Simplify fractions: Irreducible fractions

Now we are going to explain how to simplify fractions and when to do it, with examples and exercises solved.

What does it mean to simplify fractions?

Simplifying fractions means converting one fraction into as simple an equivalent as possible. For example:

5/10 is equivalent to 1/2.

How to simplify fractions step by step

Next I’m going to explain you two methods to explain fractions:

Method 1

To simplify fractions with this method, divide the numerator and denominator by the same number until it is no longer possible to continue. To do this, we must know the divisibility rules.

We begin testing with the smallest prime numbers: 2, 3 and 5, that is, we divide by 2 until we can no longer, then we continue with 3 and finally with 5.

If the numerator and denominator end in 0, you can first divide by 10 and save us steps.

Method 2

Another method of simplifying fractions is to break down the numerator and denominator into prime factors and then cancel up and down the factors that are repeated.

How do we know that the fraction is already simplified?

The fraction is simplified when it can no longer be divided up and down by the same number.

Resolute exercise to simplify fractions

Method 1

We have to simplify the following fraction:

simplifying fractions exercises

Now we think:

72 and 108 Can they be divided by 2? Yes.

72/2 = 36

108/2 = 54

We place the results in the fraction and see if it can be further simplified:

how to simplify fractions

36 and 54 can they be divided by 2? Yes.

36/2 = 18

54/2 = 27

Same as in the previous step, we place the results in the fraction and check if it can be further simplified:

fraction simplification exercises

18 and 27 can they be divided by 2? 18 yes, but 27 no. Then we have to change the question:

18 and 27 Can they be divided by 3? Yes

18/3 = 6

27/3 = 9

We put them in the fraction and follow:

simplification of fractions exercises solved

6 and 9 Can they be divided by 3? Yes

6/3 = 2

9/3 = 3

We put them in the fraction and follow:

simplification exercises

At this point we have finished simplifying, because 2 and 3 are prime numbers. We also finished simplifying when we can no longer divide the numerator and denominator by the same number.

Method 2

simplification of fractions exercises

We break down into prime factors 72 and 108.

Once factored we are left with:

72 = 2.2.2.2.3

108 = 2.2.2.3.3

simplify fractions solved exercises

Now, all the factors that are repeated up and down are annulled, and what cannot be annulled, will be our simplified fraction:

How to simplify a fraction

EYE!:

  • Factors can only be overridden when multiplying the rest of the expression in the numerator and denominator. It is equivalent to multiplying by 1.
  • If there’s nothing left when you override factors, you’re really left with 1.
  • If more than one factor remains after ringing, they are multiplied back to each other.

This method is faster than the previous one, if the fraction numbers are high.

What if the fraction is not simplified?

Sometimes you may wonder what happens if the fraction is not simplified. That has consequences depending on the exercise and the resolution point we are at.

In the case of an operation where the final result is a fraction, if it is not simplified, the result is almost correct, because although it is practically fine, your teacher will always ask you to simplify it and may not value the exercise with all its score on an exam.

But apart from that, the real objective of simplifying fractions is to operate with the smallest possible numbers, since in a somewhat more complicated operation, if it is not simplified, we can end up working with very high numbers, which can lead us to error.

So simplify fractions for your sake! 🙂