Mathematics · Quantitative Aptitude

Real Number System

277 Questions

The real number system encompasses rational and irrational numbers, forming the basis of arithmetic and number theory. It is a key area in the quantitative aptitude and mathematics sections of school level and competitive exams. Practice these questions to master the properties and identification of different types of numbers.

Identifying irrational numbersOrdering rational numbersProperties of square rootsPerfect number propertiesRational and irrational products

Real Number System Questions

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

The rational number lying between $\displaystyle \frac{5}{6}$ and $\displaystyle \frac{6}{7}$ is :

  1. $\displaystyle \frac{1}{2}$
  2. $\displaystyle \frac{15}{21}$
  3. $\displaystyle \frac{35}{42}$
  4. $\displaystyle \frac{71}{84}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The rational number lying between $\displaystyle \frac{5}{6}$ and $\displaystyle \frac{6}{7}$ is
 $\displaystyle \frac{1}{2}\left ( \frac{5}{6}+\frac{6}{7} \right )=\frac{1}{2}\left ( \frac{35+36}{42} \right )=\frac{71}{84}$

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

The pair of rational numbers lying between $\displaystyle \frac{1}{4}$ and $\displaystyle -\frac{3}{4}$ is ?

  1. $\displaystyle \frac{262}{1000}$, $\displaystyle \frac{752}{1000}$
  2. $\displaystyle \frac{63}{250}$, $\displaystyle \frac{187}{250}$
  3. $\displaystyle \frac{13}{50}$, $\displaystyle \frac{264}{350}$
  4. $\displaystyle \frac{9}{50}$, $\displaystyle \frac{31}{40}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The rational numbers $\displaystyle \frac{1}{4}$ and $\displaystyle \frac{3}{4}$ can be written
as $\displaystyle \frac{250}{1000}$ and $\displaystyle \frac{750}{1000}$ Therefore $\displaystyle \frac{63}{250}=\frac{252}{1000}$

and $\displaystyle \frac{187}{250}=\frac{748}{1000}$ satisfy this condition

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

Find $9$ rational numbers between $-\displaystyle\frac{1}{9}\;and\;\displaystyle\frac{1}{5}$.

  1. $\displaystyle\frac{-7}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$
  2. $\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{16}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$
  3. $\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{83}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$
  4. $\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Convert the rational numbers into equivalent rational numbers with the same denominator.


LCM of $9\;and\;5$ is $45$.

$-\displaystyle\frac{1}{9}=\displaystyle\frac{-1\times5}{9\times5}=\displaystyle\frac{-5}{45}\;and\;\displaystyle\frac{1}{5}=\displaystyle\frac{1\times9}{5\times9}=\displaystyle\frac{9}{45}$

The integers between $-5\;and\;9$ are
$-4,\,-3,\,-2,\,-1,\,0,\,1,\,2,\,3,\,4,\,5,\,6,\,7,\,8$.

The corresponding rational numbers are $\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{0}{45},\,\displaystyle\frac{1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{6}{45},\,\displaystyle\frac{7}{45},\,\displaystyle\frac{8}{45}$

On selecting any $9$ of them, we get $9$ rational numbers between $-\displaystyle\frac{1}{9}\;and\;\displaystyle\frac{1}{5}$

$\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

State TRUE or FALSE
The three rational number between $\dfrac{1}{3}$ and $\dfrac{1}{2}$ are $\displaystyle\frac{9}{24},\frac{10}{24},\frac{11}{24}$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$x=\frac { 1 }{ 3 } =\frac { 8 }{ 24 } \quad and\quad y=\frac { 1 }{ 2 } =\frac { 12 }{ 24 } ,\ $

$ \frac { 9 }{ 24 } ,\frac { 10 }{ 24 } \ and\ \frac { 11 }{ 24 } are\quad three\quad rational\quad numbers\quad lying\quad between\quad x\quad and\quad y.\quad \quad \ $

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

A rational number between $\displaystyle \frac{1}{4}$ and $\displaystyle \frac{1}{3}$ is

  1. $\displaystyle \frac{7}{24}$
  2. $0.29$
  3. $\displaystyle \frac{13}{48}$

  4. All the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\dfrac{1}{4} = \dfrac{6}{24} = \dfrac{12}{48} = 0.25$


$\dfrac{1}{3} = \dfrac{8}{24} = \dfrac{16}{48} = 0.33$

From this, we can see $\dfrac{7}{24} , \dfrac{13}{48}, 0.29$ all lie between $\dfrac{1}{4}$ and $\dfrac{1}{3}$

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

Find $3$ rational numbers between $0$ and $1$.

  1. $\displaystyle\frac{3}{2},\,\displaystyle\frac{1}{4}$ and $\displaystyle\frac{3}{4}$
  2. $\displaystyle\frac{1}{2},\,\displaystyle\frac{1}{4}$ and $\displaystyle\frac{3}{4}$
  3. $\displaystyle\frac{1}{2},\,\displaystyle\frac{5}{4}$ and $\displaystyle\frac{3}{4}$
  4. $\displaystyle\frac{1}{2},\,\displaystyle\frac{1}{4}$ and $\displaystyle\frac{7}{4}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
The mean of $0 $ and $ 1$ is $\cfrac{0+1}{2}=\cfrac{1}{2}$.

