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 fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

Identify a rational number between $\sqrt{2}$ and $\sqrt{3}$.

  1. $\dfrac{\sqrt{2}.\sqrt{3}}{2}$
  2. $1.5$
  3. $1.8$
  4. $\dfrac{\sqrt{2}+\sqrt{3}}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

As $\sqrt 6$ is irrational therefore option A is wrong.

$1.5$ is rational and it lies between $\sqrt 2$ & $\sqrt 3$ hence Option B is correct.
$1.8$ is rational but it doesn't lies between $\sqrt {2}$ & $\sqrt 3$ implies option C is wrong.
As sum of $\sqrt 2$ & $\sqrt 3$ is irrational therefore option D is also wrong.

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

Which are three rational numbers between $-2$ and $-1$?

  1. $\dfrac { -1 }{ 2 } ,\dfrac { -1 }{ 3 } ,\dfrac { -1 }{ 5 } $
  2. $\dfrac { -3 }{ 2 } ,\dfrac { -7 }{ 4 } ,\dfrac { -5 }{ 4 } $
  3. $\dfrac { -12 }{ 5 } ,\dfrac { -22 }{ 5 } ,\dfrac { 12 }{ 5 } $
  4. $\dfrac { 3 }{ 2 } ,\dfrac { 7 }{ 4 } ,\dfrac { 5 }{ 4 } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In option B,


$\dfrac{-3}{2} = -1.5$

$\dfrac{-7}{4} = -1.75$

$\dfrac{-5}{4} = -1.25$

All these numbers lie in between $(-2,-1)$

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

The rational number between the pair of number $\dfrac{1}{2}$ and $\sqrt 1$ is:

  1. $\dfrac{9}{4}$
  2. $\dfrac{3}{4}$
  3. $\dfrac{5}{4}$
  4. $\dfrac{7}{4}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The rational number between $\dfrac12$ and $\sqrt1$ :

Since, $\sqrt1=1$
So. the rational number between $\dfrac12$ and $1=\dfrac12\times \left(\dfrac12+1\right)$
$=\dfrac12 \times \dfrac32$
$=\cfrac34$
So, $B$ is the correct option.

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

The rational number which is not lying between $\displaystyle\frac{5}{16}$ and $\displaystyle\frac{1}{2}$ is _________.

  1. $\displaystyle\frac{3}{8}$
  2. $\displaystyle\frac{7}{16}$
  3. $\displaystyle\frac{1}{4}$
  4. $\displaystyle\frac{13}{32}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We know $\dfrac{5}{16} =0.3125$

and $\dfrac{1}{2}= 0.5$
Option A: $\dfrac{3}{8} =0.375$
lying between the gven numbers

Option B: $\dfrac{7}{16}= 0.4375$
lying between the given numbers.

Option C: $\dfrac{1}{4}=0.25$
NOT lying between the given numbers.

Option D: $\dfrac{13}{32}=0.40625$
lying between the given numbers.
So, option C is correct.

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

A rational number lie 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 of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We know $\dfrac{1}{4} = 0.25$

and $\dfrac{1}{3}=0.3333333$

Option A: $\dfrac{7}{24}$
$=0.291666$
lies between the given numbers

Option B: $ 0.29$
lies between the given numbers

Option C: $\dfrac{13}{48}$
$=0.27083333$
lies between the given numbers.

All the options are correct.

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

Number of rational numbers between $15$ and $18$ is:

  1. infinite

  2. finite

  3. zero

  4. one
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Between any two rational numbers we can find infinitely many rational numbers. 

Thus, number of rational numbers between $15$ and $18$ is infinite.
Hence, the answer is infinite.

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

A rational number -2/3 ______ .

  1. Lies to the left side of 0 on the number line.

  2. Lies to the right side of 0 on the number line.

  3. It is not possible to represent on the number line.

  4. Cannot be determined on which side the number lies.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

rational number always lies on the line.

 this rational number is $\dfrac{-2}{3}$ which is negative  hence it is always lies to left side of $0$ on the number line.
hence option $A$ is correct.

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers
Among the following 
$-\frac{3}{2},-1,3,0,\frac{1}{2}$
find the rational numbers less than $2.$
  1. $0$
  2. $-\frac{3}{2}$
  3. $-1$
  4. $\frac{1}{2}$
Reveal answer Fill a bubble to check yourself
A,B,C,D Correct answer
Explanation

$\dfrac{-3}{2},-1,3,0,\dfrac{1}{2}$


$-1.5,-1,3,0,0.5$


$\implies $ Among five rational numbers $-1.5,-1,0,0.5$ are lesser than $2$ expect $3$.


All options are correct.

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

Choose the rational number which does not lie between rational numbers $-\dfrac{2}{5}$ and $-\dfrac{1}{5}$.

