- Simplification of a fraction means reducing a fraction to its simplest form.

**Simplest form**

- A fraction is in its simplest form if the numerator and the denominator don't have any common factor other than 1.
- It is also called as reduced form.

**For Example: **\(\dfrac {1}{2}\)

1 and 2 don't have any common factor other than 1.

\(\therefore\,\dfrac {1}{2}\) is in its simplest form.

- To understand it easily, let us consider an example:

**Example:** Simplify \(\dfrac {32}{24}\)

**Step-1 :** Find G.C.F (greatest common factor) of the numerator and the denominator.

G.C.F of 32 and 24 can be calculated as:

Factors of \(32=\underline{2×2×2}×2×2\)

Factors of \(24=\underline {2×2×2}×3\)

G.C.F \(=2×2×2=8\)

**Step-2 : **Divide the numerator and the denominator by the G.C.F, and the fraction obtained is in its simplest form.

In the example: \(\dfrac {32÷8}{24÷8}=\dfrac {4}{3}\)

4 and 3 don't have any common factor other than 1. Thus, the fraction \(\dfrac{4}{3}\) is in its simplest form.

A \(\dfrac {6}{5}\)

B \(\dfrac {5}{6}\)

C \(\dfrac {7}{12}\)

D \(\dfrac {12}{7}\)

**Simplest form**

- A fraction is in its simplest form if the numerator and the denominator don't have any common factor other than 1.
- It is also called as reduced form.

**For Example: **\(\dfrac {1}{2}\)

1 and 2 don't have any common factor other than 1.

\(\therefore\,\dfrac {1}{2}\) is in its simplest form.

To convert a decimal into its simplest form, consider the example of 0.2.

**Convert the decimal into a fraction**

**Step-1: **Write the decimal in the place value chart.

Tens | Ones | Decimal point | Tenths | Hundredths | Thousandths | Ten Thousandths |

0 | . | 2 |

**Step-2: **Since, the position of 2 represents the tenths place so, it can be read as 'two tenths'.

**Step-3: **Numerator = 2

Denominator = Place value of tenths, i.e. 10

**Step-4: **Write the fraction.

Fraction \(=\dfrac {2}{10}\)

**Simplify the Fraction**

Simplify \(\dfrac {2}{10}\)

**Step-1: **The** **G.C.F of 2 and 10 = 2 [ G.C.F = greatest common factor ]

**Step-2:** Divide numerator and denominator by G.C.F \(\Rightarrow\dfrac {2÷2}{10÷2}=\dfrac {1}{5}\)

1 and 5 don't have any common factor other than 1.

Therefore, \(\dfrac {1}{5}\) is in its simplest form.

A \(\dfrac{3}{5}\)

B \(\dfrac{2}{5}\)

C \(\dfrac{4}{5}\)

D \(\dfrac{2}{10}\)

- To convert the unlike fractions into like fractions, we should follow the following steps:
- Let us consider an example:

Convert \(\dfrac{1}{2}\;and\;\dfrac{3}{8}\) into like fractions.

**Step 1:** Find the least common multiple (L.C.M) of the denominators of the fractions.

Fractions \(=\dfrac{1}{2},\;\dfrac{3}{8}\)

L.C.M of \(2\) and \(8\) can be calculated as:

Multiples of \(2=2,\;4,\;6,\;8\;...\)

Multiples of \(8=8,\;16\,...\)

L.C.M of \(2\) and \(8=8\)

**Step 2:** L.C.M becomes the lowest common denominator, i.e.

L.C.D \(=8\)

**Step 3:** Find equivalent fractions having L.C.D as the denominator.

The equivalent fraction of \(\dfrac{1}{2}=\dfrac{1×4}{2×4}=\dfrac{4}{8}\)

- \(\dfrac{3}{8}\) does not need to be converted as it already has \(8\) as its denominator.

\(\dfrac{4}{8}\) and \(\dfrac{3}{8}\) have same denominators.

\(\therefore\) These are like fractions.

A \(\dfrac{18}{12},\;\dfrac{3}{12},\;\dfrac{20}{12}\)

B \(\dfrac{4}{12},\;\dfrac{6}{12},\;\dfrac{15}{12}\)

C \(\dfrac{12}{4},\;\dfrac{1}{4},\;\dfrac{5}{4}\)

D \(\dfrac{18}{3},\;\dfrac{2}{4},\;\dfrac{5}{6}\)

- Fractions can be converted into decimals and vice versa.
- There are two possible cases of conversion of fractions into decimals:

- By using place value
- By division

**Case 1: **When the denominator has base ten values

- If the denominator of a fraction has base ten values like \(10,100,1000 \space or\space10000\), then we can convert the fraction into decimals as follows:

Let us consider the fraction \(\dfrac{25}{100}\).

- The given fraction can be expressed as 25 out of 100, i.e. twenty-five hundredths.
- We know that a hundredth is represented by two decimal digits.
- So, the decimal form of the fraction \(\dfrac{25}{100}\) is \(0.25\).

**Case 2:** When the denominator does not have base ten values

- If the denominator of a fraction does not have base ten values, for eg:\(\Big(\dfrac{1}{2}, \dfrac{3}{4},\dfrac{7}{3}\Big)\), then convert the fraction into decimals by dividing the numerator by the denominator.