The mean of $0 $ and $ \cfrac{1}{2}$ is $\begin{pmatrix}0+\cfrac{1}{2}\end{pmatrix}\div2=\cfrac{1}{2}\div2=\cfrac{1}{2}\times\cfrac{1}{2}=\cfrac{1}{4}$

The mean of $\cfrac{1}{2} $ and $ 1$ is $\begin{pmatrix}\cfrac{1}{2}+1\end{pmatrix}\div2$

$=\begin{pmatrix}\cfrac{1+2}{2}\end{pmatrix}\div2=\cfrac{3}{2}\div2=\cfrac{3}{2}\times\cfrac{1}{2}=\cfrac{3}{4}$.

So, the $3$ rational numbers between $0 $ and $ 1$ are $\cfrac{1}{2},\,\cfrac{1}{4} $ and $ \cfrac{3}{4}$.
Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

Write five rational numbers greater than $-2$.

  1. $-\displaystyle\frac{3}{2},\,\displaystyle\frac{-1}{2},\,-1,\,\sqrt5,\,\displaystyle\frac{1}{2}$
  2. $-\displaystyle\frac{3}{2},\,\displaystyle\frac{-1}{2},\,-1,\,0,\,\displaystyle\frac{1}{2}$
  3. $-\displaystyle\frac{3}{2},\,\displaystyle\frac{-1}{2},\,-\sqrt3,\,0,\,\displaystyle\frac{1}{2}$
  4. $-\displaystyle\frac{5}{2},\,\displaystyle\frac{-1}{2},\,-1,\,0,\,\displaystyle\frac{1}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Five rational numbers greater than $-2$ may be taken as: 
$-\displaystyle\frac{3}{2},\,-1,\,\displaystyle\frac{-1}{2},\,0,\,\displaystyle\frac{1}{2}$
There can be many more such rational numbers.

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

There are ..........  rational numbers between two rational numbers. 

  1. infinite

  2. two

  3. one

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

There are infinite rational numbers between two rational numbers.


For example take two fractions  $\dfrac{3}{5},\dfrac{4}{5}$


we can have infinite rationals like $\dfrac{3.1}{5},\dfrac{3.2}{5},\dfrac{3.3}{5}......$  betwen these two given rationals.

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

Which of the following rational numbers lies between $\dfrac {3}{2}$ and $4$ ?

  1. $\dfrac {1}{2}$
  2. $3$
  3. $\dfrac {8}{2}$
  4. $\dfrac {9}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We have to find a rational number between $\dfrac {3}{2}$ and $4$. The L.C.M. of the denominators of both numbers is $2$.
$\therefore \dfrac {3}{2} = \dfrac {3}{2}$ and $4\times \dfrac {2}{2} = \dfrac {8}{2}$.
$\therefore$ From the above given options, only $\dfrac {6}{2}$
i.e. $=3$ lies between $\dfrac {3}{2}$ and $4$.

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

The rational number between $\cfrac{1}{2}$ and $0.6$ is

  1. $\cfrac{1}{4}$
  2. $\cfrac{3}{4}$
  3. $\cfrac{21}{40}$
  4. $\cfrac{33}{100}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\dfrac{1}{2}=0.5$

$\therefore 0.5$ and $0.6$ can be written as 
$\dfrac{5}{10}$ and $\dfrac{6}{10}$
Multiplying the denominator and numerator by $4$ we get,
$\dfrac{5}{10}\times 4=\dfrac{20}{40}$ and
$\dfrac{6}{10}\times 4=\dfrac{24}{40}$
Number between $\dfrac{20}{40}$ and $\dfrac{24}{40}$ is $\dfrac{21}{40}$

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

There exists ....... number of rational numbers between $\dfrac {2}{5}$ and $\dfrac {4}{5}.$

  1. $0$
  2. $1$
  3. $5$

  4. infinite
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

There exists infinite number of rational numbers between any two rational numbers. i.e. in this case between $\dfrac {2}{5}$ and $\dfrac {4}{5}$.

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

Which of the following rational numbers lies between $0$ and $-1$?

  1. $0$
  2. $-1$
  3. $\dfrac {-1}{4}$
  4. $\dfrac {1}{4}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Clearly, $0$ and $-1$ cannot lie between $0$ and $-1$. 

Also, 
$0=\dfrac{0}{4}$ and $-1=\dfrac{-4}{4}$
We can clearly see that $\dfrac {-1}{4}$ lies between $0$ and $-1$.

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

If we divide a positive integer by another positive integer, what is the resulting number?

  1. It is always a natural number

  2. It is always an integer

  3. It is a rational number

  4. It is an irrational number
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

When a positive integer is divided by another positive integer will yield a rational number