Adding fractions

After you’ve mastered operations with integers (because we know you did!), the next step is to master operations with fractions. It’s natural to start with addition like we did with integers, so let’s talk about how to add fractions!

As we already know, a fraction represents a part of a whole, so we need to figure out how to add two (or more) parts of a whole!

What does it mean to add fractions?

Adding fractions means finding the sum of at least two fractions. But before we can add fractions, we need to make sure you’re familiar with the least common denominator — trust us, that LCD will be your best friend when working with fractions!

The least common denominator (commonly known as “LCD”) is the least common multiple (”LCM”) of all the denominators. That means it’s the lowest number you can possibly find that’s still a multiple of your numbers. For example, if your denominators are $$2$$ and $$3$$, the LCD is $$6$$, because $$6$$ is the least common multiple of those two numbers. Is that making sense?

Why is adding fractions so useful?

Working with integers is great, but unfortunately we can’t always work with nice whole numbers! For example, let’s say we need to divide integers that are not easily divisible, like $$8 ÷ 3$$. There’s a need for another set of numbers: rational numbers.

Rational numbers — a.k.a. fractions — have so many real-life applications! Here’s an important example: When baking cookies, you want to double the batch (because why not?). If the original recipe calls for $$\frac14$$ cup of sugar, you’ll need to add an additional $$\frac14$$ cup of sugar to double the recipe.

We certainly don’t want to hold you back from having more cookies in your future, so let’s learn how to add fractions!

How to add fractions

We know why adding fractions is important (hint: the answer is always cookies), so now it’s time to see it in action!

Let’s walk through a problem together.

Add the fractions:

Hmmm, the fractions don’t have the same denominator, so we can’t add just yet. First, we have to determine the least common denominator(LCD). To find the LCD, let’s start by copying the denominators:

$$4,5$$

Then, write the prime factorizations of the numbers:

$$\begin{gathered}4=2\times 2, &&
5=5 \end{gathered}$$

Notice that $$4$$ and $$5$$ don’t have common prime factors. That means the LCD will be the product of both numbers’ prime factors!

$${2\times 2 \times 5}=20$$

So, our least common denominator is $$20$$. Nice!

Now that we have our LCD, let’s use it! However, we can’t just change the denominators, because then we’d just have two completely new fractions. To keep our fractions equal to the originals, we have to multiply each numerator by whatever we use in the denominator to get $$20$$:

$$\frac{5\times5}{4\times5}+\frac{3\times4}{5\times4}$$

Multiply the numbers in the numerators and denominators to get our new equivalent fractions:

$$\frac{25}{20}+\frac{12}{20}$$

Now that we have fractions with equal denominators, write the numerators above their common denominator and carry over the plus sign:

$$\frac{25+12}{20}$$

Finally, we just add the numbers in the numerator:

Since the fraction cannot be simplified, this is our final result!

Add the fractions:

Both fractions have the same denominator, $$5$$. Score! We’ll just write the numerators above their common denominator:

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

Add the numbers in the numerator:

$$\frac{5}{5}$$

Oh hey, we can simplify! We know $$5$$ divided by $$5$$ is $$1$$, so let’s simplify the fraction:

That wasn’t so bad, right? Now that we’ve walked through two examples, let’s review the overall process that you can use it for any problem:

Do it yourself!

You might not think practicing math is the most glamorous way to spend your time, but the repetition really helps cement the method in your mind. So, when you’re ready, we’ve got some practice problems for you!

Add the fractions:

1. $$\frac{1}{7}+\frac{4}{7}$$
2. $$\frac{1}{5}+\frac{7}{15}$$
3. $$\frac{1}{3}+\frac{2}{7}$$
4. $$\frac{9}{2}+\frac{1}{3}+\frac{7}{12}$$

Solutions:

1. $$\frac{5}{7}$$
2. $$\frac{2}{3}$$
3. $$\frac{13}{21}$$
4. $$\frac{65}{12}$$

If you’re still struggling through the solving process, that’s totally okay! Stumbling a few times is actually good for learning. If you get too stuck or lost, scan the problem using your Photomath app and we’ll walk you through to the other side!

Here’s a sneak peek of what you’ll see: