Adding and Subtracting Fractions

Adding and Subtracting Fractions 🍰

A fraction has two parts:

• The numerator is the top number — it tells you how many pieces you have.
• The denominator is the bottom number — it tells you what size the pieces are.

When two fractions have the same denominator, the pieces are all the same size, so you can just add or subtract the tops:

$\frac{2}{8} + \frac{3}{8} = \frac{5}{8}$

But what about $\frac{1}{2} + \frac{1}{3}$? The pieces are different sizes — halves and thirds. You can't just add $1 + 1$, because a half and a third are not the same thing!

Unlike denominators means the bottom numbers are different. Before we can add or subtract, we have to make the pieces match.

The Big Idea: A Common Denominator 🔑

To add or subtract fractions with unlike denominators, we rewrite them so they share the same denominator. This shared bottom number is called a common denominator.

The trick is to use equivalent fractions — fractions that look different but have the same value. If you multiply the top and bottom by the same number, the value does not change:

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

Now both fractions are written in sixths, so the pieces are the same size and we can add them:

$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

A quick way to find a common denominator is to multiply the two denominators together. For $\frac{1}{2}$ and $\frac{1}{3}$, that is $2 \times 3 = 6$.

The 4 Steps ✏️

Here is the full recipe for adding or subtracting fractions with unlike denominators:

Remember: you only add or subtract the top numbers. The common denominator stays the same — you never add the bottoms together!

Quick Check ✅

Let's make sure the main idea is clear.

Worked Example: Adding 📝

Solve $\frac{1}{3} + \frac{1}{4}$

• Step 1 — Common denominator: multiply the bottoms, $3 \times 4 = 12$.
• Step 2 — Equivalent fractions:
  • $\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$
  • $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
• Step 3 — Add the tops: $4 + 3 = 7$, keep the denominator $12$.
• Step 4 — Simplify: $\frac{7}{12}$ is already in lowest terms.

$\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$

Worked Example: Subtracting ➖

Solve $\frac{3}{4} - \frac{1}{6}$

• Step 1 — Common denominator: $4 \times 6 = 24$.
• Step 2 — Equivalent fractions:
  • $\frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24}$
  • $\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}$
• Step 3 — Subtract the tops: $18 - 4 = 14$, keep the denominator $24$.
• Step 4 — Simplify: both $14$ and $24$ divide by $2$, so $\frac{14}{24} = \frac{7}{12}$.

$\frac{3}{4} - \frac{1}{6} = \frac{18}{24} - \frac{4}{24} = \frac{14}{24} = \frac{7}{12}$

Tip: Subtraction uses the exact same steps as addition — just subtract the numerators instead of adding them.

Your Turn — Fill in the Blanks ✍️

Solve each one and type your answer as a fraction (for example: 7/12). Find a common denominator first!

1. $\frac{1}{2} + \frac{1}{4} =$ ______

2. $\frac{2}{3} - \frac{1}{6} =$ ______ (simplify your answer)

Guided Practice — Choose the Answer 🎯

Find a common denominator, then add or subtract the tops.

Build the Answer 🧩

You are solving $\frac{1}{4} + \frac{2}{3}$. Choose the correct common denominator, then choose the final answer.

Fractions in the Real World 🌎

Adding and subtracting fractions with unlike denominators happens all the time.

🥣 Baking

A recipe needs $\frac{1}{2}$ cup of sugar and $\frac{1}{4}$ cup of brown sugar. How much sugar in all?

$\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \text{ cup}$

🍕 Sharing Pizza

You had $\frac{3}{4}$ of a pizza and ate $\frac{1}{3}$ of the whole pizza. How much is left?

$\frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \text{ of the pizza}$

The key move every time: make the denominators match first, then add or subtract the tops.

Word Problem Practice ✍️

Read each story, then type your answer as a fraction (for example: 5/6). Simplify when you can.

1. Mia walked $\frac{1}{2}$ mile in the morning and $\frac{1}{3}$ mile after lunch. How far did she walk in all?

2. A water bottle was $\frac{7}{8}$ full. Leo drank $\frac{1}{4}$ of the bottle. How much is left?

3. Sam read $\frac{2}{3}$ of a book and his sister read $\frac{1}{6}$ of it. How much did they read together?

One More Story 📖

Review: What You Learned 🌟

You can now add and subtract fractions even when the denominators are different. Here is the whole idea in one table:

Golden rules to remember:

• The denominator tells the size of the pieces — make them match before adding or subtracting.
• Only the numerators get added or subtracted; the common denominator stays the same.
• Always check whether your answer can be simplified by dividing the top and bottom by the same number.

Challenge — Mix It All Together 🏆

These problems combine common denominators, subtracting, and simplifying.