Reducing Fractions — Process & Practice

Divide the numerator and the denominator by the same number

The best way to simplify, or reduce, a fraction is to divide the top (the numerator) and the bottom (the denominator) by the largest common factor (in other words, the biggest number that they can both be divided by evenly). That can be tricky if your numerator and denominator are really big, so here are some steps that might make simplifying fractions, well, simpler. We call this reducing a fraction to lowest terms. That's just a fancy way of saying "reduce". You might see it in your textbook called that.

Step 1: Check if the denominator is evenly divisible by the numerator

Sometimes, the bottom number of the fraction will be evenly divisible by the top number. If it is, simplifying the fraction is easy! Just divide the numerator and the denominator by the numerator, and you've got your simplified fraction. Let's try it on an example: $\frac{8}{32}$

We get a whole number, $4$, when we divide $32$ by $8$, so it works! Next, divide both the top and bottom of $\frac{8}{32}$ by $8$ to get the fully simplified fraction.

The most simplified form of $\frac{8}{32}$ is $\frac{1}{4}$. Since the numerator of this fraction is $1$, it can't be divided anymore—you're done!

Step 2: If the bottom isn't divisible by the top, try dividing by a small common factor.

If the denominator isn't evenly divisible by the numerator, you might be able to divide both numbers by something else. Let's try it out on the fraction $\frac{12}{32}$, starting from Step 1 just in case.

$32$ divided by $12$ is $2.6666666$, which isn't a whole number—so $32$ is not evenly divisible by $12$. However, $32$ and $12$ are both even, so let's try dividing them both by $2$.

$32$ divided by $2$ is $16$ and $12$ divided by $2$ is $6$, so our new fraction is $\frac{6}{16}$.

Step 3: Repeat until you can't divide one side anymore

$\frac{6}{16}$ is closer to simplified, but it's not quite there yet. Try dividing both sides by 2 again.

$16$ divided by $2$ is $8$ and $6$ divided by $2$ is $3$, so our new fraction is $\frac{3}{8}$. Let's try it one more time.

$3$ isn't evenly divisible by $2$, so we can't do that anymore. Let's try Step 1 again, just to be sure.

$8$ isn't evenly divisible by $3$, and since $3$ isn't divisible by $2$, it isn't evenly divisible by anything. So we're done!

(Video) Simplifying Fractions

by Math Antics

In this video by Math Antics, he goes over what simplifying fractions means, how you simplify them, and then some examples.

$\large{\text{Summary}}$

So how do you find the simplest form of a fraction? Well, you need to look at the prime factorizations of the numerator and the denominator.

In the previous example, we looked at how $\frac36$ and $\frac12$ are equivalent fractions. We also stated that $\frac12$ was the simplest form of $\frac36$.

To show this, we write out the prime factorization of both the numerator and denominator. $$\frac36=\frac{1\times3}{2\times3}$$ They both have a common factor of $3$. We can rewrite this expression as the product of two fractions: $$\frac{1\times3}{2\times3}=\frac12\times\frac33$$ And since $\frac33=1$, this becomes: $$\frac12\times\frac33=\frac12\times1$$ And anything times $1$ stays the same. $$\frac12\times1=\frac12$$

Let's look at another example: $\frac{30}{36}$. Start by writing out the prime factorization of both the top and bottom of the fraction. $$\frac{30}{36}=\frac{2\times3\times5}{2\times2\times3\times3}$$ As you can see, both the numerator and denominator have common factors of $2$ and $3$. $$\frac{\color{#29A603}{2}\times\color{#8925AE}{3}\times5}{\color{#29A603}{2}\times2\times\color{#8925AE}{3}\times3}$$ Because these are common factors, we can just cross them out. $$\frac{\color{#29A603}{\cancel{2}}\times\color{#8925AE}{\cancel{3}}\times5}{\color{#29A603}{\cancel{2}}\times2\times\color{#8925AE}{\cancel{3}}\times3}$$ This leaves us with: $$\frac{5}{2\times3}=\frac56$$ Therefore, $\frac56$ is the simplest form of the fraction $\frac{30}{36}$.

Notes on Reducing Improper Fractions

Everything we just said will still work if your original fraction is still improper. There is one extra step, though. If you reduce an improper fraction, it'll still be improper. That means it can be written as a mixed number. You might want to review converting fractions to mixed numbers.

Prime Numerators and Denominators

There's something else to be said for reduced fractions. We can check if the numerators and denominators are prime numbers. If both of these numbers are prime, then we know the fraction is fully reduced. The only exception is if the numbers are the same. For example, $3$ is a prime number, but

$$\frac{3}{3}$$

can be reduced to $1$. If the fraction is a mixed number, this trick won't work. You'll have to convert back to an improper fraction.

Simplifying Fractions

When we simplify a fraction, we put it in its simplest form. This is also called reducing a fraction.

The whole idea of simplification in math is the same as in the rest of life—to reduce clutter, and make things easier to see and use.

Think of messy fractions like a messy coat rack. Yes, it's still a coat rack no matter how many coats you fling on it. But, the clutter makes it darn near impossible to find the coat you want when you want it.

Messy fractions are like that. Yes, they are still fractions as long as they've got a numerator and a denominator. But, the more cluttered and un-simplified they are, the harder they are to work with.

So, let's get down to the business of how we can simplify fractions!

For both proper and improper fractions:

The fractions below are all examples of fractions that are not simplified:

The easiest way to simplify a fraction in the smallest number of steps is to divide the numerator and denominator by their greatest common factor (GCF). Of course, one must find the greatest common factor.

There are a couple of hints to keep in mind when you're simplifying fractions:

• If your numerator and denominator are both even numbers, you can always start simplifying by dividing each by $2$.
• If your numerator and denominator both end in $0$ or $5$, you can start simplifying by dividing each by $5$.
• It's a good idea to check first if your denominator can be evenly divided by your numerator. (For example, in the fraction $\frac{2}{6}$, our denominator $6$ can be divided by our numerator $2$). This will bring you to the simplified form of your fraction quickly.