Informative line

Conversion Of Rational Numbers

Converting a Fraction, a Mixed Number, a Ratio and a Percent into a Rational Number

  • A number that can be written in the form of \(\dfrac{a}{b}\), is called rational number, where \(a\) and \(b\) are integers and  \(b\neq\) zero.

Rational numbers can be written in different forms.

(1) Rational numbers as fractions:

A rational number is also a fraction, as it can be written in \(\dfrac{a}{b}\) form, where \(b\ne0\).

For example: \(\dfrac{3}{4}\)

(2) Rational numbers as mixed numbers:

Mixed numbers can be converted into improper fractions.

For example: \(3\dfrac{1}{2}\)

\(=\dfrac{(3×2)+1}{2}=\dfrac{6+1}{2}=\dfrac{7}{2}\)

\(\therefore\;\dfrac{7}{2}\) is a rational number.

(3) Rational numbers as ratio:

A ratio can be written as a fraction.

\(\therefore\) It is also a rational number.

For example: \(3:2\) \(=\dfrac{3}{2}\)

\(\dfrac{3}{2}\) is a rational number.

(4) Rational numbers as percent:

A percent can be changed into a fraction by putting it over \(100\).

For example: \(25\,\%\)

\(=\dfrac{25}{100}\)

\(=\dfrac{25\div25}{100\div25}=\dfrac{1}{4}\)

Thus, \(\dfrac{1}{4}\) is a rational number.

Illustration Questions

Which one of the following is true?

A \(65\,\%=\dfrac{13}{20}\)

B \(3:4=\dfrac{6}{2}\)

C \(-9=\dfrac{1}{9}\)

D \(\dfrac{6}{5}=20\,\%\)

×

Option (A):

Given: \(65\,\%=\dfrac{13}{20}\)

Converting \(65\,\%\) into a fraction by putting it over \(100\),

\(=\dfrac{65}{100}\)

Simplifying the fraction obtained,

\(=\dfrac{65\div5}{100\div5}\)

\(=\dfrac{13}{20}\)

Thus, we can see that

\(65\,\%=\dfrac{13}{20}\)

Hence, option (A) is true.

Option (B):

Given: \(3:4=\dfrac{6}{2}\)

Converting the ratio into a fraction \(3:4\),

\(=\dfrac{3}{4}\)

Thus, we can see that

\(3:4\neq\dfrac{6}{2}\)

Hence, option (B) is false.

Option (C):

Given: \(-9=\dfrac{1}{9}\)

Converting \(-9\) into fraction by putting it over \(1\),

\(=\dfrac{-9}{1}\)

Thus, we can see that

\(-9\neq\dfrac{1}{9}\)

Hence, option (C) is false.

Option (D):

Given: \(\dfrac{6}{5}=20\,\%\)

Converting \(20\,\%\) into fraction by putting it over \(100\),

\(=\dfrac{20}{100}\)

Simplifying the fraction obtained,

\(=\dfrac{20\div20}{100\div20}\)

\(=\dfrac{1}{5}\)

Thus, we can see that

\(\dfrac{6}{5}\neq20\,\%\)

Hence, option (D) is false.

Which one of the following is true?

A

\(65\,\%=\dfrac{13}{20}\)

.

B

\(3:4=\dfrac{6}{2}\)

C

\(-9=\dfrac{1}{9}\)

D

\(\dfrac{6}{5}=20\,\%\)

Option A is Correct

Introducing Decimals as Rational and Irrational Numbers

  • A rational number is in the form of \(\dfrac{a}{b}\) where \(b\neq0\) and, \(a\) and \(b\) are integers.

Example: \(\dfrac{3}{2}\)

  • An irrational number can not be written in the form of \(\dfrac{a}{b}\).

Example: \(\sqrt {15}\)

  • If we need to represent a decimal as a rational or an irrational number, we should know about the types of decimals.

The types are described as follows:

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

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

(2) Repeating decimals:  The decimals which do not end but one or more digits keep on repeating after decimal 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: \(0.6262...=0.\overline{62}\)

  • We can only put the bar over the digits which repeat.

\(0.888=0.\overline8\)

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

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

The decimals which neither end with a finite number of digits nor the digits keep on repeating are called non-terminating and non-repeating decimals.

For example: \(2.42152434...\)

Note : Those squares 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 if it is-
  1. Terminating or
  2. Repeating
  • A decimal is an irrational number if it is non-terminating and non-repeating.

Illustration Questions

Which is NOT true?

