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 exactly in between the numbers $\displaystyle \frac { 1 }{ 5 } $ and $\displaystyle \frac { 1 }{ 3 } $ is

  1. $\displaystyle \frac { 1 }{ 2 } $
  2. $\displaystyle \frac { 1 }{ 4 } $
  3. $\displaystyle \frac { 2 }{ 15 } $
  4. $\displaystyle \frac { 4 }{ 15 } $
  5. $\displaystyle \frac { 8 }{ 15 } $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Required number $=$ $\dfrac{1}{2}\left(\dfrac{1}{5}+\dfrac{1}{3}\right)$

$\Rightarrow \dfrac{1}{2}\left(\dfrac{3+5}{15}\right)$
$\Rightarrow \dfrac{1}{2}\times \dfrac{8}{15}=\dfrac{4}{15}$
Hence, $\dfrac{4}{15}$ is a rational number lying between $\dfrac{1}{5}$ 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

Identify a rational number between $\dfrac {1}{3}$ and $\dfrac {4}{5}$

  1. $\dfrac {1}{4}$
  2. $\dfrac {9}{10}$
  3. $\dfrac {17}{30}$
  4. $\dfrac {7}{10}$
Reveal answer Fill a bubble to check yourself
C,D Correct answer
Explanation

Converting the given fraction in decimal format we get $\frac { 1 }{ 3 } =0.33\quad and\quad \frac { 4 }{ 5 } =0.8$

Now converting all option in decimal we get
A.  $\frac{1}{4}$=0.25
B. $\frac{9}{10}$=0.9
C. $\frac{17}{30}$=0.567
D.  $\frac{7}{10}$ =0.7
So it is clear that option C and D lies between 0.33 and 0.8 and both are rational numbers so correct answer will be option C and D

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:

Five rational numbers between.
$\dfrac{2}{3}$ and $\dfrac{4}{5}$ are $\dfrac{41}{60},\dfrac{42}{60},\dfrac{43}{60},\dfrac{44}{60},\dfrac{45}{60}$

  1. True

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

To get the rational numbers between $\displaystyle\frac{2}{3}$ and $\displaystyle\frac{4}{5}$

Take an LCM of these two numbers: $\displaystyle\frac{10}{15}$ and $\displaystyle\frac{12}{15}$

Multiply numerator and denominator by 4: $\displaystyle\frac{40}{60}$ and $\displaystyle\frac{48}{60}$

All the numbers between $\displaystyle\frac{40}{60}$ and $\displaystyle\frac{48}{60}$ form the answer

Some of these numbers are $\displaystyle\frac{41}{60}$, $\displaystyle\frac{42}{60}$, $\displaystyle\frac{43}{60}$, $\displaystyle\frac{44}{60}$, $\displaystyle\frac{45}{60}$


Hence the statement is true

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

Three rational numbers between $\frac{2}{5}$ and $\frac{3}{5}$ is  $\frac{9}{20},\frac{10}{20},\frac{11}{20}$
If true then enter $1$ and if false then enter $0$

  1. True

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

2/5 = 8/20 and 3/5 = 12/20. The numbers 9/20, 10/20, and 11/20 are all strictly between 8/20 and 12/20. Thus, the statement is true.

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 $\displaystyle \frac{1}{3}$ and $\displaystyle \frac{1}{2}$ is _________.

  1. $\displaystyle \frac{2}{5}$
  2. $\displaystyle \frac{1}{5}$
  3. $\displaystyle \frac{3}{5}$
  4. $\displaystyle \frac{4}{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\dfrac{1}{3} = 0.3333.....$


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

Option A :
$\dfrac{2}{5} = 0.4$

Option B :
$\dfrac{1}{5} = 0.2$

Option C :
$\dfrac{3}{5} = 0.6$

Option D :
$\dfrac{4}{5} = 0.8$

$\therefore$ Option A lies in between $\dfrac{1}{3}$ and $\dfrac{1}{2}$

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

Choose the rational number, which does not lie, between the rational numbers, $\frac{-2}{3}$ and $\frac{-1}{5}$

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

Both given rational numbers are negative rational number. So, a rational number between both these rational numbers will be negative. But option B is a positive. So, option B will not lie between the given rational numbers. So, correct answer is option B.

