When we talk about fractions, the denominator is the bottom number that tells us how many parts the whole is divided into. When the denominators are the same, we can easily add or subtract the fractions because we are working with the same sized parts. But when the denominators are different, we have to find a way to make them the same before we can combine the fractions.
Like Denominators
Example: Imagine you have \( \frac{1}{4} \) of a pizza and \( \frac{2}{4} \) of another pizza. Since both pizzas are cut into 4 equal slices, you can simply add them together:
\[ \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \]
Now you know you have \( \frac{3}{4} \) of a pizza!
Unlike Denominators
Example: Now, think about a different situation where you have \( \frac{1}{4} \) of a pizza and \( \frac{1}{2} \) of another pizza. Here, the fractions have different denominators (4 and 2), so they are made of different-sized pieces. To add them, you need to make the pieces the same size.
To do that, you can convert \( \frac{1}{2} \) (which is the same as \( \frac{2}{4} \) because \( 2 \) is half of \( 4 \)):
\[ \frac{1}{2} = \frac{2}{4} \]
Now you can add them:
\[ \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \]
Why We Can't Add Unlike Denominators Directly
If we tried to add \( \frac{1}{4} \) and \( \frac{1}{2} \) without converting, we would be mixing slices of different sizes—it wouldn’t make sense to add \( 1 \) piece of \( 4 \) equal parts with \( 1 \) piece of \( 2 \) equal parts. Just like if someone gave you \( 1 \) apple (cut into \( 4 \) pieces) and \( 1 \) banana (cut into \( 2 \) pieces), you wouldn't know how to combine them without having them be the same type of piece first.
In summary, you can only add or subtract fractions when their denominators are the same (like having pieces of the same pizza). When they are different (like slicing apples and bananas differently), you need to change them to have the same denominator first before combining!
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