Tag: rational numbers and their decimal expansions

Questions Related to rational numbers and their decimal expansions

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}$.

The rational number $\displaystyle -\frac{18}{5}$ lies between the consecutive integers

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

  1. $- 2\  and\  - 3$

  2. $- 3 \ and\  - 4$

  3. $- 4 \ and\  - 5$

  4. $- 5\  and \ - 6$

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Correct Option: B
Explanation:

$\displaystyle -\frac{18}{5}= - 3.6 $ which lies between - 3 and - 4

Write five rational numbers greater than $-2$.

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

  1. $-\displaystyle\frac{3}{2},\,\displaystyle\frac{-1}{2},\,-1,\,\sqrt5,\,\displaystyle\frac{1}{2}$

  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. $-\displaystyle\frac{5}{2},\,\displaystyle\frac{-1}{2},\,-1,\,0,\,\displaystyle\frac{1}{2}$

Show answer
Correct Option: B
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.

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

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

  1. infinite

  2. two

  3. one

  4. none of these

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Correct Option: A
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.

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

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

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

  2. $3$

  3. $\dfrac {8}{2}$

  4. $\dfrac {9}{2}$

Show answer
Correct Option: B
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 $4$.

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

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

  1. $\cfrac{1}{4}$

  2. $\cfrac{3}{4}$

  3. $\cfrac{21}{40}$

  4. $\cfrac{33}{100}$

Show answer
Correct Option: C
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 $\dfrac{21}{40}$

If $\frac {4+3\sqrt 5}{\sqrt 5}=a+b\sqrt 5$ then, the value of b is

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

  1. $\frac {3}{5}$

  2. $\frac {4}{5}$

  3. $\frac {3\sqrt 5}{5}$

  4. $\frac {2}{5}$

Show answer
Correct Option: B
Explanation:

$\frac {4+3\sqrt 5}{\sqrt 5}=a+b\sqrt 5$
$\frac {(4+3\sqrt 5)\sqrt 5}{\sqrt 5\times \sqrt 5}=\frac {4\sqrt 5+15}{5}=\frac {4\sqrt 5}{5}+\frac {15}{5}$
$=3+\frac {4\sqrt 5}{5}$
Clearly, $a=3$
$b=\frac {4}{5}$.

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

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

  1. $2$

  2. $5$

  3. none

  4. infinite

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Correct Option: D
Explanation:

There can be infinite number of rational numbers between two given rational numbers.

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

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

  1. $0$

  2. $1$

  3. $5$

  4. infinite

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Correct Option: D
Explanation:

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

Which of the following numbers lies between $2\dfrac {1}{7}$ and $3\dfrac {1}{7}$?

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

  1. $\dfrac {37}{7}$

  2. $\dfrac {14}{7}$

  3. $\dfrac {37}{14}$

  4. $2$

Show answer
Correct Option: C
Explanation:

$Mean = \dfrac{\left (\dfrac {15}{7} + \dfrac {22}{7}\right )}{2} = \left (\dfrac {15 + 22}{7}\right ) \times \dfrac {1}{2} = \dfrac {37}{7}\times \dfrac {1}{2}$
$= \dfrac {37}{14}$

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