Consider an example: \(\dfrac{7}{2}\)

- Divide the numerator by the denominator.
- Thus, we can write \(\dfrac{7}{2}=3.5\)

A \(1.33\)

B \(0.2\)

C \(0.5\)

D \(0.8\)

To convert a decimal into a mixed number, consider the example of 1.5.

**Step-1: **Write the decimal in the place value chart.

Tens |
Ones |
Decimal point |
Tenths |
Hundredths |
Thousandths |
Ten Thousandths |

1 | . | 5 |

**Step-2: **From the place value chart, the decimal is read as 'one and five tenths'.

**Step-3: **Whole number = 1

Numerator of the fraction = 5

Denominator of the fraction = Place value of tenths, i.e. 10

**Step-4: **Write the fraction.

Fraction \(=\dfrac {5}{10}\)

**Step-5: **Simplify the fraction.

The G.C.F of 5 and 10 = 5

Divide numerator and denominator with G.C.F \(\Rightarrow\dfrac {5÷5}{10÷5}=\dfrac {1}{2}\)

1 and 2 don't have any common factor other than 1.

Therefore, \(\dfrac {1}{2}\) is in its simplest form.

**Step-6: **Write the mixed number.

Mixed number \(=1\dfrac {1}{2}\)

A \(2\dfrac {1}{2}\)

B \(2\dfrac {1}{10}\)

C \(1\dfrac {2}{5}\)

D \(1\dfrac {5}{10}\)

- To convert a mixed fraction into an improper fraction, we should follow the following steps:
- Let us consider an example:

Convert \(5\dfrac{1}{3}\) into an improper fraction.

**Step 1:** Multiply the whole number by the denominator and add the numerator,

\(=5×3+1\)

\(=15+1\)

\(=16\)

**Step 2:** Put the result over the original denominator,

\(=\dfrac{16}{3}\)

**Step 3:** The fraction obtained is an improper fraction.

As \(16>3\)

\(\therefore\;\dfrac{16}{3}\) is an improper fraction.

- Thus, \(\dfrac{16}{3}\) is the answer.

A \(\dfrac{18}{5}\)

B \(\dfrac{42}{5}\)

C \(\dfrac{16}{5}\)

D \(\dfrac{10}{5}\)

- To convert an improper fraction into a mixed number, we should follow the following steps:
- Let us consider an example:

Convert \(\dfrac{11}{2}\) into a mixed number.

**Step 1:** Rewrite the given fraction as a division problem and solve it.

Fraction \(=\dfrac{11}{2}\)

**Step 2:** To get a mixed number from division:

- The quotient becomes the whole number.
- The remainder becomes the numerator of the fraction.
- The divisor becomes the denominator of the fraction.

Quotient \(=5\longleftarrow\) Whole number

Remainder \(=1\longleftarrow\) Numerator

Divisor \(=2\longleftarrow\) Denominator

**Steps 3:** Write the required mixed number.

Whole number \(=5\)

Fraction \(=\dfrac{1}{2}\)

Mixed number \(=5\dfrac{1}{2}\)

**Step 4:** Simplify the fraction part of the mixed number.

\(1\) and \(2\) do not have any common factor other than \(1\).

\(\therefore\;\dfrac{1}{2}\) is in its simplest form.

Thus, \(5\dfrac{1}{2}\) is the answer.

A \(3\dfrac{1}{2}\)

B \(3\dfrac{1}{3}\)

C \(2\dfrac{1}{3}\)

D \(5\dfrac{1}{2}\)

- Mixed numbers can be converted into decimals.
- There are two possible cases for conversion of mixed fractions into decimals.

Case 1: When the denominator has base ten values

Consider an example: \(3\dfrac{1}{10}\)

- To change a mixed number into a decimal, the whole number goes to the left of the decimal point.
- In \(3\dfrac{1}{10}\), \(3\) is the whole number and \(\dfrac{1}{10}\) is the fraction part.
- Convert the fraction part into decimal.
- \(10\) is the denominator of the fraction part \(\dfrac{1}{10}\), which implies one tenth.
- We know that a tenth number is represented by one decimal digit.
- Here, one is in the numerator.
- So \(\dfrac{1}{10}\) \(=0.1\)
- Thus, \(3\dfrac{1}{10}=3.1\)

Case 2: When the denominator does not have base ten values

Consider an example: \(4\dfrac{1}{5}\)

- In \(4\dfrac{1}{5}\) , \(4\) is the whole number and \(\dfrac{1}{5}\) is the fraction part.
- Convert the fraction part into decimal.
- Since \(\dfrac{1}{5}\) does not have base ten value in its denominator, so divide the numerator by the denominator.
- We can write \(\dfrac{1}{5}=0.2\)
- The whole number goes to the left of the decimal point.

So, \(4\dfrac{1}{5}=4.2\)

Case 2: When the denominator does not have base ten values

Consider an example: \(4\dfrac{1}{5}\)

- In \(4\dfrac{1}{5}\) , \(4\) is the whole number and \(\dfrac{1}{5}\) is the fraction part.
- Convert the fraction part into decimal.
- \(\dfrac{1}{5}\) does not have base ten value in its denominator, so divide the numerator by the denominator.
- We can write \(\dfrac{1}{5}=0.2\)
- The whole number goes to the left of the decimal point.

So, \(4\dfrac{1}{5}=4.2\)

A \(1.25\)

B \(3.89\)

C \(7.25\)

D \(1.20\)