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:

Five rational numbers between.
$\dfrac{1}{4}$ and $\dfrac{1}{2}$ are $\displaystyle\frac{9}{32},\frac{10}{32},\frac{11}{32},\frac{12}{32},\frac{13}{32}$

  1. True

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

To get the rational numbers between $\displaystyle\frac{1}{4}$ and $\displaystyle\frac{1}{2}$

Take an LCM of these two numbers: $\displaystyle\frac{1}{4}$ and $\displaystyle\frac{2}{4}$

Multiply numerator and denominator by 8: $\displaystyle\frac{8}{32}$ and $\displaystyle\frac{16}{32}$

All the numbers between $\displaystyle\frac{8}{32}$ and $\displaystyle\frac{16}{32}$ form the answer

Some of these numbers are $\displaystyle\frac{9}{32}$, $\displaystyle\frac{10}{32}$, $\displaystyle\frac{11}{32}$, $\displaystyle\frac{12}{32}$, $\displaystyle\frac{13}{32}$

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:

Five rational numbers between.
$\dfrac{-3}{2}$ and $\dfrac{5}{3}$ are $\dfrac{-8}{6},\dfrac{-7}{6},0,\dfrac{1}{6},\dfrac{2}{6}$

  1. True

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

To get the rational numbers between $\displaystyle\frac{-3}{2}$ and $\displaystyle\frac{5}{3}$

Take an LCM of these two numbers: $\displaystyle\frac{-9}{6}$ and $\displaystyle\frac{10}{6}$

All the numbers between $\displaystyle\frac{-9}{6}$ and $\displaystyle\frac{10}{6}$ form the answer

Some of these numbers are $\displaystyle\frac{-8}{6}$, $\displaystyle\frac{-7}{6}$, $\displaystyle{0}$, $\displaystyle\frac{1}{6}$, $\displaystyle\frac{2}{6}$


Hence the statement is true

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:

Ten rational numbers between $\dfrac{3}{5}$ and $\dfrac {3}{4}$ are 

$\displaystyle\frac{97}{160},\frac{98}{160},\frac{99}{160},\frac{100}{160},\frac{101}{160},\frac{102}{160},\frac{103}{160},\frac{104}{160},\frac{105}{160},\frac{106}{160}$

  1. True

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

To get the rational numbers between $\displaystyle\frac{3}{5}$ and $\displaystyle\frac{3}{4}$

Take an LCM of these two numbers: $\displaystyle\frac{12}{20}$ and $\displaystyle\frac{15}{20}$


So now to make the denominator $(160)$ as per the question we need to
multiply numerator and denominator by $8$: $\displaystyle\frac{96}{160}$ and $\displaystyle\frac{120}{160}$

All the numbers between $\displaystyle\frac{96}{160}$ and $\displaystyle\frac{120}{160}$ form the answer

Some of these numbers are $\displaystyle\frac{97}{160}$, $\displaystyle\frac{98}{160}$, $\displaystyle\frac{99}{160}$, $\displaystyle\frac{100}{160}$, $\displaystyle\frac{101}{160}$, $\displaystyle\frac{102}{160}$, $\displaystyle\frac{103}{160}$, $\displaystyle\frac{104}{160}$, $\displaystyle\frac{105}{160}$, $\displaystyle\frac{106}{160}$

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

Two rational numbers between $\dfrac{1}{5}$ and $\dfrac{4}{5}$ are :

  1. 1 and $\dfrac{3}{5}$
  2. $\dfrac{2}{5}$ and $\dfrac{3}{5}$
  3. $\dfrac{1}{2}$ and $\dfrac{2}{1}$
  4. $\dfrac{3}{5}$ and $\dfrac{6}{5}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since the denominator of both rational numbers are same. So, for getting the rational numbers between the given rational numbers, we only have to consider the numerators of the rational numbers.

Two numbers between 1 & 4 are 2 and 3.
So, two rational numbers between the given rational numbers will be $\dfrac { 2 }{ 5 }$ and $ \dfrac { 3 }{ 5 } $
So, correct answer is option B.

