Oh No! Not Fractions! Part 2

This is part two in a series of posts on fraction arithmetic. The other posts can be found here:

Previously, in part one, I had a very quick look at the basic structure of a fraction and introduced a little bit of terminology. Now we’ll look at the idea of equivalent fractions, and fractions in lowest terms.

Equivalent Fractions

In the last post I mentioned that you can read a fraction as one number divided by another. Let’s have a look at these two different fractions and think about them as divisions:

$\large{\boldsymbol{\frac{6}{3}=6\div 3=2}}$

$\large{\boldsymbol{\frac{8}{4}=8\div 4=2}}$

Here you have two fractions that look different, $\boldsymbol{\frac{6}{3}}$ and $\boldsymbol{\frac{8}{4}}$ , but actually represent the same value. They are equivalent fractions. Here are a few more (check them on a calculator if you like):

$\large{\boldsymbol{\frac{21}{28}=21\div 28=0.75}}$

$\large{\boldsymbol{\frac{9}{12}=9\div 12=0.75}}$

$\large{\boldsymbol{\frac{3}{4}=3\div 4=0.75}}$

Generally speaking, smaller numbers are easier to use in calculations and easier to read. For example, it’s much easier to think about doing multiplication and subtraction and so on with the numbers 3 and 4, than it is with 21 and 28. So when we use fractions we prefer to use the equivalent fraction with the smallest possible numbers. This is often referred to as reducing a fraction to lowest terms.

Fractions in lowest terms

To reduce a fraction to lower terms you look for factors that both the numerator and denominator have in common and then divide both of them by that factor. For example:

21 has the factors 1,3,7,21

28 has the factors 1,2,4,7,14,28

So 21 and 28 have common factors of 1 and 7. However 1 is not going to be any use to us here since, when we divide by 1, we don’t reduce the value, e.g.: $\boldsymbol{21\div 1 = 21}$ . Let’s have a look at what happens when we divide both numerator and denominator by 7:

$\large{\boldsymbol{\frac{21}{28}=\frac{21\div 7}{28\div 7}=\frac{3}{4}}}$

We get the equivalent fraction $\boldsymbol{\frac{3}{4}}$ . The fraction is certainly now in lower terms, but how can we tell if it is in lowest terms? We look at the factors of 3 and 4:

3 has the factors 1,3

4 has the factors 1,2,4

Except for 1, there are no longer any common factors. So we’re not going to be able to divide them again to make them smaller. If there were still common factors (greater than 1) we would repeat the process until there weren’t.

That’s all for now.

Oh No! Not Fractions! Part 1

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