Mathematics · Quantitative Aptitude

Real Number System

261 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
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

The rational number lies between $\dfrac{3}{7}$ and $\dfrac{2}{3}$ is

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

Converting the fractions to decimals or common denominators helps locate a rational number between 3/7 (approx 0.428) and 2/3 (approx 0.667). Option B, 4/7, is approximately 0.571, which correctly lies between these two values.

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 $\frac {1}{3} ,\frac {7}{24}, \frac {13}{48}, \frac {1}{4}$

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 $\displaystyle -\dfrac{3}{4}$ and $\displaystyle \dfrac{1}{2}.$

  1. $\dfrac{-7}{16}, \dfrac{-1}{8}, \dfrac{9}{16}$
  2. $\dfrac{-15}{16}, \dfrac{-1}{8}, \dfrac{3}{16}$
  3. $\dfrac{-7}{16}, \dfrac{-1}{8}, \dfrac{3}{16}$

  4. none of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
A rational number between two numbers $ a $ and $ b = \dfrac {(a + b)}{2} $ 

So,
a rational number between $ - \dfrac {3}{4} $ and $ \dfrac {1}{2} $
$= \dfrac {-\dfrac {3}{4} + \dfrac {1}{2}}{2} = - \dfrac {1}{8} $

Now,
another rational number between $ - \dfrac {3}{4} $ and $ - \dfrac {1}{8} $
$=\dfrac {- \dfrac {3}{4} - \dfrac {1}{8}}{2} = - \dfrac {7}{16} $ 

Another rational number between $ - \dfrac {1}{8} $ and $ \dfrac {1}{2} =$

$\dfrac { - \dfrac {1}{8} + \dfrac {1}{2}} {2} =  \dfrac {3}{16} $ 

Hence, required three rational numbers between $ - \dfrac {3}{4} $ and $  \dfrac {1}{2} $ are $ - \dfrac {3}{4}, - \dfrac {7}{16},  - \dfrac {1}{8}, \dfrac {3}{16}, \dfrac {1}{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

The rational number lying between the numbers $\displaystyle \frac{1}{3}$ and $\displaystyle \frac{3}{4}$ are

  1. $\displaystyle \frac{97}{300}$,$\displaystyle \frac{299}{500}$
  2. $\displaystyle \frac{99}{300}$,$\displaystyle \frac{301}{400}$
  3. $\displaystyle \frac{95}{300}$,$\displaystyle \frac{301}{400}$
  4. $\displaystyle \frac{117}{300}$,$\displaystyle \frac{287}{400}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To insert rational numbers between $2$ numbers, we will arrange the options and check if they are in ascending order.
$\dfrac { 1 }{ 3 } { ? }\dfrac { 117 }{ 300 } \ \Longrightarrow 300<351\ \qquad \dfrac { 3 }{ 4 } { ? }\dfrac { 287 }{ 400 } \ \Longrightarrow 1200>1148$
They are in ascending order, i.e., $\dfrac { 1 }{ 3 } ,\dfrac { 117 }{ 300 } ,\dfrac { 287 }{ 400 } ,\dfrac { 3 }{ 4 } $
From the given options only option $D$ satisfies this condition. Hence, $D$ is the answer.

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

Let a, b, c be positive integers such that $\frac {a\sqrt 2+b}{b\sqrt 2+c}$ is a rational number, then which of the following is always an integers?

  1. $\frac {2a^2+b^2}{2b^2+c^2}$
  2. $\frac {a^2+b^2-c^2}{a+b-c}$
  3. $\frac {a^2 _2b^2}{b^2+2c^2}$
  4. $\frac {a^2+b^2+c^2}{a+c-b}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For (a*sqrt(2)+b)/(b*sqrt(2)+c) to be rational, the irrational parts must cancel. This implies a/b = b/c, so b^2 = ac. Testing option D: (a^2+b^2+c^2)/(a+c-b) is a standard algebraic identity related to this condition.

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 of these is true?
$(I)$ $5\sqrt {3}$ is not a rational number
$(II)$ $1$ is not the cube of a rational number
$(III)$ If a is rational and $n$ is an integer greater than $1$, then ${a}^{n}$ is rational.

  1. $I$ and $II$
  2. $II$ and $III$
  3. $III$ and $I$

  4. all three
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

(I) In $5\sqrt{3}$

5 ia s rational number and $\sqrt{3}$ is an Irrational number
As we know, The product of a rational and irrational number is an irrational number.
So, $5\sqrt{3}$ is not a rational number.
Hence, the option (I) is true
(II) 1 is a rational number 
and cube of 1 is 1 only, which is a rational number
Hence the option (II) is False
(III) We know that product of two rational number is always a rational number
Hence if a is a rational number and n is greater than one 
Then,
a2 = a x a is a rational number.

a3 = a2 x a is a rational number,

a4 = a3x a is a rational number,

......

......

∴ an = an-1 x a is a rational number.

So, the option (III) is true

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 of the following represents a rational number between $-6$ and $-7$?

  1. $\dfrac {-6 - 7}{2}$
  2. $\dfrac {-6 + 7}{2}$
  3. $\dfrac {6 + 7}{2}$
  4. $-6 - 7$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Mean $= \dfrac {(-6) + (-7)}{2} = \dfrac {-6 -7}{2}$.

Mean of two numbers always lies between the two numbers.
So, answer is option $A.$

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 between $\dfrac {-9}{10}$ and $\dfrac {4}{5}$ is:

  1. $\left (\dfrac {-9}{10} + \dfrac {4}{5}\right ) \times \dfrac {1}{2}$
  2. $\left (\dfrac {-9}{10} - \dfrac {4}{5}\right ) + \dfrac {1}{2}$
  3. $\left (\dfrac {-9}{10} + \dfrac {4}{5}\right ) \times 2$

  4. All above are correct
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Mean of two numbers always lies between the two numbers.

Mean 

$= \dfrac{\left (\dfrac {-9}{10} + \dfrac {4}{5}\right )}{2}$

$= \left (\dfrac {-9}{10} + \dfrac {4}{5}\right )\times \dfrac {1}{2}$.

So, answer is option $A.$
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 of the following rational numbers lies between $\dfrac {3}{4}$ and $\dfrac {13}{8}$?

  1. $\dfrac {11}{16}$
  2. $\dfrac {12}{16}$
  3. $\dfrac {19}{16}$
  4. $\dfrac {26}{16}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$Mean = \dfrac{\left (\dfrac {3}{4} + \dfrac {13}{8}\right )}{2} = \dfrac {6 + 13}{8} \times \dfrac {1}{2} = \dfrac {19}{16}$.

Mean of two numbers always lies between the two numbers.
So, answer is option $C.$