Equal Fractions

Fractions

In this video we look at how to find equal fractions. We explain how to get equivalent fractions and look at several situations where this might be necessary. Several examples are given including: reducing fractions, comparing fractions, and how to find how much of the original object you have.

Equal/Equivalent Fractions

There are many ways to write the same fraction, and, at times, you may want different denominators.

So, let’s look at this situation where we have $\tfrac{1}{3}$ of a pizza left. Well, that is a very large piece, so we may decide to cut it into 3 slices.

To figure out how big each slice is compared to the whole pizza, we need to look at how many equal pieces the whole pizza can be cut into. As we can see, we’ve cut it into 9 pieces. So, we have $\tfrac{3}{9}$ of the original pizza, therefore $\tfrac{3}{9}$ must be the same as $\tfrac{1}{3}$

Those pieces seem a little small, so perhaps we would rather cut each slice into half, giving us $\tfrac{2}{6}$ of a pizza.
Well it does not matter what we call it: $\tfrac{1}{3}$,$\tfrac{3}{9}$ or $\tfrac{2}{6}$. All the fractions are equivalent. We still have $\tfrac{1}{3}$ of the pizza left.

That is all fine, but let’s now look at what we actually did to change the fraction from $\tfrac{1}{3}$ into $\tfrac{2}{6}$. To do this we need to multiply both the numerator and the denominator of the fraction by 2.

But, wait, why are we allowed to do that?

Why can I multiply the numerator and the denominator by the same number and not change the value of the fraction? It is very simple, the only number you can multiply something by without changing it is 1, but $\tfrac{2}{2}$ is 1.

So, as long as we are multiplying or dividing the numerator and the denominator by the same number, we will not change the value of the fraction.

Can you see why this does not work with adding and subtracting?

Simplifying Fractions.

To start, let’s simplify the fraction $\tfrac{8}{12}$,
First, we need to figure out what number goes into both 8 and 12.
Since both numbers are even, we know that they are divisible by 2.
8 divided by 2 is 4, and 12 divided by 2 is 6, giving us $\tfrac{4}{6}$.
Wait a second– those numbers are both even, as well, so I’m not done. We will have to divide them both by 2 again, giving us $\tfrac{2}{3}$.
Maybe you noticed that 4 went into both 8 and 12 and saved yourself a step. But, either way, we get $\tfrac{2}{3}$ as an answer.

How to find how much of the Cake.

What if I had $\tfrac{3}{4}$ of a cake left, and I wanted to share the cake with 8 friends? That would mean that there are 9 of us who want to eat cake.
How much of the original cake did each person get?
Well, first, I need to know how many pieces the original cake was cut into.
We can set this up as the ratio $\tfrac{3}{4}$ is equal to 9 over “some number” of slices
In math when we have an unknown, we let a letter represent that quantity. Let’s use “n” to represent the unknown. Ex. $\tfrac{3}{4} = \tfrac{9}{n}$

Well, we know that $3 * 3 = 9$. If we don’t want to change the value of our fraction, we need to multiply the numerator and the denominator by the same number. So, we will need to multiply 4 by 3 giving us 12.
In this case, the cake would have been cut into 12 pieces and each of us would have gotten 1/12 of the cake.

Comparing Fractions.

Look! Sam and I found some baby tree frogs. My frog $\tfrac{4}{9}$ of an adult frog, and Sam’s frog is $\tfrac{5}{12}$ of an adult frog. Which of our baby tree frogs is larger?
To compare these fractions, we need to find a common denominator.
Usually, we do this by writing out the multiples of 9 and the multiples of 12
Then, we look for the smallest number they both go into.
In this case, the smallest number they both go into is 36
If we multiply $\tfrac{4}{9}$ by $\tfrac{4}{4}$ we get $\tfrac{16}{36}$
And if we multiply $\tfrac{5}{12}$ by $\tfrac{3}{3}$ we get $\tfrac{15}{36}$
So my frog is a little bit larger.