How To Math With Fractions - Education in Context

If I’ve learned anything during my time working as a math tutor (with students from 5^thgrade all the way through college), it’s this:

The #1 Big Bad Math Enemy that makes it hard to do math is fractions. Seriously. When I was working in the Math Center at the University of Utah, college students would come in all the time saying they needed help with Calculus — but upon further inspection the Calculus was fine. The problem was fractions.

I know, I know — we learned fractions clear back in elementary school and we feel silly struggling with them later. But the truth is, most students could use a refresher on how to math with fractions.

The Good News: Fractions Are Not As Scary As They Look

Secretly, fractions are just division. Read that again:

Secretly, fractions are just division.

What that means is that $\frac{4}{5}$ is secretly the exact same thing as $4\div5$.

Okay, cool, so fractions are just division. Great.

So then why would we use fractions instead of just dividing the numbers?

Well, there are a few good reasons:

If we divide the numbers in certain fractions, the result will be a really gross decimal which will:
- make us sad
- possibly lose precision due to rounding.
Sometimes keeping numbers as fractions rather than turning them into decimals is easier and faster even if the decimal isn’t that bad. Especially if you’re doing math in your head or on an exam that does not allow a calculator.
- For example, $\frac{4}{5}=0.80$, which is not an insane decimal, but there are cases where it will be easier to keep it as $\frac{4}{5}$.
Sometimes keeping numbers in fractions will allow us to cancel numbers out so we don’t have to do as much algebra or work with as big of numbers (always good).
We can use fractions for secret math tricks (like multiplying by one). This sounds ridiculous, but it’s probably the number one thing that sets budding mathematicians apart from struggling math students. Seriously. Go read my Secret Math Tricks post.
If at some point you decide to pursue becoming a serious mathematician, some silly tricks with fractions will be important to make your proofs work.

Before We Go Any Further: Some Quick Vocabulary

We have fancy words for the thing in the top of the fraction and the thing in the bottom of the fraction.

The numerator is the thing in the top of the fraction, and the denominator is the thing in the bottom of the fraction. (So in the fraction $\frac{4}{5}$, 4 is the numerator and 5 is the denominator.)

Fractions That Are Secretly Just One (1)

This is probably the most important concept in fractions. Luckily, it’s not too complicated.

Okay, so you know how $4\cdot1=4$? Meaning that $4\div4=1$?

Since fractions are secretly just division, $4\div4=1$ is secretly the exact same thing as $\frac{4}{4}=1$.

We can generalize this to say $\frac{x}{x}=1$, meaning that any fraction that has the exact same thing in the numerator as in the denominator is secretly just 1.

All Whole Numbers Are Secretly Fractions

See? Another reason fractions aren’t even that scary. The number 3 is secretly a fraction, and you don’t think the number 3 is scary!

This is another fractions concept that is important, and that students easily forget.

Any time that you’re working with fractions and you have a whole number, you’ll probably need to represent it as a fraction in order to do things like add it to a fraction or multiply it by a fraction. Luckily, this is very simple.

You can represent any whole number as a fraction by putting it over 1. For example, $3=\frac{3}{1}$.

We can generalize this as $x=\frac{x}{1}$, meaning that any whole number is equal to that same whole number over 1.

This makes sense if you think about it, because fractions are secretly just division, and dividing by 1 does nothing. So you can divide something by 1 any time you want — which is why you can put a whole number over 1 to make it into a fraction any time you want.

Multiplying Fractions

Multiplying fractions is simple and easy. Just multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction.

For example, $\frac{4}{5}\cdot\frac{2}{3}=\frac{8}{15}$, because $4\cdot2=8$ and $5\cdot3=15$.

Dividing Fractions

If you can multiply fractions, you can divide them. The trick is to flip the numerator and denominator of the second fraction, and then multiply.

So $\frac{4}{5}\div\frac{2}{3}=\frac{4}{5}\cdot\frac{3}{2}=\frac{12}{10}$

See? Not so tricky.

Adding and Subtracting With Fractions

Adding and subtracting with fractions is…. worse. But it’s not too scary. Let’s do it.

In order to add and subtract with fractions, we need to find the common denominator. This sounds scary, but it’s actually not that bad.

Finding The Common Denominator

Fractions that have a common denominator are just fractions that have the same thing in the bottom.

For example, $\frac{4}{5}$ and $\frac{3}{5}$ have a common denominator because they both have 5 in the bottom.

However, $\frac{4}{5}$ and $\frac{4}{7}$ do not have a common denominator because $\frac{4}{5}$ has a 5 in the bottom but $\frac{4}{7}$ has a 7 in the bottom (even though they both have a 4 in the top).

In order to add or subtract $\frac{4}{5}$ and $\frac{4}{7}$, we will need to find the common denominator.

