Mathematics » Introducing Fractions » Add and Subtract Fractions with Different Denominators

# Key Concepts

## Key Concepts

• Find the least common denominator (LCD) of two fractions.
1. Factor each denominator into its primes.
2. List the primes, matching primes in columns when possible.
3. Bring down the columns.
4. Multiply the factors. The product is the LCM of the denominators.
5. The LCM of the denominators is the LCD of the fractions.
• Equivalent Fractions Property
• If $$a,b$$, and $$c$$ are whole numbers where $$b\ne 0$$, $$c\ne 0$$ then

$$\frac{a}{b}=\frac{a\cdot c}{b\cdot c}$$ and $$\frac{a\cdot c}{b\cdot c}=\frac{a}{b}$$

• Convert two fractions to equivalent fractions with their LCD as the common denominator.
1. Find the LCD.
2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2.
4. Simplify the numerator and denominator.
• Add or subtract fractions with different denominators.
1. Find the LCD.
2. Convert each fraction to an equivalent form with the LCD as the denominator.
3. Add or subtract the fractions.
4. Write the result in simplified form.
• Summary of Fraction Operations
• Fraction multiplication: Multiply the numerators and multiply the denominators.

$$\cfrac{a}{b}\cdot \cfrac{c}{d}=\cfrac{ac}{bd}$$

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

$$\cfrac{a}{b}+\cfrac{c}{d}=\cfrac{a}{b}\cdot \cfrac{d}{c}$$

• Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

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

• Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

$$\cfrac{a}{c}-\cfrac{b}{c}=\cfrac{a-b}{c}$$

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

## Glossary

### least common denominator (LCD)

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

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