# How to simplify fractions

**Why do you need to simplify fractions?**

Fractions are numbers that address a part of the whole. Some examples of the fractions are $$\frac{7}{13}$$, $$\frac{10}{18}$$.

It is essential to ensure that everything is in its least complex structure. Changing the fraction into the simplest form is necessary before continuing further in the question.

Also, examiners like to see answers simplified as possible. You may even lose marks if your answer is not in its simplest form.

**E1.5:** **Use the language and notation of simple vulgar fractions in appropriate contexts.**

Before figuring out how to change the fractions into their simplest form, we need to know what equivalent fractions are. Equivalent fractions can be characterised as fractions with various numerators and denominators; however, they address a similar value. For instance, $$\frac{12}{15}$$ and $$\frac{24}{30}$$ are equivalent fractions because both are equivalent to $$\frac{4}{5}$$.

Simplifying a fraction suggests decreasing a fraction to its simplest form. If both the numerator and the denominator of a fraction have no common factor except $$1$$, then it is said to be in its simplest form. The fraction $$\frac{4}{5}$$ is in the simplest form because $$4$$ and $$5$$ have no common factors.

**Worked examples**

**Example 1:** Change $$\frac{10}{50}$$ into its simplest form.

**Step 1: Write all the factors of the numerator and the denominator.**

The factors of $$10$$ are $$1$$, $$2$$, $$5$$, $$10$$.

The factors of 50 are $$1$$, $$2$$, $$5$$, $$10$$, $$25$$, $$50$$.

**Step 2: Figure out the common factors of the numerator and the denominator.**

The common factors of $$10$$ and $$50$$ are $$1$$, $$2$$, $$5$$, $$10$$.

**Step 3: Figure out the greatest common factor of the numerator and the denominator.**

The greatest common factor of $$10$$ and $$50$$ is $$10$$.

**Step 4: Divide the numerator and the denominator by their greatest common factor and change the fraction into its simplest form.**

$$10\div10=1$$, $$50\div10=5$$.

$$\frac{10}{50}=\frac{1}{5}$$

**Example 2: **Raul has $$20$$ pens in his box. $$5$$ of them are red, $$3$$ of them are blue and rest of them are black. Find the fractional value of the black pens then write it into its simplest form.

**Step 1: Find the number of black pens in the box.**

Total number of pens is $$20$$.

Number of red and blue pens is $$8$$.

Black pens are $$12$$.

**Step 2: Make fraction of black pens in which the total is out of $$20$$.**

So, the fraction is $$\frac{12}{20}$$.

**Step 3: Figure out the factors of the numerator and the denominator of the fraction.**

The factors of $$12$$ are $$1$$, $$2$$, $$3$$, $$4$$, $$6$$, $$12$$.

The factors of $$20$$ are $$1$$, $$2$$, $$4$$, $$5$$, $$10$$, $$20$$.

**Step 4: Figure out the common factors of the numerator and the denominator**

The common factors of $$12$$ and $$20$$ are $$1$$, $$2$$, $$4$$.

**Step 5: Figure out the greatest common factor of the numerator and the denominator**

The greatest common factor of $$12$$ and $$20$$ is $$4$$.

**Step 6: Divide the numerator and the denominator by their greatest common factor and change the fraction into its simplest form**

$$12\div 4=3$$, $$20\div 4=5$$.

$$\frac{12}{20}=\frac{3}{5}$$.

The fractional value of the black ball is $$\frac{3}{5}$$.

**Example 3: **Rachael has $$36$$ tomatoes in her bag. $$12$$ of them are rotten and the rest are fresh. Find the simplest fractional value of the fresh tomatoes in the bag.

**Step 1: Find the number of fresh tomatoes in the bag**

$$\text{Total number of tomatoes in bag} =36$$

$$\text{Rotten tomatoes} = 12$$

$$\text{Fresh tomatoes}=36-12$$ which is equal to $$24$$.

**Step 2: Make the proper fraction for the fresh tomatoes in which the whole value is $$36$$.**

The fraction is $$\frac{24}{36}$$.

**Step 3: Figure out the factors of the numerator and the denominator of the fraction.**

The factors of $$24$$ are $$1$$, $$2$$, $$3$$, $$4$$, $$6$$, $$8$$, $$12$$, $$24$$.

The factors of $$36$$ are $$1$$, $$2$$, $$3$$, $$4$$, $$9$$, $$12$$, $$18$$, $$36$$.

**Step 4: Figure out the common factors of the numerator and the denominator.**

The common factors of $$24$$ and $$36$$ are $$1$$, $$2$$, $$3$$, $$4$$, $$12$$.

**Step 5: Figure out the greatest common factor of the numerator and the denominator**

The greatest common factors of $$24$$ and $$36$$ is $$12$$.

**Step 6: Divide the numerator and the denominator by their greatest common factor and change the fraction into its simplest form**

$$24\div 12=2$$, $$36\div 12=3$$. So, $$\frac{24}{36} = \frac{2}{3}$$.

The fractional value of the fresh tomatoes is $$\frac{2}{3}$$.