{20} + \frac{15}{20} = \frac{23}{20}$.

• To subtract $\frac{5}{6} - \frac{7}{8}$, the LCD is 24. This becomes $\frac{5 \cdot 4}{6 \cdot 4} - \frac{7 \cdot 3}{8 \cdot 3} = \frac{20}{24} - \frac{21}{24} = -\frac{1}{24}$.

Property

If $a$, $b$, $c$ are whole numbers where $b \neq 0$, $c \neq 0$, then

Examples

• To give $\frac{3}{4}$ a new denominator of 12, you must multiply the denominator by 3. The property says you must also multiply the numerator by 3: $\frac{3 \cdot 3}{4 \cdot 3} = \frac{9}{12}$.

• The fractions $\frac{5}{6}$ and $\frac{20}{24}$ are equivalent because you can multiply the numerator and denominator of $\frac{5}{6}$ by 4: $\frac{5 \cdot 4}{6 \cdot 4} = \frac{20}{24}$.

Property

Fraction multiplication: Multiply the numerators and multiply the denominators.

Fraction division: Multiply the first fraction by the reciprocal of the second.

Fraction addition/subtraction: Convert to equivalent forms with the LCD, then add/subtract the numerators.

Examples

• Subtraction needs an LCD: $\frac{7y}{12} - \frac{5}{18}$ becomes $\frac{21y}{36} - \frac{10}{36} = \frac{21y - 10}{36}$.

• Multiplication does not need an LCD: $\frac{5x}{6} \cdot \frac{3}{10} = \frac{15x}{60}$, which simplifies to $\frac{x}{4}$ by removing common factors.

Property

Simplify complex fractions.
Step 1. Simplify the numerator.
Step 2. Simplify the denominator.
Step 3. Divide the numerator by the denominator.
Step 4. Simplify if possible.

Examples

• To simplify $\frac{\left(\frac{2}{5}\right)^2}{6+2^2}$, first simplify the top to $\frac{4}{25}$ and the bottom to $10$. The problem is now $\frac{4}{25} \div 10$, which equals $\frac{4}{25} \cdot \frac{1}{10} = \frac{4}{250} = \frac{2}{125}$.

• To simplify $\frac{2+\frac{3}{4}}{\frac{5}{6}-\frac{1}{3}}$, the numerator becomes $\frac{11}{4}$ and the denominator becomes $\frac{1}{2}$. The problem is now $\frac{11}{4} \div \frac{1}{2} = \frac{11}{4} \cdot \frac{2}{1} = \frac{11}{2}$.