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Ordering Of Fractions

Conversion of Fractions into Terminating and Repeating Decimals

  • Fractions can be converted into decimals using division method.

For this, we follow the given steps:

  • The fraction should be in its simplest form.
  • To convert a fraction into a decimal, divide the numerator by the denominator.

When we divide, we get two types of decimals, which are as follows:

(1) Terminating decimals: The decimals that end with a finite (limited) number of digits are known as terminating decimals.

For example: \(0.8,\;0.25,\;0.46374\) etc.

NOTE : The fractions having the denominators that can be written in the form of powers of \(2\;\text{[ example: 4, 8, 16....]}\) or \(5\;\text{[ example: 5, 25, 125...]}\) result in terminating decimals.

  • It means that the denominator should have only \(2\) or \(5\) or both as its factors.
  • For example : \(\dfrac{5}{8}= 6.25\),
  • Here, 8 has 2 as a factor.

(2) Repeating decimals: The decimals which do not end but one or more digits keeps on repeating are known as repeating decimals or non-terminating but repeating decimals.

  • We can also write a repeating decimal by using a 'bar' \((-)\).

Here, 'bar' represents the repetition.

For example: \(0.6262...=0.\overline{62}\)

NOTE : We can only put the bar over the digits which repeat.

For example: \(0.888=0.\overline8\)

\(2.1575757...=2.1\overline{57}\)

(3) Non-terminating and non-repeating decimals:

The decimals which neither end with a finite number of digits nor do they repeat are called non-terminating and non-repeating decimals.

For example: \(2.42152434...\)

NOTE: Those square roots which are not perfect square numbers, always give non-terminating and non-repeating decimals.

For example: \(\sqrt 3=1.732.....\)

  • A decimal is a rational number only if it is-
  1. terminating or
  2. repeating
  • A decimal is an irrational number if it is non-terminating and non-repeating.

Illustration Questions

In an interview, one candidate is selected out of \(9\) candidates for the job. Which decimal is equivalent to the fraction of candidates selected?

A \(0.\overline2\)

B \(0.\overline1\)

C \(0.\overline4\)

D \(0.\overline5\)

×

Given: 

\(1\) candidate is selected from \(9\) candidates, i.e. \(\dfrac{1}{9}\)

\(\dfrac{1}{9}\) is already in its simplest form.

The denominator \(9\) cannot be written in the form of powers of \(2\) or \(5\) or in other words, we can say that \(9\) does not have \(2\) or \(5\) or both as its factors.

\(\therefore\) \(\dfrac{1}{9}\) results in a repeating decimal.

Now, convert \(\dfrac{1}{9}\) into a decimal.

Divide numerator by the denominator,

\(1\div9=0.111.........\)

In the quotient, \(1\) is repeating.

So, we make a horizontal line above \(1\).

Thus, \(\dfrac{1}{9}\) can be written as \(0.\overline1\)

Hence, option (B) is correct.

In an interview, one candidate is selected out of \(9\) candidates for the job. Which decimal is equivalent to the fraction of candidates selected?

A

\(0.\overline2\)

.

B

\(0.\overline1\)

C

\(0.\overline4\)

D

\(0.\overline5\)

Option B is Correct

Repeating Decimal into Fraction

  • Repeating decimals are those in which one or more digits keep on repeating themselves.

For example: \(1.2323.....\)

\(0.\overline4\)

  • To understand the conversion of repeating decimals into fractions, consider the following procedure:

For example: Converting \(0.\overline{09}\) into a fraction.

Step 1. Consider the decimal as a variable.

Let \(x=0.\overline{09}\)         \(.......(1)\)

Step 2. Multiply both the sides of the equation by \(10\) if one digit repeats, by \(100\) if two digits repeat and so on.

In \(0.\overline{09}\), two digits are repeating every time.

\(\therefore\) We multiply both the sides of equation \((1)\) by \(100\).

\(x=0.\overline{09}\)

\(\Rightarrow x=0.090909....\)

\(\Rightarrow100×x=100×(0.0909....)\)

\(\Rightarrow100x=09.0909.....\)

\(\Rightarrow100x=9.0909.....\)

We can write \(9.0909.....=9+0.0909......\)

\(\therefore\;\;100x=9+0.0909.....\)

\(\Rightarrow100x=9+0.\overline{09}\;\;\;\;\;\;\;\;..........(2)\)

Putting \(x\) in place of \(0.\overline{09}\) in equation \((2)\) from equation \((1)\).

