Informative line

# Simplification of a Fraction

• 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.

#### Which one of the following fractions represents the simplest form of $$\dfrac {35}{60}$$?

A $$\dfrac {6}{5}$$

B $$\dfrac {5}{6}$$

C $$\dfrac {7}{12}$$

D $$\dfrac {12}{7}$$

×

Given fraction $$=\dfrac {35}{60}$$

Finding the greatest common factor (G.C.F) of 35 and 60,

Factors of 35 = 5 × 7

Factors of 60 = 2 × 2 × 3 × 5

Here, 5 is G.C.F of 35 and 60.

$$\therefore$$ G.C.F = 5

Dividing 35 and 60 by 5,

$$\dfrac {35÷5}{60÷5}=\dfrac {7}{12}$$

Resulting fraction $$=\dfrac {7}{12}$$

7 and 12 don't have any common factor other than 1.

$$\therefore\,\dfrac {7}{12}$$ is the simplest form of  $$\dfrac {35}{60}$$.

Hence, option (C) is correct.

### Which one of the following fractions represents the simplest form of $$\dfrac {35}{60}$$?

A

$$\dfrac {6}{5}$$

.

B

$$\dfrac {5}{6}$$

C

$$\dfrac {7}{12}$$

D

$$\dfrac {12}{7}$$

Option C is Correct

# Conversion of Decimal into Simple Fraction

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.

#### Which fraction has the same value as 0.4?

A $$\dfrac{3}{5}$$

B $$\dfrac{2}{5}$$

C $$\dfrac{4}{5}$$

D $$\dfrac{2}{10}$$

×

Given: 0.4

Converting 0.4 into a fraction.

 Tens Ones Decimal point Tenths Hundredths Thousandths 0 . 4

Since the position of 4 represents the tenths place so, it can be read as 'four tenths'.

Numerator = 4

Denominator = Place value of 4, i.e. 10

Fraction = $$\dfrac {4}{10}$$

Simplifying $$\dfrac {4}{10}$$

G.C.F of 4 and 10 = 2  [ G.C.F = greatest common factor ]

Now, dividing numerator and denominator by G.C.F $$\Rightarrow\dfrac {4÷2}{10÷2}=\dfrac {2}{5}$$

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

Therefore, $$\dfrac {2}{5}$$ is in its simplest form.

Hence, option (B) is correct.

### Which fraction has the same value as 0.4?

A

$$\dfrac{3}{5}$$

.

B

$$\dfrac{2}{5}$$

C

$$\dfrac{4}{5}$$

D

$$\dfrac{2}{10}$$

Option B is Correct

# Conversion of Unlike Fractions into Like Fractions

• 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.

#### Which one of the following options represents the like fractions of  $$\dfrac{3}{2},\;\dfrac{1}{4}$$ and $$\dfrac{5}{3}$$  respectively?

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 $$\dfrac{18}{3},\;\dfrac{2}{4},\;\dfrac{5}{6}$$

×

Given fractions:

$$\dfrac{3}{2},\;\dfrac{1}{4},\;\dfrac{5}{3}$$

Finding the least common multiple (L.C.M) of denominators.

Denominators $$=2,\;4,\;3$$

Multiples of $$2=2,\;4,\;6,\;8,\;10,\;12\,...$$

Multiples of $$3=3,\;6,\;9,\;12,\;15\,...$$

Multiples of $$4=4,\;8,\;12,\;16\,...$$

L.C.M of $$2,\;3,\;4=12$$

L.C.M becomes the lowest common denominator, i.e.

L.C.D $$=12$$

Finding equivalent fractions of each fraction having L.C.D as the denominator.

Equivalent fraction of $$\dfrac{3}{2}=\dfrac{3×6}{2×6}=\dfrac{18}{12}$$

Equivalent fraction of $$\dfrac{1}{4}=\dfrac{1×3}{4×3}=\dfrac{3}{12}$$

Equivalent fraction of $$\dfrac{5}{3}=\dfrac{5×4}{3×4}=\dfrac{20}{12}$$

$$\dfrac{18}{12},\;\dfrac{3}{12},\;\dfrac{20}{12}$$ have same denominators.

$$\therefore\;\dfrac{18}{12},\;\dfrac{3}{12},\;\dfrac{20}{12}$$ are like fractions of $$\dfrac{3}{2},\;\dfrac{1}{4},\;\dfrac{5}{3}$$ respectively.

Hence, option (A) is correct.

### Which one of the following options represents the like fractions of  $$\dfrac{3}{2},\;\dfrac{1}{4}$$ and $$\dfrac{5}{3}$$  respectively?

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

$$\dfrac{18}{3},\;\dfrac{2}{4},\;\dfrac{5}{6}$$

Option A is Correct

# Conversion of Fractions to Decimals

• Fractions can be converted into decimals and vice versa.
• There are two possible cases of conversion of fractions into decimals:
1. By using place value
2. 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 $$\dfrac{7}{2}=3.5$$  #### Which one of the following options has the same value as $$\dfrac{4}{8}$$?

A $$1.33$$

B $$0.2$$

C $$0.5$$

D $$0.8$$

×

Given fraction: $$\dfrac{4}{8}$$

The G. C. F. of $$4$$ and $$8$$ is $$4$$.

Simplifying the fraction,

$$\dfrac{4\div4}{8\div 4}\space=\space\dfrac{1}{2}$$

$$1$$ and $$2$$ do not have any common factor other than 1.

So, $$\dfrac{1}{2}$$ is in its simplest form.

