Fractions: How to Divide, Multiply, Add & Subtract a Fraction

Fractions are an important part of the math learning journey and, really, an important lesson in life. Sometimes, things aren’t as easy as we’d like; sometimes, we have to put pieces together and work with what we’ve got.

The moral of the story: Math is like real life — because math is real life! — and real life is not always nice, whole numbers.

But you know what? You can handle it. And we’re here to show you how.

Let’s get started!

What is a fraction?

A fraction is a mathematical quantity that is not a whole number.

Even though fractions might not be equal to one whole, fractions do tell you how many parts of a whole you have. For example, if you have $$3$$ pieces left of an $$8$$-slice pizza, you have $$\frac{3}{8}$$ of the pizza.

However, you can also represent the whole number $$1$$ as fraction by putting any number over itself, like $$\frac{5}{5}$$. Think of it this way: If you have $$5$$ pieces of a $$5$$-slice apple pie, you have one whole apple pie.

You can also represent a whole number as a fraction by placing that number over 1, like $$\frac{5}{1}$$ (remember that for later!).

So, in very specific instances, a fraction can represent a whole. Generally speaking, however, a fraction shows you a quantity that is less (or sometimes more) than one whole.

Fraction form

How do you know you’re working with a fraction? You can tell by its form!

For example, this is a whole number (an “integer”): $$3$$

But this is a fraction: $$\frac{1}{3}$$

You could read that fraction as either “one over three” or “one third.”

Before we move on, let’s take a closer look at each piece of a fraction’s form:

Parts of a fraction

A typical fraction will have three parts: a top, a bottom, and a dividing line. Usually you’ll see it stacked, like this:

That horizontal line in the middle is often referred to as the “fraction bar.”

Now, as you might guess, we don’t just refer to the numbers as the “top number” or “bottom number.” So, you may find yourself asking:

What is the top number of a fraction called?

The top number of a fraction — that is, the number above the line — is called the “numerator.”

For example, in the fraction $$\frac{2}{3}$$, the numerator is $$2$$.

The numerator tells us how many parts we have.

What is the bottom number of a fraction called?

The bottom number of a fraction, beneath the fraction bar, is called the “denominator.”

In the example $$\frac{2}{3}$$, our denominator is $$3$$.

The denominator tells us how many parts we’d need to make a whole.

Solving problems with fractions

So, we know how to recognize and reference fractions now — but how do we actually work with them? That all depends on your operator!

When you do solve, you’ll want to make sure that your result is simplified, or reduced down to its simplest terms. For example, if your result is $$\frac{4}{6}$$, you’d notice that both the numerator ($$4$$) and denominator ($$6$$) are divisible by $$2$$. So, you’d divide each by $$2$$ to get the fraction’s simplest form: $$\frac{2}{3}$$.

But how do we get to our result? Read on for our how-to guides!

How to: Divide fractions

If one operation with fractions is going to confuse you, it’s probably division. So don’t feel like you’re alone! Luckily, we’re here to explain how to navigate it with confidence.

You can divide a fraction by a whole number, a whole number by a fraction, or even a fraction by a fraction!

When dividing with fractions, there’s a very important term to remember: reciprocal.

The reciprocal is a fraction that’s turned upside-down — in other words, we swap the numerator and denominator. For example, the reciprocal of $$\frac{2}{3}$$ is $$\frac{3}{2}$$.

Let’s start with dividing a whole number by a fraction, like this:

Working through the problem mathematically is actually simpler than you think! Here’s how we do it:

1. First, we actually have to turn our whole number into a fraction. Remember when we said any number over $$1$$ equals itself? That’s why we can turn $$2$$ into $$\frac{2}{1}$$. So, our equation will actually read $$\frac{2}{1} ÷ \frac{1}{4}$$
2. Here’s the thing: we don’t actually divide by a fraction. We multiply by its reciprocal! Remember: a reciprocal is just the upside-down version of the fraction. That’s why our next step looks like $$\frac{2}{1} \times \frac{4}{1}$$
3. Then we just multiply across, so multiply the numerators in your top row and the denominators in your bottom row.
4. If you can simplify the result of your multiplication, reduce it to its simplest form! In our case, our result is $$\frac{8}{1}$$, which is actually just $$8$$.

