4.5 Add and Subtract Fractions with Different Denominators

Working with Fractions

Adding and subtracting fractions with different denominators requires one key step before you can operate: getting the fractions to share the same denominator. This section covers how to find that common denominator, rewrite fractions, and then add or subtract them cleanly.

Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest number that all denominators in a problem divide into evenly. You need the LCD to rewrite fractions so they share the same denominator.

To find the LCD using prime factorization:

1. Break each denominator into its prime factors.
2. For each prime factor that appears, take the highest power of it from any denominator.
3. Multiply those together. That product is your LCD.

For example, suppose you need the LCD of fractions with denominators 8, 20, and 40:

• $8 = 2^3$
• $20 = 2^2 \times 5$
• $40 = 2^3 \times 5$

The highest power of 2 is $2^3$, and the highest power of 5 is $5^1$. So the LCD is $2^3 \times 5 = 40$.

For simpler problems (like denominators of 3 and 4), you can also just list multiples of each denominator until you find the smallest one they share. Multiples of 3: 3, 6, 9, 12. Multiples of 4: 4, 8, 12. The LCD is 12.

Equivalent Fractions with Common Denominators

Once you have the LCD, rewrite each fraction so its denominator equals the LCD. You do this by multiplying both the numerator and denominator by the same number (which is really just multiplying by 1, so the fraction's value doesn't change).

Take $\frac{3}{4}$ and $\frac{5}{6}$ with an LCD of 12:

• $\frac{3}{4} \times \frac{3}{3} = \frac{9}{12}$
• $\frac{5}{6} \times \frac{2}{2} = \frac{10}{12}$

To figure out what to multiply by, divide the LCD by the original denominator. For $\frac{3}{4}$: $12 \div 4 = 3$, so multiply top and bottom by 3.

Addition and Subtraction of Unlike Fractions

Here's the full process, step by step:

1. Find the LCD of all denominators.
2. Rewrite each fraction as an equivalent fraction with the LCD.
3. Add or subtract the numerators. The denominator stays the same.
4. Simplify the result if possible (reduce to lowest terms or convert to a mixed number).

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

1. LCD of 3 and 4 is 12.
2. $\frac{2}{3} \times \frac{4}{4} = \frac{8}{12}$ and $\frac{3}{4} \times \frac{3}{3} = \frac{9}{12}$
3. $\frac{8}{12} + \frac{9}{12} = \frac{17}{12}$
4. $\frac{17}{12} = 1\frac{5}{12}$ (already in lowest terms)

A common mistake is adding the denominators together. Don't do that. The denominator stays the same after you've converted; you only add or subtract the numerators.

Simplification of Complex Fractions

A complex fraction has a fraction in its numerator, its denominator, or both. Think of it as a big division problem: the top fraction divided by the bottom fraction.

To simplify one:

1. Simplify the numerator into a single fraction (find a common denominator if it contains addition or subtraction).
2. Simplify the denominator into a single fraction the same way.
3. Divide the resulting numerator fraction by the resulting denominator fraction (multiply by the reciprocal).

Example: $\frac{\frac{3}{4} - \frac{1}{6}}{\frac{2}{3} + \frac{1}{2}}$

1. Numerator: LCD of 4 and 6 is 12. $\frac{3}{4} - \frac{1}{6} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12}$

2. Denominator: LCD of 3 and 2 is 6. $\frac{2}{3} + \frac{1}{2} = \frac{4}{6} + \frac{3}{6} = \frac{7}{6}$

3. Divide: $\frac{7}{12} \div \frac{7}{6} = \frac{7}{12} \times \frac{6}{7} = \frac{42}{84} = \frac{1}{2}$

Problem-Solving with Fraction Operations

Word problems with fractions follow the same steps as any other fraction problem. The tricky part is translating the words into math.

1. Identify what you're given and what you need to find.
2. Set up the expression or equation (look for clue words: "total" suggests addition, "how much more" suggests subtraction).
3. Convert to a common denominator if the fractions are unlike.
4. Solve, then simplify.
5. Check that your answer makes sense in context.

Example: A recipe calls for $1\frac{1}{4}$ cups of flour and $\frac{3}{4}$ cup of sugar. How many cups of dry ingredients total?

• Convert the mixed number: $1\frac{1}{4} = \frac{5}{4}$
• The denominators already match, so add: $\frac{5}{4} + \frac{3}{4} = \frac{8}{4} = 2$ cups

Expressions with Fractions and Variables

When an expression contains variables, substitute the given values first, then simplify using the same fraction rules.

Example: Evaluate $\frac{3x}{4} - \frac{y}{6}$ when $x = \frac{2}{3}$ and $y = \frac{1}{2}$

1. Substitute: $\frac{3 \cdot \frac{2}{3}}{4} - \frac{\frac{1}{2}}{6}$

2. Simplify each piece: $3 \cdot \frac{2}{3} = 2$, so the first term becomes $\frac{2}{4} = \frac{1}{2}$. The second term: $\frac{1}{2} \div 6 = \frac{1}{12}$.

3. Find the LCD (2 and 12): LCD is 12. $\frac{1}{2} = \frac{6}{12}$

4. Subtract: $\frac{6}{12} - \frac{1}{12} = \frac{5}{12}$

Fraction Arithmetic and Simplification Techniques

A few useful tools to keep in your back pocket:

• Reciprocal: Flip the numerator and denominator. The reciprocal of $\frac{3}{5}$ is $\frac{5}{3}$. You use reciprocals whenever you divide fractions (multiply by the reciprocal instead).
• Divisibility rules: Quick checks to help you simplify. If both the numerator and denominator are even, you can divide by 2. If their digits sum to a multiple of 3, you can divide by 3. These save time when reducing fractions to lowest terms.
• Simplify early: Whenever possible, cancel common factors before you multiply. This keeps numbers small and reduces errors.