A \(\sqrt 3\) is an irrational number.

B \(0.\overline2\) is a rational number.

C \(0.232\overline5\) is a non-terminating and non-repeating decimal.

D \(0.2\overline3\) is a repeating decimal.

×

To choose the correct answer we will check each option.

Option (A):

\(\sqrt 3=1.732\,.....\)

\(\sqrt3\) is a non-terminating and also non-repeating decimal.

\(\therefore\) It is an irrational number.

Hence, option (A) is true.

Option (B):

\(0.\overline2=0.222\,.....\)

\(0.\overline2\) is a non-terminating but a repeating decimal.

\(\therefore\) It is a rational number.

Hence, option (B) is true.

Option (C):

\(0.232\overline5=0.232555\,.....\)

\(0.232\overline5\) is a non-terminating but a repeating decimal.

Hence, option (C) is false.

Option (D):

\(0.2\overline3=0.2333\,.....\)

\(0.2\overline3\) is a repeating decimal.

Hence, option (D) is true.

Which is NOT true?

A

\(\sqrt 3\) is an irrational number.

.

B

\(0.\overline2\) is a rational number.

C

\(0.232\overline5\) is a non-terminating and non-repeating decimal.

D

\(0.2\overline3\) is a repeating decimal.

Option C is Correct

Converting Rational Numbers into Non-Terminating but Repeating Decimals

  • Non-terminating but repeating decimals are the decimals which do not end but one or more digits keep on repeating.
  • We can convert a rational number into a decimal by dividing the numerator by the denominator.
  • If the remainder always repeats or does not come out to be zero, the quotient will be a non-terminating but repeating decimal.
  • Consider the two cases:

Case I: When only \(1\) digit after the decimal repeats.

For example: \(\dfrac{16}{45}\)

Denominator \(45\) cannot be changed into \(10\) or \(100\) or \(1000\,.....\), thus, we should use long division method.

Here, \(25\) is the repeating remainder and \(0.3555\,.....\) is a repeating decimal or repetend.

Thus, \(\dfrac{16}{45}=0.3555\,.....\)

\(=0.3\overline5\)

Case II: When more than \(1\) digit repeat.

For example:

\(\dfrac{1}{11}\)

Here, repeating remainder is \(1\) and \(.090909\,.....\) is a repeating decimal or repetend.

Thus, \(\dfrac{1}{11}=0.090909\,.....\)

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

Here, repeating remainder is \(1\) and \(.090909\,.....\) is a repeating decimal or repetend.

Thus, \(\dfrac{1}{11}=0.90909\,.....\)

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

Illustration Questions

Which one of the decimals has the same value as \(\dfrac{15}{9}\)?

A \(0.67\)

B \(1.\overline{36}\)

C \(1.\overline6\)

D \(1.602\)

×

Convert \(\dfrac{15}{9}\) into a decimal by using long division method.

image

Here, \(6\) is a repeating remainder and \(1.66\,.....\) is a repetend.

Thus, \(\dfrac{15}{9}=1.66...\)

or \(1.\overline6\)

Hence, option (C) is correct.

Which one of the decimals has the same value as \(\dfrac{15}{9}\)?

A

\(0.67\)

.

B

\(1.\overline{36}\)

C

\(1.\overline6\)

D

\(1.602\)

Option C is Correct

Terminating and Non-terminating Decimals on a Number Line

  • We can plot a terminating decimal at the exact place on the number line but in case of non-terminating, it is quite different.
  • Consider the following two cases:

Case 1: Terminating decimals on the number line:

Example: \(0.3\)

\(0.3\) means \(3\) tenths.

\(\therefore\) Draw a number line from \(0\) to \(1\) and divide the interval into \(10\) equal segments.

Thus, count \(3\) segments and plot \(0.3.\)

Case 2: Non-terminating decimals on the number line:

Example: \(0.\overline3\)

\(0.\overline3\) means \(0.333\,...\)

\(\therefore\;0.3<0.\overline3<0.4\), which means \(0.\overline3\) lies in between \(0.3\) and \(0.4\) on the number line.

Thus, plotting \(0.\overline3\) between \(0.3\) and \(0.4.\)

Illustration Questions

Which one of the number lines represents \(0.5\) and \(0.\overline7\) correctly?

A

B

C

D

×

\(0.5\) is terminating decimal and \(0.5\) means \(5\) tenths.

\(\therefore\) Draw a number line from \(0\) to \(1\) and divide the interval into \(10\) equal segments.

