How to simplify fractions - Equivalent and irreducible fractions

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What it means simplify a fraction? Simplifying a fraction means writing an equivalent fraction that has a smaller numerator and denominator than the original fraction.

equivalent fractions are nothing more than fractions that represent the same amount or part of a whole.

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To understand how this works, consider the following example:

In an 8-piece pizza, the fraction representing half of the pizza is $\dpi{120} \frac{4}{8}$ . However, if that same pizza is divided into just 4 pieces, half of it is represented by the fraction $\dpi{120} \frac{2}{4}$ . See the figure below: $equivalent fractions$

Despite having different numerator and denominator, fractions $\dpi{120} \frac{4}{8}$ It is $\dpi{120} \frac{2}{4}$ represent the same amount of pizza. Therefore, these fractions are equivalent, which means that:

$\dpi{120} \frac{4}{8} \frac{2}{4}$

Question 1: Is there a fraction equivalent to fraction $\dpi{120} \frac{2}{4}$ , with even smaller terms?

yes, the fraction $\dpi{120} \frac{1}{2}$ is equivalent to fraction $\dpi{120} \frac{2}{4}$ , as it also represents half of the pizza, when it is divided into just two pieces.

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So these three fractions are equivalent: $\dpi{120} \frac{4}{8} \frac{2}{4}\frac{1}{2}$ Question 2: Is there a fraction equivalent to fraction $\dpi{120} \frac{1}{2}$ , with even smaller terms?

No, these are the smallest possible values, that is, we can no longer simplify this fraction.

In this case, we say that the fraction $\dpi{120} \frac{1}{2}$ and the irreducible form of the fraction $\dpi{120} \frac{4}{8}$ .

The pie design made it easy to find equivalent fractions with smaller terms to represent half of the pie.

Now, let's see a practical and easy way to simplify fractions without needing the help of an illustrative image.

How to simplify a fraction?

To simplify a fraction, divide the numerator and denominator of the fraction by a same number greater than 1.

Example 1: Let's simplify the fraction $\dpi{120} \frac{3}{12}$ .

The first number greater than 1 is the number 2. Are both terms of this fraction divisible by 2?

3 is not divisible by 2, as the remainder of the division is not zero;
12 is divisible by 2 since the remainder of the division is zero.

The two numbers must be divisible! If both are not, let's go to the next number.

The next number is 3. Are both terms of this fraction divisible by 3?

3 is divisible by 2, because the remainder of the division is zero;
12 is divisible by 3 because the remainder of the division is zero.

So, to simplify the fraction, let's divide the numerator and denominator by 3.

Hence, we have to $\dpi{120} \frac{3}{12}\frac{1}{4}$ .

Question: It is possible to simplify the fraction $\dpi{120} \frac{1}{4}$ ? No, these are the smallest possible terms of this fraction.

Then, $\dpi{120} \frac{1}{4}$ is the irreducible form of the fraction $\dpi{120} \frac{3}{12}$ .

Example 2:Let's simplify the fraction $\dpi{120} \frac{45}{120}$ .

Then, $\dpi{120} \frac{45}{120}\frac{3}{8}$ .

Example 3:Let's simplify the fraction $\dpi{120} \frac{60}{84}$ .

We have to $\dpi{120} \frac{60}{84}\frac{5}{7}$ .

Read too:

Activities with fractions
How to add and subtract fractions

How to simplify fractions - Equivalent and irreducible fractions

How to simplify a fraction?

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