A fraction divided by another fraction

You may have noticed that, even when whole numbers are involved, we end up dividing two fractions. That’s because we need to have like terms in order to perform the operation correctly!

So, dividing a fraction by a fraction is actually one step less than dividing a whole number by a fraction!

Take the example $$\frac{3}{8} ÷ \frac{2}{3}$$

1. Both our terms are fractions, so we already have like terms! Don’t you love it when the first step is a freebie?
2. Remember that we don’t actually divide by a fraction — we multiply by its reciprocal! That means our first real step is to transform the expression into $$\frac{3}{8} \times \frac{3}{2}$$
3. The next part is simple: Just multiply across the top and bottom. So, we’ll multiply the numerators and, separately, multiply the denominators to get $$\frac{9}{16}$$
4. You won’t always be able to reduce your result down to simpler terms, but if you can, you would do that now. In our case, $$\frac{9}{16}$$ is already as simple as it can get, so we’re done!

How to: Multiply fractions

Multiplication is one of the easiest operations you can perform with fractions!

All you have to do is stay in your lane — meaning, multiply the top row and bottom row separately.

What do we mean by that?

Let’s say we want to multiply $$\frac{2}{5} \times \frac{3}{4}$$. We would:

1. Carry over the fraction bar
2. Multiply the numerators above it $$2 \times 3$$
3. Multiply the denominators below it  $$5 \times 4$$
4. Reduce the result if you can!  $$\frac{6}{20}$$ can be reduced to $$\frac{3}{10}$$

How to: Subtract fractions

In order to subtract fractions, you need to make sure both your fractions have the same denominator (called the “least common denominator,” or LCD for short).

Once you have your denominators figured out, it’s really simple! Still need more? We can show you all the details of subtracting fractions.

How to: Addition of fractions

Adding fractions also requires matching denominators, so you may have to find the LCD before you can begin your operation.

Need help? Let’s add fractions together.

Fraction examples

It’s one thing to learn about fractions from a textbook. It’s another to actually put them into context!

Luckily, we’ve compiled a few examples of fractions in real life — and what better topic is there than food?

• You want to double your favorite pasta sauce recipe (because who wouldn’t?). The original recipe calls for $$\frac{1}{2}$$ cup of crushed tomatoes. To double the recipe, you’ll need to add fractions: $$\frac{1}{2}+\frac{1}{2}$$
• You’ve made smoothies for a pool party. Your batch is $$2$$ cups of liquid, and you need to divide it evenly into glasses that hold $$\frac{1}{4}$$ cup. If you want to know how many glasses you’ll get, you would solve $$2 ÷ \frac{1}{4}$$
• You just had a pizza feast and you have $$3$$ boxes of pizza left. Each box has $$\frac{1}{3}$$ of a pizza remaining. To find out how many whole pizzas you have left, you’d multiply $$3 \times \frac{1}{3}$$.

Fraction practice problems

The best way to master a skill is practice, so here are some problems you can work through in a safe space:

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

If you need help along the way, scan the problem with your Photomath app and we’ll put you on the right path!

Here’s how we solve the first practice problem in the app:

FAQ
What are the 3 types of fraction?

Fractions can be proper, improper, or part of a mixed number. A proper fraction has a lower number in the numerator than the denominator, like $$\frac{2}{3}$$. An improper fraction has a higher numerator than denominator, as in $$\frac{6}{2}$$.

How do you solve and simplify fractions?

Once you’ve solved an equation with fractions, you can simplify your result if the numerator and denominator are both divisible by the same number.

How do you solve fractions with mixed numbers?

Solving an equation with mixed numbers depends on the operation, so scan the problem with your Photomath app so we can explain in detail!