# How to do the Decomposition of Fractions

Decomposition of fractions involves breaking down a given fraction into a sum of smaller fractions with different denominators. There are several methods to decompose fractions, and in this explanation, we will focus on the method using common denominators and the method using prime factorization.

## A Step-by-step Guide to the Decomposition of Fractions

Here is a step-by-step guide to decomposing fractions:

### Step 1: Identify the fraction to be decomposed:

Determine the fraction you wish to decompose. For example, let’s choose the fraction \(\frac{5}{6}\).

Reduce the fraction, if possible: Simplify the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, \(\frac{5}{6}\) is already in its simplest form.

Choose the method of decomposition: You can decompose fractions in different ways, but two common methods are decomposition using common denominators and decomposition using prime factorization. In this example, we’ll use the common denominators method.

### Step 2: Break the fraction into smaller fractions:

Identify two or more smaller fractions that add up to the given fraction. You can use whole numbers or other fractions with the same denominator. For example, you can decompose \(\frac{5}{6}\) into \(\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}, \frac{3}{6} + \frac{2}{6}\), or \(\frac{4}{6} + \frac{1}{6}\).

### Step 3: Simplify the smaller fractions, if necessary:

If any of the smaller fractions can be further simplified, do so.

### Step 4: Choose the decomposition that best suits your needs:

Depending on the context or the purpose of the problem, select the decomposition that is most useful or appropriate.

Remember that there are multiple ways to decompose a fraction, and your choice of decomposition depends on the problem you’re solving or the context in which you’re working. Practice decomposing different fractions to get comfortable with the process and become familiar with different decompositions.

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