Secretly, all we have to do to find the common denominator (or get the same thing in the bottom of both fractions) is multiply by one (which is always allowed because $4\cdot1=4$ (or, in general, $x\cdot1=x$) meaning that multiplying by 1 doesn’t technically change anything. You can think of it as changing the font — it looks different, but it says the exact same thing).

The only tricky part of finding a common denominator is deciding on the correct 1 by which to multiply…. this sounds ridiculous, but bear with me. Let’s look at an example and then talk about what it means:

An Example of Finding The Common Denominator

For $\frac{4}{5}$ and $\frac{4}{7}$, we are going to multiply $\frac{4}{5}$ by $\frac{7}{7}$ (which is secretly just 1) and $\frac{4}{7}$ by $\frac{5}{5}$ (which is also secretly just 1).

That will give us:

$\frac{4}{5}\cdot\frac{7}{7}=\frac{28}{35}$ and $\frac{4}{7}\cdot\frac{5}{5}=\frac{20}{35}$.

$\frac{28}{35}$ and $\frac{20}{35}$ have a common denominator (because they both have 35 in the bottom), so we can add them together.

Discussion of How To Choose The Correct 1 by Which to Multiply

As you can see in the example above, when we are trying to add two fractions together, we can choose which 1 to multiply by each fraction by looking at the denominator of the other fraction.

This is because if we multiply each denominator by the other denominator, we will get the same number ($5\cdot7=35$ and $7\cdot5=35$). So our goal is to multiply each denominator by the other denominator.

In order to do this math legally (and avoid math jail and incorrect answers), we have to turn that into a secret 1 so that we’re not actually changing our fractions, which is how we get $latex\frac{7}{7}$ and $\frac{5}{5}$ as our 1s (these 1s will result in multiplying the denominator each fraction by the denominator of the other fraction, as you can see in the example above).

So, the general rule for which 1 to use when finding the common denominator is this:

When finding the common denominator of two fractions, $\frac{a}{x}$ and $\frac{b}{y}$, multiply by 1 as follows: $\frac{a}{x}\cdot\frac{y}{y}$ and $\frac{b}{y}\cdot\frac{x}{x}$.

The Actual Adding and Subtracting Part

Once you have your common denominator, adding and subtracting fractions is easy. All you do is add or subtract the numerators, and leave the denominators alone.

Again for those of you in the back, because this is important: DO NOT CHANGE, ADD, OR SUBTRACT THE DENOMINATORS when adding or subtracting fractions.

So, using the fractions from our example above:

$\frac{28}{35}+\frac{20}{35}=\frac{48}{35}$,

$\frac{28}{35}-\frac{20}{35}=\frac{8}{35}$, and

$\frac{20}{35}-\frac{28}{35}=\frac{-8}{35}$. NOTE THAT THE DENOMINATORS STAY THE SAME.

Simplifying Fractions

Simplifying fractions is important for a few reasons:

Sometimes simplifying will help you see what you’re actually working with, because our brains have an easier time comprehending smaller numbers.
Some exams will require you to leave your answer in simplest form.
Standardized tests will often give their answer choices in simplest form, so you will need to simplify to know which answer to choose.

When we simplify fractions, our goal is to cancel anything that’s in both the top and the bottom, to make the fraction easier to understand. Generally, we can accomplish this goal by factoring the top and the bottom to see if there are any sneaky factors that are the same.

For example, if we have the fraction $\frac{27}{45}$, we can factor the numerator and the denominator, which will give us $\frac{3\cdot9}{5\cdot9}$. Since $\frac{9}{9}=1$, we can cancel the 9s (giving us $\frac{3\cdot9}{5\cdot9}=\frac{3}{5}\cdot\frac{9}{9}=\frac{3}{5}\cdot1=\frac{3}{5}$).

Note that the reason we can separate $\frac{3\cdot9}{5\cdot9}$ into $\frac{3}{5}\cdot\frac{9}{9}$ is the commutative property of multiplication: since there is no addition or subtraction, everything is multiplication or division (which is secretly just multiplication by a fraction), so we can rearrange/regroup as we please without actually changing the equation.

Simplifying More Complicated Fractions

There are a few more weird things that can happen with fractions:

Simplifying Improper Fractions and Mixed Numbers

An improper fraction is a fraction where the numerator is greater than the denominator, meaning that the number on top is bigger than the number on bottom. For example, $\frac{10}{3}$ is an improper fraction.

A mixed number is a whole number and fraction combo number. For instance, $3\frac{1}{3}$ is the mixed number form of $\frac{10}{3}$.

All improper fractions are greater than 1, meaning you can write them as mixed numbers if you so desire.

Are improper fractions or mixed numbers better?

