Explore Fractions

# Subtracting fractions

We’ve mastered operations with integers, and you even know how to add fractions! But, of course, we can’t stop there.

Now it’s time to learn how to subtract fractions — and you might be more prepared than you think!

Let’s get started.

## What does it mean to subtract fractions?

Subtracting fractions means finding the difference of at least two fractions. Just like in adding fractions, we’ll need to use our old friend: the LCD.

The least common denominator, known by his friends as “LCD,” is the least common multiple (LCM) of all the denominators. So, for example, the LCD of numbers $$2$$ and $$3$$ is $$6$$, since $$6$$ is the least common multiple of those two numbers. Are you getting the hang of it?

### Why is subtracting fractions so useful?

Unfortunately, math isn’t all nice whole numbers. When we can’t use integers, we need another set of numbers: rational numbers. Rational numbers (otherwise known as fractions) are a necessity of life, so you’ll start noticing them all around you. For example, you’re baking a cake and measured out $$\frac{6}{7}$$ cups of flour. The recipe says you need $$\frac{3}{4}$$ cup. Do you have enough flour for the cake? (We sure hope so!) You can calculate this by subtracting fractions:

$$\frac{6}{7}-\frac{3}{4}$$

## How to subtract fractions

Now that we know why we need to be able to subtract fractions, let’s walk through some problems together.

### Example 1

Subtract the fractions:

$$\frac{3}{2} - \frac{1}{2}$$

Our fractions both have the same denominator, $$2$$ — nice! That means we can just write the numerators above that common denominator, including our operator:

$$\frac{3-1}{2}$$

Subtract the numbers in the numerator:

$$\frac{2}{2}$$

Oh, hey: $$2$$ divided by $$2$$ is $$1$$, so let’s simplify our result:

$$1$$

Awesome! Let’s try another example.

### Example 2

Subtract the fractions:

$$\frac{2}{5}-\frac{1}{6}$$

Oh no, our fractions don’t have the same denominator. Don’t worry, we can still subtract — we’ll just need to find the least common denominator (LCD) first! To find the LCD, write the denominators of both fractions:

$$5, ~6$$

Write the prime factorizations of the numbers:

$$\begin{gathered}5=5, && 6=2\times3 \end{gathered}$$

Hmmm, the numbers $$5$$ and $$6$$ don’t have any common factors. That means our least common denominator will be the product of each number’s prime factors:

$${2\times3\times5}=30$$

Okay, so our LCD is $$30$$. Remember from adding fractions that we can’t just change the denominator to $$30$$ — we have to update the numerators, too:

$$\frac{2\times6}{5\times6}-\frac{1\times5}{6\times5}$$

Multiply the numbers within the fractions to get our updated numerators and denominators:

$$\frac{12}{30}-\frac{5}{30}$$

Now that the fractions have equal denominators, we can write the numerators above a single denominator, remembering to include the subtraction sign:

$$\frac{12-5}{30}$$

All that’s left to do is subtract the numbers in the numerator:

$$\frac{7}{30}$$

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

Nice job! Based on those examples, let’s review the overall process so that you can solve more problems on your own:

## Study summary

1. Find the least common denominator (LCD), if needed.
2. Write all numerators above the common denominator.
3. Subtract the numbers in the numerator.
4. If possible, simplify the fraction.

## Do it yourself!

We know, we know — you don’t really want to do any more math than you have to. But if you want to get good at subtracting fractions, the best way to accomplish that is by practicing! Whenever you’re ready, try your hand at these problems:

Subtract the fractions:

1. $$\frac{1}{5} – \frac{1}{11}$$
2. $$\frac{5}{3} – \frac{2}{3}$$
3. $$\frac{2}{7} – \frac{3}{4}$$
4. $$\frac{11}{10} – \frac{1}{2}$$

Solutions:

1. $$\frac{6}{55}$$
2. $$1$$
3. $$-\frac{13}{28}$$
4. $$\frac{3}{5}$$

If you get stuck during the solving process, don’t get upset. (Plot twist: Mistakes actually help you learn!) Whenever you need help, you can just 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:

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