Tag: operations on rational numbers

Questions Related to operations on rational numbers

A rational number between $\dfrac {-2}{3}$ and $\dfrac {1}{2}$ is

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

  1. $-\dfrac {3}{6}$

  2. $\dfrac {-1}{12}$

  3. $\dfrac {-5}{6}$

  4. $\dfrac {5}{6}$

Show answer
Correct Option: A

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

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

  1. $\cfrac{3}{2}$

  2. $\cfrac{4}{2}$

  3. $\cfrac{5}{2}$

  4. $5$

Show answer
Correct Option: A
Explanation:

$\sqrt{2} = 1.414 ... $ and $\sqrt{3} = 1.732 ... $.

Clearly, $1.5 =\dfrac{15}{10} =\dfrac{3}{2}$ is the required number.

The three rational number between $5$ and $6$ are $ [\displaystyle\frac{21}{4},\frac{22}{4},\frac{23}{4}]$.

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

  1. True

  2. False

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Correct Option: A
Explanation:

$5=\frac { 20 }{ 4 } \quad and\quad 6=\frac { 24 }{ 4 } ,\ \frac { 21 }{ 4 } ,\frac { 22 }{ 4 } and\frac { 23 }{ 4 } \quad are\quad three\quad rational\quad numbers\quad lying\quad between\quad 5\quad and\quad 6.\quad \quad \ $

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

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

  1. $\displaystyle \frac{1}{2}$

  2. $\displaystyle \frac{15}{21}$

  3. $\displaystyle \frac{35}{42}$

  4. $\displaystyle \frac{71}{84}$

Show answer
Correct Option: D
Explanation:

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

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

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

  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. $\displaystyle \frac{9}{50}$, $\displaystyle \frac{31}{40}$

Show answer
Correct Option: B
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

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

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

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

Show answer
Correct Option: D
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 $-\displaystyle\frac{1}{9}\;and\;\displaystyle\frac{1}{5}$

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

$\displaystyle \frac{11}{4}$ is a number between

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

  1. $1 \ and \ 2$

  2. $2 \ and\ 3$

  3. $3 \ and \ 4$

  4. $11 \ and\ 12$

Show answer
Correct Option: B
Explanation:

$\displaystyle \frac{11}{4} = 2\frac{3}{4}$
So $\displaystyle \frac{11}{4}$ lies between 2 and 3.

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

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

  1. True

  2. False

Show answer
Correct Option: A
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 \ $

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

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

  1. $\displaystyle \frac{7}{24}$

  2. $0.29$

  3. $\displaystyle \frac{13}{48}$

  4. All the above

Show answer
Correct Option: D
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 $\dfrac{1}{3}$

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

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

  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 $\displaystyle\frac{7}{4}$

Show answer
Correct Option: B
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}$.