Thus, count \(5\) segments and plot \(0.5.\)

image

\(0.\overline7\) is a non-terminating decimal and \(0.\overline7\) means \(0.777\,...\)

\(\therefore\;0.7<0.\overline7<0.8\), which means \(0.\overline7\) lies in between \(0.7\) and \(0.8\) on the number line.

Thus, plotting \(0.\overline7\) between \(0.7\) and \(0.8\).

image

Hence, option (B) is correct.

Which one of the number lines represents \(0.5\) and \(0.\overline7\) correctly?

A image
B image
C image
D image

Option B is Correct

Converting Rational Numbers into Terminating Decimals

  • Terminating decimals are the decimals that end with a finite (limited) number of digits after decimal(point).
  • We can convert a rational number into a decimal by dividing the numerator by the denominator.
  • After dividing, if the remainder comes out to be zero, then the quotient will be a terminating decimal.
  • There are two methods of converting a rational number into a decimal.

I. By Using Long Division Method

For example: \(\dfrac{14}{8}\)

Divide \(14\) by \(8.\)

Thus, \(\dfrac{14}{8}\) gives the quotient as \(1.75\).

Note:

This method can be used in both terminating and non-terminating decimals.

II. By Using Equivalent Fraction Method

  • This method is used:

(i) When we have \(10\) or \(100\) or \(1000\,.....\) (multiples of \(10\)) in the denominator.

(ii) Fractions having the denominators that can be written in the form of powers of  \(2\,[like\,\,4,8,16,...]\) or \(5\,[like\,\,25,625,...]\) result in terminating decimals.

  •  It means that the denominator should have only 2 or 5 or both as its factors. 

For example: \(\dfrac{13}{25}\)

For the multiples of \(10\) in the denominator, we multiply both numerator and the denominator by \(4.\)

\(=\dfrac{13×4}{25×4}\)

\(=\dfrac{52}{100}\)

\(=0.52\)

Thus, \(\dfrac{13}{25}\) can be written as \(0.52.\)

  • This method always gives results as terminating decimals.

Note:

If we have a mixed number to be converted into a decimal, consider the following two methods:

Method 1:

First convert it into an improper fraction and then convert it into a decimal.

For example: \(1\dfrac{1}{2}\)

\(=\dfrac{(2×1)+1}{2}\)

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

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

\(=1.5\)

Method 2:

Keep aside the whole number and convert only the fraction part into a decimal.

For example: \(-2\dfrac{2}{3}=-\left(2+\dfrac{2}{3}\right)\)

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

\(\implies-\dfrac{2}{3}=-0.66\)

Write the whole number and the decimal part together.

\(=-2.66\)

Illustration Questions

Which one of the decimals has the same value as \(\dfrac{41}{20}\)?

A \(2.05\)

B \(0.05\)

C \(-2.05\)

D \(1.25\)

×

Given: \(\dfrac{41}{20}\)

Convert \(\dfrac{41}{20}\) into a decimal by using the equivalent fraction method.

For the multiples of \(10\) in the denominator, we multiply both numerator and the denominator by \(5.\)

\(=\dfrac{41×5}{20×5}\)

\(=\dfrac{205}{100}\)

\(=2.05\)

Thus, \(\dfrac{41}{20}\) can be written as \(2.05.\)

Hence, option (A) is correct.

Which one of the decimals has the same value as \(\dfrac{41}{20}\)?

A

\(2.05\)

.

B

\(0.05\)

C

\(-2.05\)

D

\(1.25\)

Option A is Correct

Converting a Terminating Decimal into a Rational Number

  • As we can convert a rational number into a decimal number, we can also convert a decimal number into a rational number.
  • Here, we are converting a terminating decimal into a rational number.
  • Consider the following two cases:

Case 1: Terminating decimal as a rational number:

For example: \(0.36\)

According to the place value chart, \(0.36\) is thirty-six hundredths.

So, it can be written as-

\(=\dfrac{36}{100}\)

Now, simplify the fraction obtained.

\(=\dfrac{36\div4}{100\div4}\)

\(=\dfrac{9}{25}\)

\(9\) and \(25\) do not have any common factor other than \(1\).

\(\therefore\;\dfrac{9}{25}\) is in its simplest form.

Thus, \(0.36\) can be written as \(\dfrac{9}{25}\) in the rational form.

Case 2: Terminating decimal as mixed numbers:

For example:

\(6.05\)

According to the place value chart, \(6.05\) is six and five hundredths.