\(100x=9+x\)

Step 4. Solving the equation to find the value of \(x\).

\(100x=9+x\)

Subtracting \(x\) from both the sides,

\(\Rightarrow100x-x=9+x-x\)

\(\Rightarrow(100-1)x=9\)                 \((x-x=0)\) 

\(99x=9\)

Dividing both the sides by \(99\)

\(\Rightarrow\dfrac{99x}{99}=\dfrac{9}{99}\)

\(\Rightarrow x=\dfrac{1}{11}\)                               \(\left( \dfrac{9\div9}{99\div9}=\dfrac{1}{11} \right)\) 

Therefore, the repeating decimal \(0.\overline{09}=\dfrac{1}{11}\)

Illustration Questions

What is the fraction form of \(0.\overline7\)?

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

B \(\dfrac{3}{8}\)

C \(\dfrac{7}{9}\)

D \(\dfrac{8}{5}\)

×

Given: \(0.\overline7\)

Consider the decimal as a variable.

Let 

\(x = 0.\overline7\)          \(.......(1)\)

In \(0.\overline7\), one digit is repeating every time.

\(\therefore\) We multiply both the sides of equation \((1)\) by \(10\).

\(x=0.\overline7\)

\(\Rightarrow x=0.777......\)

\(10×x=10×0.777....\)

\(10x=7.777....\)              \(....(2)\)

We can write \(7.777=7+0.7777\)

\(\therefore\) Equation \((2)\) can be written as 

\(10x=7+0.777\)

\(10x=7+0.\overline7\)              \(......(3)\)

Substitute the value of \(0.\overline7\) in equation \((3)\) from equation \((1)\), i.e.

\(\because\;\;\; x=0.\overline7\)

\(\therefore\) In equation \(3\)

\(10x=7+x\)            \(.......(4)\)

Solving equation \((4)\) to find the value of \(x\), i.e.

\(10x=7+x\)            \(........(4)\)

Subtracting \(x\) from both the sides,

\(10x-x=7+x-x\)

\((10-1)x=7\)

\(9x=7\)

Dividing both sides by \(9\)

\(\dfrac{9x}{9}=\dfrac{7}{9}\)

\(x=\dfrac{7}{9}\)

Therefore, the repeating decimal,

\(0.\overline7=\dfrac{7}{9}\)

Hence, option (C) is correct.

What is the fraction form of \(0.\overline7\)?

A

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

.

B

\(\dfrac{3}{8}\)

C

\(\dfrac{7}{9}\)

D

\(\dfrac{8}{5}\)

Option C is Correct

Comparison of Fractions through Representation on a Number Line

  • Consider two fractions as two points on the number line.

  • On a number line, a fraction which is placed to the left is smaller than the fraction which is placed to the right side.

For example: Consider \(\dfrac{1}{4}\) and \(\dfrac{5}{4}\).

\(\dfrac{1}{4}\) is placed to the left of \(\dfrac{5}{4}\).

\(\therefore\) \(\dfrac{1}{4}<\dfrac{5}{4}\)

Note: While comparing two or more fractions, make sure that each one of them has the same denominator.

Illustration Questions

Two points namely, \(P\) and \(R\), are plotted on the given number line. Determine the values of \(P\)  and \(R\)  in the fraction form and choose their relation from the following options.

A \(P\)= \(\dfrac{9}{6}\), \(R\)= \(\dfrac{5}{6}\), \(P>R\)

B \(P\)= \(\dfrac{5}{6}\), \(R\)= \(\dfrac{9}{6}\), \(P>R\)

C \(P\)= \(\dfrac{9}{6}\), \(R\)= \(\dfrac{5}{6}\), \(R>P\)

D \(P\)= \(\dfrac{5}{6}\), \(R\)= \(\dfrac{9}{6}\), \(R>P\)

×

Given number line:

image

We can observe that each interval from 0 to 1 and 1 to 2, is subdivided into 6 equal parts.

