You can't directly add or subtract fractions with different denominators. Think of it like trying to add apples and oranges. If you have 1/2 of a pizza and your friend has 1/3 of another, you can't just add the tops and bottoms to say you have 2/5 of a pizza. The pieces are different sizes.

To perform the operation, you first need to find a common language. In fractions, this means finding a common denominator. This process involves converting the fractions into equivalent forms that have the same bottom number, so all the 'pieces' are the same size.

The most precise way to find a common denominator is by identifying the Least Common Multiple (LCM) of the two denominators. The LCM is the smallest positive number that is a multiple of both numbers. Using the LCM keeps the numbers you're working with as small as possible, which simplifies the calculation later.

Let's try adding $1/4 + 5/6$. The denominators are 4 and 6. The LCM of 4 and 6 is 12. Now, we convert each fraction into an equivalent fraction with a denominator of 12.

For $1/4$, we ask: what do we multiply 4 by to get 12? The answer is 3. So we multiply both the numerator and the denominator by 3. This gives us $3/12$.

For $5/6$, we do the same. To get 12, we multiply 6 by 2. So we multiply the top and bottom by 2. This gives us $10/12$.

Now that the denominators are the same, we can add the numerators.

Our answer is $13/12$, which is an improper fraction. It can also be written as the mixed number $1 \, 1/12$.

Finding the LCM is solid, but it can sometimes feel slow. A faster, more visual trick is the butterfly method. It's a form of cross-multiplication that works every time, though it might leave you with a larger fraction to simplify at the end.

Here's how it works for adding $a/b + c/d$:

1. Multiply the numerator of the first fraction by the denominator of the second ($a \times d$).
2. Multiply the numerator of the second fraction by the denominator of the first ($b \times c$).
3. Add those two products together to get the new numerator.
4. Multiply the two denominators ($b \times d$) to get the new common denominator.

Let's use our example, $1/4 + 5/6$, with this method.

1. Multiply $1 \times 6 = 6$.
2. Multiply $4 \times 5 = 20$.
3. Add them for the numerator: $6 + 20 = 26$.
4. Multiply denominators: $4 \times 6 = 24$.

The result is $26/24$. This is correct, but it needs to be simplified. Both numbers are divisible by 2, so it simplifies to $13/12$, the same answer we got before.

The great news is that subtraction works exactly the same way. You find a common denominator using either the LCM or butterfly method, convert the fractions, and then subtract the numerators.

Let's try subtracting $2/3 - 1/5$. The denominators, 3 and 5, are coprime, meaning their only common positive factor is 1. In cases like this, the LCM is simply their product: $3 \times 5 = 15$.

Before you test your skills, let's review the main ideas.

Ready to try a few problems?

Mastering these methods gives you the power to combine any fractions, no matter how different they seem at first. It's all about finding that common ground.