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 lying between $\sqrt{2}$ and $\sqrt{3}$ is :

  1. $\dfrac{\sqrt{2}+\sqrt{3}}{2}$
  2. $\sqrt{6}$

  3. 1.6

  4. 1.9
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\sqrt {2  } \cong $ 1.41....and $\sqrt {3  } \cong $ 1.73..

Now we see that option A is irrational number so it is incorrect , Option B is also irrational number so it is also incorrect ,Option C is rational number and lie between the given number so it is correct , Option D is rational number but it does not lie between the given number so it is incorrect.

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 rational number lies between $-\dfrac { 2}{ 3} $ and $\dfrac {1 }{4}$

  1. $ {\dfrac{ - 5} {24}}$
  2. ${\dfrac {25} {12}}$
  3. ${\dfrac {51} {24}}$
  4. ${\dfrac {5} {12}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given the fractions $-\dfrac { 2}{ 3} $ and $\dfrac {1 }{4}$ can be written as $-\dfrac{8}{12}$ and $\dfrac{3}{12}$ or, $-\dfrac{16}{24}$ and $\dfrac{6}{24}$.

Now it is clear that $-\dfrac{5}{24}$ lies between the given fractions.

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 five rational numbers between $\displaystyle\frac{-3}{2}$ and $\displaystyle\frac{5}{3}$.

  1. $\displaystyle\frac{-8}{6},\,\displaystyle\frac{-13}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$ and $\displaystyle\frac{2}{6}$
  2. $\displaystyle\frac{-8}{6},\,\displaystyle\frac{-7}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$ and $\displaystyle\frac{13}{6}$
  3. $\displaystyle\frac{-8}{6},\,\displaystyle\frac{-7}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$ and $\displaystyle\frac{11}{6}$
  4. $\displaystyle\frac{-8}{6},\,\displaystyle\frac{-7}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$ and $\displaystyle\frac{2}{6}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Converting the given rational numbers with the same denominators
$\cfrac{2}{3}=\cfrac{2\times5}{3\times5}=\cfrac{10}{15}$ and $\cfrac{4}{5}=\cfrac{4\times3}{5\times3}=\cfrac{12}{15}$

Also, $\cfrac{2}{3}=\cfrac{10}{15}=\cfrac{10\times4}{15\times4}=\cfrac{40}{60}$ and $\cfrac{4}{5}=\cfrac{12}{15}=\cfrac{12\times4}{15\times4}=\cfrac{48}{60}$

We know that $40,\,<\,41\,<\,42\,<\,43\,<44\,<45\,<\,46\,<\,47\,<\,48$
$\Rightarrow\cfrac{40}{60}\,<\,\cfrac{41}{60}\,<\,\cfrac{42}{60}\,<\,\dots\,<\,\cfrac{47}{60}\,<\,\cfrac{48}{60}$
Thus, we have the following five rational numbers between $\cfrac{2}{3}$ and $\cfrac{4}{5}$
$\cfrac{41}{60},\,\cfrac{43}{60},\,\cfrac{43}{60}\,\cfrac{44}{60}$ and $\cfrac{45}{60}$.

Converting the given rational numbers with the same denominators 
$\cfrac{-3}{2}=\cfrac{-3\times3}{2\times3}=\cfrac{-9}{6}$ and $\cfrac{5}{3}=\cfrac{5\times2}{3\times2}=\cfrac{10}{6}$

We know that $-9\,<\,-8\,<\,-7\,<\,-6\,<\,\dots\,<\,0\,<\,1\,<\,2\,<\,8\,<\,9\,<\,10$
$\Rightarrow\cfrac{-9}{6}\,<\,\cfrac{-8}{6}\,<\,\cfrac{-7}{6}\,<\,\cfrac{-6}{6}\,<\,\dots\,<\,\cfrac{0}{6}\,<\,\cfrac{1}{6}\,<\,\cfrac{2}{6}\,<\,\dots\,<\,\cfrac{8}{6}\,<\,\cfrac{9}{6}\,<\,\cfrac{10}{6}$.

Thus, we have the following five rational numbers between $\cfrac{-3}{2}$ and $\cfrac{5}{3}$ 
$\Rightarrow \cfrac{-8}{6},\,\cfrac{-7}{6},\,\cfrac{0}{6},\,\cfrac{1}{6}and\cfrac{2}{6}$