The denominator of $$\dfrac{1}{2}$$ does not have base ten value, so we divide the numerator by the denominator.

Thus, $$\dfrac{1}{2}=0.5$$ Hence, option (C) is correct.

### Which one of the following options has the same value as $$\dfrac{4}{8}$$?

A

$$1.33$$

.

B

$$0.2$$

C

$$0.5$$

D

$$0.8$$

Option C is Correct

# Conversion of Decimal into Mixed Fraction

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 $$=1\dfrac {1}{2}$$

#### Which one of the following options has the same value as 2.5?

A $$2\dfrac {1}{2}$$

B $$2\dfrac {1}{10}$$

C $$1\dfrac {2}{5}$$

D $$1\dfrac {5}{10}$$

×

Given: 2.5

Converting 2.5 into a fraction:

 Tens Ones Decimal point Tenths Hundredths Thousandths Ten Thousandths 2 . 5

Using the place value chart, the decimal is read as 'two and five tenths'.

Whole number = 2

Numerator = 5

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

$$\therefore$$ Fraction $$=\dfrac {5}{10}$$

Simplifying $$\dfrac {5}{10}$$

The G.C.F of 5 and 10 = 5

Dividing numerator and denominator by 5 $$\Rightarrow\dfrac {5÷5}{10÷5}=\dfrac {1}{2}$$

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

$$\therefore\,\;\dfrac {1}{2}$$ is in its simplest form.

Writing the mixed number,

Mixed number $$=2\dfrac {1}{2}$$

Hence, option (A) is correct.

### Which one of the following options has the same value as 2.5?

A

$$2\dfrac {1}{2}$$

.

B

$$2\dfrac {1}{10}$$

C

$$1\dfrac {2}{5}$$

D

$$1\dfrac {5}{10}$$

Option A is Correct

# Conversion of a Mixed Fraction into an Improper Fraction

• 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.

#### Convert $$8\dfrac{2}{5}$$ into an improper fraction.

A $$\dfrac{18}{5}$$

B $$\dfrac{42}{5}$$

C $$\dfrac{16}{5}$$

D $$\dfrac{10}{5}$$

×

Given: $$8\dfrac{2}{5}$$

Multiplying $$8$$ with $$5$$ and adding $$2$$ to it,

$$=8×5+2$$

$$=40+2$$

$$=42$$

Putting the result over the original denominator,

$$=\dfrac{42}{5}$$

As $$42>5$$

$$\therefore\;\dfrac{42}{5}$$ is an improper fraction.

Thus, $$\dfrac{42}{5}$$ is the answer.

Hence, option (B) is correct.

### Convert $$8\dfrac{2}{5}$$ into an improper fraction.

A

$$\dfrac{18}{5}$$

.

B

$$\dfrac{42}{5}$$

C

$$\dfrac{16}{5}$$

D

$$\dfrac{10}{5}$$

Option B is Correct

# Conversion of an Improper Fraction into a Mixed Number

• 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.  #### Convert $$\dfrac{10}{3}$$ into a mixed number.

A $$3\dfrac{1}{2}$$

B $$3\dfrac{1}{3}$$

C $$2\dfrac{1}{3}$$

D $$5\dfrac{1}{2}$$

×

Given: $$\dfrac{10}{3}$$

Rewriting $$\dfrac{10}{3}$$ as a division problem, From the division,

Quotient $$=3\longleftarrow$$ The whole number

Divisor $$=3\longleftarrow$$ Denominator

Remainder $$=1\longleftarrow$$ Numerator

Whole number $$=3$$

Fraction $$=\dfrac{1}{3}$$

The required mixed fraction $$=3\dfrac{1}{3}$$

$$1$$ and $$3$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{1}{3}$$ is in its simplest form.

Thus, $$3\dfrac{1}{3}$$ is the answer.

Hence, option (B) is correct.

### Convert $$\dfrac{10}{3}$$ into a mixed number.

A

$$3\dfrac{1}{2}$$

.

B

$$3\dfrac{1}{3}$$

C

$$2\dfrac{1}{3}$$

D

$$5\dfrac{1}{2}$$

Option B is Correct

# Conversion of Mixed Number into Decimal

• 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, $$4\dfrac{1}{5}=4.2$$  #### Which one of the following options has the same value as $$7\dfrac{2}{8}$$?

A $$1.25$$

B $$3.89$$

C $$7.25$$

D $$1.20$$

×

Given : $$7\dfrac{2}{8}$$

In $$7\dfrac{2}{8}$$$$7$$ is the whole number and $$\dfrac{2}{8}$$ is the fraction part.

$$\dfrac{2}{8}$$ is not in its simplest form.

The G. C. F. of $$2$$ and $$8$$ is $$2$$

$$\dfrac{2\div2}{8\div2}=\dfrac{1}{4}$$

The fraction which we obtained is $$\dfrac{1}{4}$$.

$$1\space and \space 4$$ do not have any common factor other than $$1$$.

$$\therefore$$ $$\dfrac{1}{4}$$ is in its simplest form.

Converting the fraction part $$\dfrac{1}{4}$$ into decimal.

Since $$\dfrac{1}{4}$$ does not have base ten value in its denominator, so divide the numerator by the denominator.

Thus, $$\dfrac{1}{4}=0.25$$ The whole number goes to the left of the decimal point.

$$\therefore \space 7\dfrac{2}{8}=7.25$$

Hence, option (C) is correct.

### Which one of the following options has the same value as $$7\dfrac{2}{8}$$?

A

$$1.25$$

.

B

$$3.89$$

C

$$7.25$$

D

$$1.20$$

Option C is Correct