\(\therefore\) Each small segment = \(\dfrac{1}{6}\)

 

 

Thus, the value of point \(R\) = \(\dfrac{5}{6}\)

and the value of point \(P\) = \(\dfrac{9}{6}\)

image

On a number line, a fraction which is placed to the left is smaller than the fraction which is placed to the right side.

\(\dfrac{5}{6}\) is placed to the left of \(\dfrac{9}{6}\).

\(\therefore\) \(\dfrac{9}{6}>\dfrac{5}{6}\)

image

Thus, \(P>R\)

Hence, option (A) is correct.

Two points namely, \(P\) and \(R\), are plotted on the given number line. Determine the values of \(P\)  and \(R\)  in the fraction form and choose their relation from the following options.

image
A

\(P\)= \(\dfrac{9}{6}\), \(R\)\(\dfrac{5}{6}\)\(P>R\)

.

B

\(P\)\(\dfrac{5}{6}\), \(R\)\(\dfrac{9}{6}\), \(P>R\)

C

\(P\)\(\dfrac{9}{6}\), \(R\)\(\dfrac{5}{6}\), \(R>P\)

D

\(P\)\(\dfrac{5}{6}\), \(R\)\(\dfrac{9}{6}\), \(R>P\)

Option A 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.

  • 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 \(\dfrac{10}{3}\) 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

Ordering of Different Forms (Fraction, Mixed Fraction and Decimal)

  • Ordering of fractions is done by comparing them.
  • To understand it, consider an example.

Arrange \(2\dfrac{1}{3}\)\(\dfrac{3}{4}\) and \(2\) from greatest to least.

First, make sure that all the numbers are in the same form. 

So, we convert them into fraction form.

Here, \(2\dfrac{1}{3}=\dfrac{2×3+1}{3}=\dfrac{7}{3}\) and \(2=\dfrac{2}{1}\)

  • Now, arranging \(\dfrac{7}{3}\)\(\dfrac{3}{4}\) and \(\dfrac{2}{1}\).

Step 1: Make sure that the denominators are equal, if they are not, convert them into like fractions.

Here, the denominators are not equal. 

So, we convert them into like fractions.

L.C.M. of \(4\)\(3\) and \(1=12\)

L.C.M. becomes the least common denominator.

\(\dfrac{3}{4}×\dfrac{3}{3}=\dfrac{9}{12}\)

\(\dfrac{7}{3}×\dfrac{4}{4}=\dfrac{28}{12}\) and \(\dfrac{2}{1}×\dfrac{12}{12}=\dfrac{24}{12}\)

Step 2: Compare the numerators.

The fraction with larger numerator has the higher value than the others.

Here,  \(28>24>9\)

So, \(\dfrac{28}{12}>\dfrac{24}{12}>\dfrac{9}{12}\)

Step 3. Arranging the original numbers from greatest to least.

 

\(2\dfrac{1}{3},\;2,\;\dfrac{3}{4}\)

Illustration Questions

Carl, Aron, and Sam have to solve one problem in \(2\) minutes individually. Carl, Aron, and Sam solve the problem in \(1.2\) minutes, \(1\dfrac{1}{2}\) minutes and \(\dfrac{3}{4}\) minutes respectively. Arrange them according to the time taken by them in the order from least to greatest.

A \(\text{Carl, Aron, Sam }\)

B \(\text{Sam, Carl, Aron}\)

C \(\text{Aron, Carl, Sam}\)

D \(\text{Carl, Sam, Aron}\)

×

In this problem, we need to order the time taken from least to greatest.

To do this, all the numbers should be in the same form.

So, we convert them into fraction form.

Here, \(1.2=\dfrac{12}{10}\) 

\(=\dfrac{12\div2}{10\div2}=\dfrac{6}{5}\)

and

 \(1\dfrac{1}{2}=\dfrac{2×1+1}{2}\)

\(=\dfrac{3}{2}\)

Make the denominators equal.

L.C.M. of \(5,2\) and \(4=20\)

L.C.M. becomes the least common denominator.

