Informative line

Conversion And Simplification Of Fractions

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.

Illustration Questions

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.

Illustration Questions

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.

Illustration Questions

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 \(\dfrac{7}{2}=3.5\)

Illustration Questions

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, \(\dfrac{1}{2}=0.5\) 

image

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 \(=1\dfrac {1}{2}\)

Illustration Questions

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.

Illustration Questions

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.

Illustration Questions

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,

image

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, \(4\dfrac{1}{5}=4.2\)

Illustration Questions

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, \(\dfrac{1}{4}=0.25\)

image

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

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