Simplifying expressions with fractions

Calculating the expression with integers $$2+3\times 4-12÷4$$ is a piece of cake (thanks, PEMDAS!). But, as we’ve learned, we can’t always work with integers. Sometimes, we need to throw fractions into the mix — but that’s no reason to panic! If you can perform arithmetic operations and PEMDAS with integers, you can do it with fractions, too!

What does it mean to simplify expressions with fractions?

Simplifying expressions with fractions means using PEMDAS and basic arithmetic operations to make an expression with fractions shorter and less complicated. Who wouldn’t want that?

Let’s not gloss over PEMDAS — one of our favorite math pneumonics. In case you forgot, PEMDAS is an acronym for the order of operations: P for Parentheses, E for Exponentiation, M for Multiplication, D for Division, A for Addition, and S for Subtraction. But, remember: Multiplication and division have the same significance, so it doesn’t really matter which of those two you do first. The same goes for addition and subtraction.

Why is simplifying expressions with fractions so useful?

Now that you’re learning how to work with fractions, you’ll start seeing them pop up in all kinds of equations and expressions — and sometimes, those expressions are too long or too complicated. It’s always easier to calculate a smaller expression or simplified fraction, so we’re really learning how to make things easier for ourselves!

But that’s not all: Simplifying expressions with fractions can help you in real life! For example, you might have $$\frac{9}{10}$$ of a basket full of apples, and then a friend takes $$\frac{2}{3}$$ of apples from the basket; after a few days, your friend gives you back twice the amount that they took! How full is the basket? This can be solved by simplifying the expression:

How to simplify expressions with fractions

Okay, it’s time to get into the process! Let’s go through some examples together so we can show you how to simplify expressions with fractions.

Simplify the expression:

Cancel out the greatest common factor $$3$$:

$$\frac{1}{6}-\frac{1}{20}\times \frac{5}{3}$$

Cancel out the greatest common factor $$5$$:

$$\frac{1}{6}-\frac{1}{4}\times \frac{1}{3}$$

According to PEMDAS, we need to multiply first, so multiply the fractions by multiplying the numerators and denominators:

$$\frac{1}{6}-\frac{1\times 1}{4\times 3}$$

Complete your multiplication in the numerator and denominator, respectively:

$$\frac{1}{6}-\frac{1}{12}$$

Well, our fractions have different denominators, so we need to find the least common denominator. And we’re in luck: the denominator $$12$$ is a multiple of the second denominator $$6$$, so we’ll rewrite the first fraction as an equivalent fraction with the denominator $$12$$:

$$\frac{2}{12}-\frac{1}{12}$$

Write the numerators above their common denominator $$12$$, carrying over the subtraction sign:

$$\frac{2-1}{12}$$

Subtract the numbers in the numerator:

That’s it! Let’s look at another one.

Simplify the expression:

According to our good friend PEMDAS, we know we need to add the fractions in the parentheses first. The denominators of the fractions in the parentheses are different, so let’s find the least common denominator. Since the prime factorizations of the denominators are $$16=2\times 2\times 2\times 2$$ and $$10=2\times 5$$, the LCD is the product $$2\times 2\times 2\times 2\times 5=80$$. So, we’ll write the fractions as equivalent fractions with the denominator $$80$$:

$$\frac{1}{10}-\left(\frac{75}{80}+\frac{24}{80}\right)÷\frac{1}{2}$$

Write the numerators above their LCD, $$80$$, including the operator:

$$\frac{1}{10}-\left(\frac{75+24}{80}\right)÷\frac{1}{2}$$

Add the numbers in the numerator:

$$\frac{1}{10}-\left(\frac{99}{80}\right)÷\frac{1}{2}$$

Now that it’s just one fraction, we can remove the parentheses:

$$\frac{1}{10}-\frac{99}{80}÷\frac{1}{2}$$

Next in line is division! To divide by a fraction, we multiply by the reciprocal of that fraction:

$$\frac{1}{10}-\frac{99}{80}\times \frac{2}{1}$$

Cancel out the greatest common factor $$2$$:

$$\frac{1}{10}-\frac{99}{40}$$

Our last operation is subtraction! Unfortunately, our fractions don’t have the same denominator, so we need to find the LCD of $$10$$ and $$40$$. Luckily, the number $$40$$ is a multiple of $$10$$, so our least common denominator is $$40$$. That means we just have to write the first fraction as an equivalent fraction with the denominator $$40$$:

$$\frac{4}{40}-\frac{99}{40}$$

Now that we have fractions with the same denominator, write the numerators and the operator above the same denominator:

$$\frac{4-99}{40}$$

Subtract the numbers in the numerator:

$$\frac{-95}{40}$$

Factor out the negative sign:

$$-\frac{95}{40}$$

We’re almost there! This fraction can be reduced by $$5$$, so let’s reduce:

$$-\frac{19}{8}$$

Our fraction can’t be simplified any further, so that’s our result!

See? Working with fractions isn’t so scary after all.

To summarize, you can simplify expressions with fractions by following this procedure:

Do it yourself!

Yes, we know you have homework, but wouldn’t it be nice to do some practice problems in a safe, un-graded space? Try these and see how you do:

Simplify the expressions:

1. $$\frac{1}{5}\times \frac{10}{7}+\frac{8}{21}$$
2. $$\frac{3}{4}÷\frac{8}{9}+(\frac{1}{6}-\frac{2}{3})$$
3. $$\frac{3}{5}\times \frac{1}{4}\times \frac{6}{10}÷\frac{5}{7}$$
4. $$\frac{9}{2}\times \frac{1}{3}+\frac{7}{12}÷\frac{1}{2}$$

Solutions:

1. $$\frac{2}{3}$$
2. $$\frac{11}{32}$$
3. $$\frac{63}{500}$$
4. $$\frac{8}{3}$$

If you’re having some trouble, that’s okay! Those roadblocks can actually help you remember the process better once you break through. 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: