加分母相同的分数

Add the numerators

In a fraction, the top number is the numerator and the bottom number is the denominator. Fractions are examples of rational numbers, alongside the whole numbers.

When two fractions have a common denominator, they have like denominators (the denominators are alike). In this case, to add the fractions together, all you have to do is add the numerators (the top numbers) together. Leave the bottom number the same!

You can visualize this by thinking of a pizza plate. The plate has space for $8$ slices of pizza, and each slice is $\frac{1}{8}$ of the pizza. Each slice of pizza is the same size.

Each new slice of pizza you put on the plate is like adding $\frac{1}{8}$. Notice that you're not changing the denominator—the plate can always hold up to $8$ slices, so nothing is changing about that.

$$\frac{2}{7} + \frac{3}{7} = \frac{5}{7}$$

Since the denominators are the same, we can find the answer by just looking at and adding the top numbers (the numerators)!

Definitions

A fraction is a part of a whole. Numbers such as $\frac{1}{2}$ and $-\frac{27}{521}$ are fractions. Some numbers are written as percents instead, but they can be turned into fractions. Finally, some numbers are written as mixed numbers, but these can also be turned into fractions.

The number on top of the fraction line is the numerator. The number below the line is the denominator. Two fractions with the same denominator, or a common denominator, have "like denominators".

Simple Rule

To add fractions with like denominators, keep the denominator and add the numerators.

Example

Compute $\frac{1}{5} + \frac{2}{5}$.

Wrong Approach

When you add two fractions with like denominators, do NOT follow the method below:

$$\begin{align} &\frac{1}{5} + \frac{2}{5} \\\ &= \frac{1+2}{5+5} \\\ &= \frac{3}{10} \end{align}$$

Correct Approach

Instead, you only add the numerators together. $$\begin{align} &\frac{1}{5} + \frac{2}{5} \\\ &= \frac{1+2}{5} \\\ &= \frac{3}{5} \end{align}$$

Let me say it again: this is for when the denominators are the same.

For a more visual approach, you can imagine you have $2$ people eating a pizza. The pizza has $5$ slices (because mathematicians like weird pizzas!). The 1st person only eats $1$ slice, while the $2$nd person eats $2$ slices. The total amount of pizza eaten would be the sum you are computing, as pictured below.

If the denominators are not the same, this technique won't work. We'd have to do some extra steps to make the denominators the same.

(Video) Adding and Subtracting Fractions

by PatrickJMT

This video by PatrickJMT covers adding and subtracting fractions with like denominators.

$\large{\text{Summary}}$

First, check if the denominators if your two fractions match. Matching denominators means that you can add or subtract the numerators. $$\begin{gather} \frac ac+\frac bc=\frac{a+b}c\\ \frac ac-\frac bc=\frac{a-b}c \end{gather}$$

He goes over this example. $$\frac4{11}-\frac3{11}$$ To solve, keep the same denominator but subtract numerators. $$\frac4{11}-\frac3{11}=\frac{4-3}{11}=\frac1{11}$$

When to Reduce

In some cases the final result can be simplified. This is called reducing the fraction. Here's a quick example of this in action.

Example: Let's simplify $\frac{5}{8} + \frac{1}{8}$.

Just like before, the denominators are the same. So, we add the numerators:

$$\frac{5}{8} + \frac{1}{8} = \frac{5 + 1}{8} = \frac{6}{8}$$

However, $6$ and $8$ share a common factor of $2$, so we pull it out and cancel:

$$\frac{6}{8} = \frac{2 \times 3}{2 \times 4} = \frac{\cancel{2} \times 3}{\cancel{2} \times 4} = \boxed{\frac{3}{4}}.$$

Subtract the numerators

When you have to subtract fractions that have the same denominator, you can subtract one numerator from the other. In this case keep the same denominator. For example, if we have $3$ pieces left of a pizza that originally had $8$ pieces, then there is $\frac{3}{8}$ of the pizza left.

One slice of pizza is $\frac{1}{8}$ of the pizza, and if you eat $1$ of $3$ slices ($3 - 1$), there will be $2$ slices left.

There'll be $\frac{2}{8}$ of the pizza left! If you simplify the fraction, you'll see that that's also $\frac{1}{4}$ of the pizza—but that's not part of the subtraction.

A note of caution!

Notice that even though we're subtracting the numerators, we aren't touching the denominator at all. That's because even if we take something out of or off a container (like the pizza plate), the plate doesn't change—fractions work the same way. Make sure not to change anything about the denominator!