Informative line

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.

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.

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, $$\dfrac{1}{11}=0.90909\,.....$$

$$=0.\overline{09}$$

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.

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 $$0.4.$$

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 $$0.5.$$

$$0.\overline7$$ 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 $$0.8$$.

Hence, option (B) is correct.

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

A
B
C
D

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.

$$=-2.66$$

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 $$6.05=6\dfrac{1}{20}$$

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 $$-9$$.

$$\therefore\;-11<-9$$

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:

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

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