4.4 Add and Subtract Fractions with Common Denominators

Adding and Subtracting Fractions with Common Denominators

When two fractions share the same denominator, adding or subtracting them is straightforward: you only need to work with the numerators. This skill comes up constantly in later math, so it's worth getting comfortable with the process now.

Addition of Fractions with Common Denominators

The rule is simple: add the numerators and keep the denominator the same.

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

Here, $a$ and $b$ are the numerators, and $c$ is the common denominator.

Example: $\frac{3}{8} + \frac{1}{8} = \frac{3+1}{8} = \frac{4}{8}$

You're adding 3 eighth-sized pieces to 1 eighth-sized piece, which gives you 4 eighth-sized pieces. The size of each piece (eighths) doesn't change.

Visual models make this click. Picture two pie charts, each divided into 6 equal slices. One has 2 slices shaded, the other has 4 shaded. Combine them and you get 6 out of 6 slices shaded: $\frac{2}{6} + \frac{4}{6} = \frac{6}{6} = 1$.

Subtraction of Fractions with Common Denominators

Same idea, but subtract the numerators instead.

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

Example: $\frac{7}{10} - \frac{3}{10} = \frac{7-3}{10} = \frac{4}{10}$

You start with 7 tenth-sized pieces and take away 3 of them, leaving 4 tenths.

With a bar model: imagine a bar split into 5 equal parts with 4 shaded. Remove 2 shaded parts, and you're left with $\frac{2}{5}$.

Steps for Solving

1. Check that both fractions have the same denominator.
2. Add or subtract the numerators (top numbers).
3. Write the result over the common denominator (bottom number stays the same).
4. Simplify the fraction if possible.

Word Problems with Fraction Operations

When you see a word problem involving fractions, follow these steps:

1. Identify the fractions in the problem and confirm they share a common denominator.
2. Determine the operation. Words like "combined," "total," or "altogether" point to addition. Words like "difference," "remaining," or "less" point to subtraction.
3. Apply the operation and simplify your result.
4. Answer in context. Don't just write a fraction; connect it back to what the problem asked.

Example: "John ate $\frac{2}{6}$ of a pizza, and Mary ate $\frac{1}{6}$ of the same pizza. What fraction of the pizza did they eat together?"

$\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$

They ate $\frac{1}{2}$ of the pizza together.

Why Common Denominators Matter

The denominator tells you the size of each piece. You can only add or subtract pieces that are the same size.

Think of it this way: $\frac{1}{4}$ and $\frac{1}{5}$ represent different-sized pieces. Combining them directly doesn't work because a quarter and a fifth aren't the same amount. You'd first need to rewrite both fractions with a common denominator so the pieces match.

For example, to subtract $\frac{3}{4}$ from $\frac{5}{6}$, you'd find a common denominator of 12 and rewrite: $\frac{5}{6} = \frac{10}{12}$ and $\frac{3}{4} = \frac{9}{12}$. Then subtract: $\frac{10}{12} - \frac{9}{12} = \frac{1}{12}$.

The least common denominator (LCD) is the smallest number that works as a common denominator for the fractions you're working with. Using the LCD keeps your numbers smaller and simplifying easier.

Types of Fractions and Related Concepts

• Improper fractions have a numerator greater than or equal to the denominator (e.g., $\frac{5}{3}$). These represent a value of 1 or more.
• Mixed numbers combine a whole number with a proper fraction (e.g., $2\frac{1}{4}$). This is another way to write an improper fraction.
• A reciprocal is a fraction flipped upside down (e.g., the reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$). Reciprocals aren't used in addition or subtraction, but they'll show up when you get to dividing fractions.