Subtracting Fractions with Common Denominators

TL;DR

When subtracting fractions with the same bottom number (denominator), you just subtract the top numbers (numerators) and keep the bottom number the same. Always simplify your answer to its smallest form if possible. Think of it like taking away pieces of the same-sized pizza.

1. The Mental Model

Imagine you have a pizza cut into 8 equal slices. If you start with 5 slices and eat 2, you're left with 3 slices. The size of the slices (the denominator) doesn't change, only the number of slices you have (the numerator).

2. The Core Material

Subtracting fractions with common denominators is straightforward because all your pieces are the same size. You don't need to worry about changing how the fractions look before you subtract.

What's a Common Denominator?

A common denominator just means the bottom number of your fractions is the same. For example, in 5/8 - 2/8, the common denominator is 8. This is important because it means you're comparing and subtracting parts of the same whole.

The Subtraction Rule

Here's the simple rule:
1. Keep the denominator: The bottom number stays exactly the same in your answer.
2. Subtract the numerators: Subtract the second top number from the first top number.
3. Simplify (if needed): Check if your final fraction can be reduced to a simpler form.

graph TD
    A["Start with fractions with common denominators"] --> B["Subtract the numerators"]
    B --> C["Keep the common denominator the same"]
    C --> D["Form the new fraction (new numerator / common denominator)"]
    D --> E{ "Can the fraction be simplified?" }
    E -- "Yes" --> F["Divide both numerator and denominator by their greatest common factor"]
    E -- "No" --> G["Done!"]
    F --> G

Example: 7/10 - 3/10

Here, both fractions have 10 as the denominator.
1. Keep the denominator: It'll be 10.
2. Subtract the numerators: 7 - 3 = 4.
3. The new fraction is 4/10.
4. Simplify: Both 4 and 10 can be divided by 2. So, 4 ÷ 2 = 2, and 10 ÷ 2 = 5.
The simplified answer is 2/5.

3. Worked Example

Let's subtract 5/6 - 1/6.

Step 1: Check for common denominators.
Both fractions have 6 as the denominator. Perfect!

Step 2: Subtract the numerators.
The numerators are 5 and 1.
5 - 1 = 4.

Step 3: Keep the common denominator.
The denominator remains 6.

Step 4: Form the new fraction.
The result is 4/6.

Step 5: Simplify the fraction.
Can 4/6 be simplified? Yes, both 4 and 6 can be divided by 2.
4 ÷ 2 = 2
6 ÷ 2 = 3
So, 4/6 simplifies to 2/3.

Final answer: 2/3.

4. Key Takeaways

• You can only directly subtract fractions when they have the same bottom number (common denominator).
• When subtracting, only the top numbers (numerators) change; the bottom number stays the same.
• Always look to simplify your final fraction by dividing both the numerator and denominator by their largest common factor.
• A common denominator means you're talking about pieces of the same size.
• Think of it as counting pieces of a whole, where the size of the pieces doesn't change.

Common Mistakes to Avoid

• Subtracting the denominators: Never subtract the bottom numbers. They define the size of the parts, not the quantity.
• Forgetting to simplify: Always reduce your fraction to its simplest form at the end.
• Trying to subtract without a common denominator: This topic is about when you do have one. If you don't, you need another step first (which we'll cover later!).
• Getting the order wrong: Always subtract the second numerator from the first.

5. Now Try It

Subtract 9/12 - 3/12. After you find your answer, make sure it's simplified as much as possible. Success looks like arriving at 1/2 as your final, simplified fraction.