  1. $-\dfrac{1}{4}$
  2. $-\dfrac{3}{10}$
  3. $\dfrac{3}{10}$
  4. $-\dfrac{7}{20}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a rational number to lie between $\dfrac{-2}{5}$ and $\dfrac{-1}{5}$,it should be less than $\dfrac{-1}{5}$ and greater than $\dfrac{-2}{5}$.
Now,$\dfrac{3}{10}$ is not less than $\dfrac{-1}{5}$.
So,$\dfrac{3}{10}$ does not lie between $\dfrac{-1}{5}$ and $\dfrac{-2}{5}$.

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

Choose the rational number which does not lie between rational numbers $\dfrac{3}{5}$ and $\dfrac{2}{3}$.

  1. $\dfrac{46}{75}$
  2. $\dfrac{47}{75}$
  3. $\dfrac{49}{75}$
  4. $\dfrac{50}{75}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

All the options have denominator $75$. Hence, let us convert into equivalent fractions having denominator $75$. 
$\dfrac{3}{5} $ $=\dfrac{3\times 15}{5\times 15} $ $=\dfrac{45}{75}$

$\dfrac{2}{3}$ $=\dfrac{2\times 25}{3\times 25}$ $=\dfrac{50}{75}$

Hence, $\dfrac{50}{75}$ does not lie between the given numbers.

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

 Rational numbers between $\displaystyle \frac{3}{8}$ and $\displaystyle \frac{7}{12}$ are

  1. $\displaystyle \frac{3}{8}, \frac{41}{96}, \frac{23}{48}, \frac{7}{12}$
  2. $\displaystyle \frac{3}{8}, \frac{41}{196}, \frac{23}{48}, \frac{7}{12}$
  3. $\displaystyle \frac{3}{8}, \frac{41}{96}, \frac{23}{148}, \frac{7}{12}$

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

A rational number between two numbers $ a $ and $ b = \dfrac {(a +

b)}{2} $
So, a rational number between $\dfrac {3}{8} $ and $ \dfrac {7}{12}$

$ = \dfrac {\dfrac {3}{8} + \dfrac {7}{12}}{2} = \dfrac {23}{48} $

Now, another rational number between $ \dfrac {3}{8} $ and $ \dfrac {23}{48} $

$= \dfrac {\dfrac {3}{8} + \dfrac {23}{48}}{2} = \dfrac {41}{96} $ 

Hence, required two rational numbers between $\dfrac {3}{8} $ and $ \dfrac {7}{12} $ are $\dfrac {3}{8} ,\dfrac {41}{96}, \dfrac {23}{48}, \dfrac {7}{12}$

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

__________ are rational numbers between between 5 and -2.

  1. $\displaystyle 5, \frac{33}{4},\ \frac{3}{2}, -\frac{1}{4}, -2$
  2. $\displaystyle \frac{13}{4},\ \frac{3}{2}, -\frac{1}{4} $
  3. $\displaystyle 5, \frac{13}{4},\ \frac{13}{2}, -\frac{1}{4}, -2$

  4. none of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A rational number between two numbers $ a $ and $ b = \dfrac {(a + b)}{2} $

So, a rational number between $ 5 $ and $ - 2 = \dfrac

{( 5 - 2 )}{2} = \dfrac {3}{2} $


Now, another rational number between $ 5 $ and $ \dfrac {3}{2} =

\dfrac {( 5 + \dfrac {3}{2})}{2} = \dfrac {13}{4} $

Another rational number between $ \dfrac {3}{2} $ and $ -2 =

\dfrac {( \dfrac {3}{2}) - 2}{2} = -\dfrac {1}{4} $

 $ \therefore \dfrac {13}{4},  \dfrac {3}{2}, - \dfrac {1}{4} $  are the rational numbers between $ 5 $ and $ -2 $

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

________ are rational numbers between $\displaystyle \frac{1}{3}$ and $\displaystyle \frac{1}{4}$

  1. $\displaystyle \frac{1}{3}, \frac{7}{64}, \frac{13}{48}, \frac{1}{4}$
  2. $\displaystyle \frac{1}{3}, \frac{7}{24}, \frac{13}{48}, \frac{1}{4}$
  3. $\displaystyle \frac{1}{3}, \frac{7}{24}, \frac{13}{68}, \frac{1}{4}$

  4. none of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A

rational number between two numbers $ a $ and $ b = \frac {(a +

b)}{2} $

So, a

rational number between $\frac {1}{3} $ and $ \frac {1}{4} = \frac {(\frac {1}{3} + \frac {1}{4})}{2} = \frac {7}{24} $

Now, another rational number between $ \frac {7}{24} $ and $ \frac {1}{4} = \frac {(\frac {7}{24} + \frac {1}{4})}{2} = \frac {13}{48} $



Hence, required two rational numbers between $\frac {1}{3} $ and $ \frac {1}{4} $ are $\frac {1}{3} ,\frac {7}{24}, \frac {13}{48}, \frac {1}{4}$