So, it can be written as-

\(6.05=6\dfrac{5}{100}\)

\(=6+\dfrac{5}{100}\)

Now, we simplify only the fraction part i.e. \(\dfrac{5}{100}.\)

\(=\dfrac{5\div5}{100\div5}\)

\(=\dfrac{1}{20}\)

Thus,

\(6+\dfrac{5}{100}=6+\dfrac{1}{20}\)

or \(6\dfrac{5}{100}=6\dfrac{1}{20}\)

or \(6.05=6\dfrac{1}{20}\)

Illustration Questions

Which one of the following represents the same value as \(15.25\)?

A \(\dfrac{61}{4}\)

B \(\dfrac{161}{8}\)

C \(15\dfrac{1}{8}\)

D \(\dfrac{45}{26}\)

×

Given: \(15.25\)

According to the place value chart, \(15.25\) is fifteen and \(25\) hundredths.

So, it can be written as-

\(15.25=15\dfrac{25}{100}\)

\(=15+\dfrac{25}{100}\)

Now, we simplify only the fraction part i.e. \(\dfrac{25}{100}.\)

\(=\dfrac{25\div25}{100\div25}\)

\(=\dfrac{1}{4}\)

Thus,

\(15+\dfrac{25}{100}=15+\dfrac{1}{4}\)

\(15\dfrac{25}{100}=15\dfrac{1}{4}\)

or \(15.25=15\dfrac{1}{4}\)

or \(\dfrac{61}{4}\)

Hence, option (A) is correct.

Which one of the following represents the same value as \(15.25\)?

A

\(\dfrac{61}{4}\)

.

B

\(\dfrac{161}{8}\)

C

\(15\dfrac{1}{8}\)

D

\(\dfrac{45}{26}\)

Option A is Correct

Comparison of Two Rational Numbers

  • Negative numbers are always smaller than the positive numbers.
  • Let's consider an example to understand the comparison of two rational numbers.

Example: Compare \(1\dfrac{3}{4}\) and \(\dfrac{5}{12}\)

(1) First, convert them into the same form i.e. rational number.

\(1\dfrac{3}{4}=1+\dfrac{3}{4}\)

\(=\dfrac{(1×4)+3}{4}\)

\(=\dfrac{4+3}{4}\)

\(=\dfrac{7}{4}\)

\(\dfrac{5}{12}\) is already a rational number.

(2) For comparing \(\dfrac{5}{12}\) and \(\dfrac{7}{4}\), convert them into their equivalent fractions having the same denominator.

\(\dfrac{7}{4}=\dfrac{7×3}{4×3}=\dfrac{21}{12}\)

\(\dfrac{5}{12}\) already has the denominator

(3) \(21>5\)

\(\therefore\;\dfrac{21}{12}>\dfrac{5}{12}\) 

Note:

On the number line, the number which is to the left of another number is always smaller than the other, even if it looks bigger.

For example:

\(-11\) is at the left of \(-9\).

\(\therefore\;-11<-9\)

Illustration Questions

Which of the following options correctly represents the relation between \(3.20\) and \(2\dfrac{4}{5}\)?

A \(3.20>2\dfrac{4}{5}\)

B \(3.20<2\dfrac{4}{5}\)

C \(3.20=2\dfrac{4}{5}\)

D \(3.20\geq2\dfrac{4}{5}\)

×

Converting \(3.20\) and \(2\dfrac{4}{5}\) into the same form i.e. decimal,

\(2\dfrac{4}{5}=2+\dfrac{4}{5}\)

\(=\dfrac{(2×5)+4}{5}=\dfrac{10+4}{5}\)

\(=\dfrac{14}{5}=\dfrac{14\div5}{5\div5}\)

\(=\dfrac{2.8}{1}=2.8\)

\(3.20\) is already in decimal form.

Comparing \(3.20\) and \(2.8\),

\(3.20>2.8\)

\(\therefore\;3.20>2\dfrac{4}{5}\)

Hence, option (A) is correct.

Which of the following options correctly represents the relation between \(3.20\) and \(2\dfrac{4}{5}\)?

A

\(3.20>2\dfrac{4}{5}\)

.

B

\(3.20<2\dfrac{4}{5}\)

C

\(3.20=2\dfrac{4}{5}\)

D

\(3.20\geq2\dfrac{4}{5}\)

Option A is Correct

Ordering Rational Numbers, Ratios, Fractions, Decimals, Percents and Whole Numbers

  • Ordering refers to a method of arranging the numbers either from least to greatest or greatest to least.
  • While ordering, first we convert all the numbers into the same form and then compare.
  • Here, we are converting all the forms into the rational numbers.
  • Let us order the following numbers from least to greatest.