There isn’t a “better” way to express fractions greater than 1: both improper fractions and mixed numbers will do just fine. Sometimes an exam will require you to put the fraction in a particular form, but otherwise, here are the pros and cons of each:

	PROS	CONS
Improper Fractions	This is a good format if you need to continue to use this fraction to do math. You will need to put mixed numbers into this format in order to multiply, divide, add, or subtract them in most cases.	It can be very difficult to see what this fraction actually means. As you practice math and strengthen your math skills, it will become easier to assess the value of improper fractions.
Mixed Numbers	It’s easy to see what these fractions mean. For example, it’s much easier to understand how much $3\frac{1}{3}$ is than it is to understand how much $\frac{10}{3}$ is.	It is difficult to use mixed numbers to do math. For instance, what is $3\frac{1}{3}\cdot7\frac{2}{5}$? It’s much easier to do $\frac{10}{3}\cdot\frac{37}{5}=\frac{370}{15}$.

So if you’re simplifying in order to keep doing math, you should simplify to an improper fraction. If you’re simplifying in order to better understand what you’re looking at, you may want to simplify to a mixed number.

How to convert between improper fractions and mixed numbers

Converting between improper fractions and mixed numbers is important, because often fractions will not appear in the form that you need them in.

Turning an Improper Fraction into a Mixed Number

To turn an improper fraction into a mixed number, all you need to do is divide, but instead of dividing in a calculator and getting a decimal answer, you keep track of the remainder and use it to form a new fraction. Like this:

If we want to turn $\frac{10}{3}$ into a mixed number, we need to divide 10 by 3. We start by asking ourselves, okay, how many times does 3 go into 10? 3 goes into 10 3 times and there is 1 left over (since $3\cdot3=9$). The 3 (from 3 going into 10 3 times) will be the whole number portion of our mixed number.
That leftover 1 is our remainder. In order to create a mixed number, we need to turn that remainder into a fraction. We will use the same denominator we had before. Since we started with ten thirds and now we have just one leftover third, the fraction portion of our mixed number will be $\frac{1}{3}$.
Then we just put them together, which gives us $3\frac{1}{3}$.

Turning a Mixed Number into an Improper Fraction

To turn a mixed number into an improper fraction, you turn the whole number portion into a fraction with the same denominator as the fraction portion, and then add them together:

If we start with $3\frac{1}{3}$, our first step is to turn the whole number portion (3) into a fraction with the same denominator as the fraction portion (also in this case 3). In order to do that, we need to treat 3 as its fraction from ($\frac{3}{1}$) and then multiply it by $\frac{3}{3}$ (which is secretly 1, since what we’re doing here is just finding the common denominator). $\frac{3}{1}\cdot\frac{3}{3}=\frac{9}{3}$, so $3=\frac{9}{3}$.
Our next step is to add the fraction we got from our whole number portion to our original fraction portion. Remember, we add the numerators and leave the denominator alone. So we get $\frac{9}{3}+\frac{1}{3}=\frac{10}{3}$, which is our improper fraction.

Simplifying Fractions Inside of Fractions

This looks at least one hundred times scarier than it actually is.

If we have a big bad fraction like $\frac{\frac{2}{3}}{\frac{5}{7}}$, all we actually have to do to simplify is remember how to divide fractions.

As you might remember from above, all we need to do to divide fractions is flip over the second fraction and then multiply the fractions.

So $\frac{\frac{2}{3}}{\frac{5}{7}}=\frac{2}{3}\cdot\frac{7}{5}=\frac{14}{15}$.

See? All simplified, and it wasn’t even that painful.

In Conclusion

Fractions are not terrible, but if you’ve forgotten how to work with them, they will make math terrible.

If you’re struggling in math, I would recommend practicing these fraction rules until they’re totally automatic.

As soon as you don’t have to think hard about what to do with fractions, math becomes about 300 times easier. (That is definitely a mathematically sound statement that I checked with math, what.)

TL;DR (I know this got long, so if you didn’t read it, these are the main points.)

These are also good study points if you did read it and want to know what to remember.

Fractions are secretly just division.
Fractions can make math easier, so we don’t want to avoid them all the time.
The top of a fraction is called the numerator, and the bottom of the fraction is called the denominator.
$\frac{x}{x}=1$, meaning that any fraction that has the exact same thing in the numerator as in the denominator is secretly just 1.
$x=\frac{x}{1}$, meaning that any whole number is equal to that same whole number over 1.
To multiply fractions, we multiply numerator by numerator and denominator by denominator.
To divide fractions, we flip the second fraction over and then multiply them.
To add or subtract fractions, we have to have the same denominator — this is called having a common denominator.
To find the common denominator, we multiply each fraction by another fraction that is secretly just 1.
When we add or subtract fractions, we add or subtract the numerators and leave the denominators alone.
We can simplify fractions by factoring the numerator and the denominator and cancelling out anything that is the same in both.
Fractions that are greater than 1 can be in improper fraction or mixed number form. Improper fractions are good for doing math and mixed numbers are good for understanding math. We convert between them using multiplication, division, and addition. (Details above).
If there are fractions inside of fractions, we flip the bottom one over and multiply them. (This is secretly just division).
If you’re struggling with math, practice these fraction rules until they’re easy for you. It will help. I promise.