\(\dfrac{6}{5}×\dfrac{4}{4}=\dfrac{24}{20}\;,\)

\(\dfrac{3}{2}×\dfrac{10}{10}=\dfrac{30}{20}\)

and \(\dfrac{3}{4}×\dfrac{5}{5}=\dfrac{15}{20}\)

Among \(\dfrac{24}{20},\dfrac{30}{20}\) and \(\dfrac{15}{20}\),

\(30\) is greater than \(24\) and \(24\) is greater than \(15\).

So, the order from least to greatest is,

\(=\dfrac{15}{20}<\dfrac{24}{20}<\dfrac{30}{20}\)

Original numbers in order,

\(=\dfrac{3}{4}<1.2<1\dfrac{1}{2}\)

Now, ordering the time taken from least to greatest, 

Sam, Carl, Aron

Hence, option (B) is correct.

Carl, Aron, and Sam have to solve one problem in \(2\) minutes individually. Carl, Aron, and Sam solve the problem in \(1.2\) minutes, \(1\dfrac{1}{2}\) minutes and \(\dfrac{3}{4}\) minutes respectively. Arrange them according to the time taken by them in the order from least to greatest.

A

\(\text{Carl, Aron, Sam }\)

.

B

\(\text{Sam, Carl, Aron}\)

C

\(\text{Aron, Carl, Sam}\)

D

\(\text{Carl, Sam, Aron}\)

Option B is Correct

Comparison of Fractions having Same Denominators

  • To compare two or more fractions, we should follow the given steps:
  • Let us consider an example.

Which one is greater between \(\dfrac{2}{5}\) and \(\dfrac{4}{5}\)?

Step 1: If the denominators are same, compare the numerators.

  • The denominator \(5\) is same in both the fractions.
  • Numerator of \(\dfrac{2}{5}=2\)

Numerator of \(\dfrac{4}{5}=4\)

Step 2: The fraction with larger numerator is greater than the fraction with smaller numerator.

\(4\) is greater than \(2\).

Thus, \(\dfrac{4}{5}\) is greater than \(\dfrac{2}{5}\).

Step 3: Write the result using greater than (>), less than (<) or equal to (=) signs.

\(4>2\)

\(\therefore\;\dfrac{4}{5}>\dfrac{2}{5}\)

  • Comparison is possible only when we convert the fractions into like fractions.
  • To compare two or more fractions having different denominators, we should follow the given steps:
  • Let us consider an example.

Compare \(\dfrac{3}{4}\) and \(\dfrac{2}{3}\).

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

L.C.M of \(4\) and \(3,\) 

Multiples of \(4=4,\;8,\;12\,...\)

Multiples of \(3=3,\;6,\;9,\;12\,...\)

L.C.M \(=12\)

Step 2: L.C.M becomes the lowest common denominator.

L.C.D \(=12\)

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

Equivalent fraction of  \(\dfrac{3}{4}=\dfrac{3×3}{4×3}=\dfrac{9}{12}\)

Equivalent fraction of  \(\dfrac{2}{3}=\dfrac{2×4}{3×4}=\dfrac{8}{12}\)

Step 4: Compare the fractions having same denominators.

Compare \(\dfrac{9}{12}\) and \(\dfrac{8}{12}\).

\(9>8\)

\(\therefore\;\dfrac{9}{12}>\dfrac{8}{12}\)

Step 5: Rewrite the original fractions for the answer.

\(\dfrac{3}{4}>\dfrac{2}{3}\)

Illustration Questions

Which one is smaller between  \(\dfrac{3}{2}\) and \(\dfrac{1}{2}\) ?

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

B \(\dfrac{3}{2}\)

C Both are equal

D

×

Given fractions: \(\dfrac{3}{2}\) and \(\dfrac{1}{2}\)

The denominator \(2\) is same in both the fractions.

Numerator of \(\dfrac{3}{2}=3\)

Numerator of \(\dfrac{1}{2}=1\)

\(1\) is smaller than \(3\)  

or  \(1<3\)

\(\therefore\;\dfrac{1}{2}\)  is smaller than  \(\dfrac{3}{2}\)

or   \(\dfrac{1}{2}<\dfrac{3}{2}\)

Hence, option (A) is correct.

Which one is smaller between  \(\dfrac{3}{2}\) and \(\dfrac{1}{2}\) ?

A

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

.

B

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

C

Both are equal

D

Option A is Correct

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