\(3:4,\;50\,\%,\;-12.5,\;2^3\) and \(\dfrac{-8}{6}\)

  • Since all the given numbers are in different forms, so first we convert them into the same form, i.e. rational number.

\(3:4\) can be written as  \(\dfrac{3}{4}\)

\(50\,\%\) can be written as \(\dfrac{50}{100}=\dfrac{1}{2}\)

\(-12.5\) can be written as \(\dfrac{-125}{10}=\dfrac{-25}{2}\)

\(2^3\) can be written as \(2^3=\dfrac{8}{1}\)

\(\dfrac{-8}{6}\) can be written as \(\dfrac{-8}{6}=\dfrac{-4}{3}\)

  • Here, the denominators of all the rational numbers are not same.
  • To compare all the rational numbers, we convert them into their equivalent rational numbers such that all have the same denominator.
  • To do so, we calculate L.C.M. of \(4,\;2\) and \(3.\)
  • Thus, \(12\) is the common denominator (L.C.M.)

  • Now converting all the rational numbers into equivalent rational numbers with denominator as \(12\).

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

\(\dfrac{1}{2}=\dfrac{1×6}{2×6}=\dfrac{6}{12}\)

\(\dfrac{-25}{2}=\dfrac{-25×6}{2×6}=\dfrac{-150}{12}\)

\(\dfrac{8}{1}=\dfrac{8×12}{1×12}=\dfrac{96}{12}\)

\(\dfrac{-4}{3}=\dfrac{-4×4}{3×4}=\dfrac{-16}{12}\)

  • Thus, we have numbers as:

\(\dfrac{9}{12},\;\dfrac{6}{12},\;\dfrac{-150}{12},\;\dfrac{96}{12},\;\dfrac{-16}{12}\)

  • Now, we arrange them from least to greatest by comparing their numerators.
  • So, the order is:

\(\dfrac{-150}{12}<\dfrac{-16}{12}<\dfrac{6}{12}<\dfrac{9}{12}<\dfrac{96}{12}\)

  • Thus, the order of original numbers is:

\(-12.5<\dfrac{-8}{6}<50\,\%<3:4<2^3\)

Illustration Questions

In a quiz competition, the correct number of questions of \(3\) students are as follows: For every \(5\) correctly answered question, Carl answered \(1\) question incorrectly. Alex answered \(20\,\%\)  questions correctly. Cooper answered \(\dfrac{3}{5}\) of the questions correctly. Which student answered the maximum number of questions correctly?

A Alex

B Cooper

C Carl

D None of these

×

To determine who answered the maximum questions correctly, we need to compare them. For that, we first need to convert them in the same form.

For every \(5\) correctly answered questions, Carl answered \(1\) questions incorrect.

Here, ratio of correct to incorrect is \(5:1=\dfrac{5}{1}\)

Thus, fraction for correctly answered questions \(=\dfrac{\text{Correct question}}{\text{Total questions}}\)

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

Alex answered \(20\,\%\) questions correctly. That means, the fraction of questions he answered correctly,

\(=20\,\%=\dfrac{20}{100}=\dfrac{1}{5}\)

The fraction of correctly answered questions by Cooper \(=\dfrac{3}{5}\)

Since all the rational numbers have different denominators, so to compare them we first calculate their L.C.Ds.

L.C.D. of \(6\) and \(5=30\)

So, the fractions for correctly answered questions with the same L.C.D. are:

For Carl \(=\dfrac{5×5}{6×5}=\dfrac{25}{30}\)

Alex \(=\dfrac{1×6}{5×6}=\dfrac{6}{30}\)

Cooper \(=\dfrac{3×6}{5×6}=\dfrac{18}{30}\)

Now, on comparing all the rational numbers and ordering them according to their correct answers, we get:

\(\dfrac{25}{30}>\dfrac{18}{30}>\dfrac{6}{30}\)

This means,

Carl > Cooper > Alex

Hence, option (C) is correct.

In a quiz competition, the correct number of questions of \(3\) students are as follows: For every \(5\) correctly answered question, Carl answered \(1\) question incorrectly. Alex answered \(20\,\%\)  questions correctly. Cooper answered \(\dfrac{3}{5}\) of the questions correctly. Which student answered the maximum number of questions correctly?

A

Alex

.

B

Cooper

C

Carl

D

None of these

Option C is Correct

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