Tag: rational numbers and their decimal expansions

Questions Related to rational numbers and their decimal expansions

State True or False.

The five rational numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$ are $ \displaystyle \frac{19}{30},\frac{20}{30},\frac{21}{30},\frac{22}{30},\frac{23}{30}$.

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:

Since we want five  numbers we write $\frac{3}{5}$ and $\frac{4}{5}$
So multiply in numerator and denominator by 5+1 = 6 we get
$\Rightarrow \frac{3}{5}=\frac{3\times 6}{5\times 6}=\frac{18}{30}$
$\Rightarrow \frac{4}{5}=\frac{4\times 6}{5\times 6}=\frac{24}{30}$
We know that $18<19<20<21<22<23<24$
$\Rightarrow \frac{18}{30}<\frac{19}{30}<\frac{20}{30}<\frac{21}{30}<\frac{22}{30}<\frac{23}{30}<\frac{24}{30}$
Hence 5 rational number between $\frac{3}{5} and  \frac{4}{5}$are
$\frac{19}{30},\frac{20}{30},\frac{21}{30},\frac{22}{30},\frac{23}{30},$

There are 50 numbers Each numbers is subtracted from 53 and the mean of the numbers so obtained is found to be -3.5 The mean of the given numbers is 

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

  1. $48.9$

  2. $49.5$

  3. $52.5$

  4. $56.5$

Show answer
Correct Option: B
Explanation:

$\Rightarrow$   Total observation is $50$

$\Rightarrow$  Let sum of $50$ number be $x$
$\therefore$    $\dfrac{x-(50\times 53)}{50}=-3.5$
$\therefore$    $x-2650=-3.5\times 50$
$\therefore$    $x-2650=-175$
$\therefore$    $x=-175+2650$
$\therefore$    $x=2475$
$\Rightarrow$   Original mean = $\dfrac{2475}{50}=49.5$

Write any $10$ rational numbers between $0\;and\;2$.

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

  1. $\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{5}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{88}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$

  2. $\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{21}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$

  3. $\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{35}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$

  4. $\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{5}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$

Show answer
Correct Option: D
Explanation:

Let us write $0$ as $\displaystyle\frac{0}{10}\;and\;2$ as $\displaystyle\frac{20}{10}$.

The rational numbers between these are


$\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{5}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10},\,\displaystyle\frac{11}{10},\,\displaystyle\frac{12}{10},\,\displaystyle\frac{13}{10},\,\displaystyle\frac{14}{10},\,\displaystyle\frac{15}{10},\,\displaystyle\frac{16}{10},\,\displaystyle\frac{17}{10},\,\displaystyle\frac{18}{10},\,\displaystyle\frac{19}{10}$

Choose the rational number which does not lie between rational numbers $ \displaystyle \frac{3}{5} $ and $ \displaystyle \frac{2}{3} $ :

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{46}{75} $

  2. $ \displaystyle \frac{47}{75} $

  3. $ \displaystyle \frac{49}{75} $

  4. $ \displaystyle \frac{50}{75} $

Show answer
Correct Option: D
Explanation:

For $\dfrac{3}{5}$ multiply numerator and denominator by $15$ to make denominator $75$ that comes into $\dfrac{45}{75}.$
Similarly doing for second then we have $\dfrac{50}{75}.$
Now question is asking about rational lying between them.

So, we need to check the numerator only that lies in between $45$ and $50$ or not.
Clearly $D$ is correct.

Find five rational numbers between $1$ and $2.$

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

  1. $\dfrac {1}{10}, \dfrac {2}{10}, \dfrac {3}{10}, \dfrac {4}{10}, \dfrac {5}{10}$

  2. $\dfrac {1}{5}, \dfrac {2}{5}, \dfrac {3}{5}, \dfrac {4}{5}, \dfrac {5}{5}$

  3. $\dfrac {1}{2}, \dfrac {1}{3}, \dfrac {1}{4}, \dfrac {1}{5}, \dfrac {1}{6}$

  4. $\dfrac {8}{7}, \dfrac {9}{7}, \dfrac {10}{7}, \dfrac {11}{7}, \dfrac {12}{7}$

Show answer
Correct Option: D
Explanation:

The given rational numbers are $1$ and $2$.

Let us multiply both the numbers by $\dfrac {7}{7}$.
$1\times \dfrac {7}{7} = \dfrac {7}{7}$ and $2\times \dfrac {7}{7} = \dfrac {14}{7}$.
Thus, five rational numbers between $\dfrac {7}{7} = 1$ and $\dfrac {14}{7} = 2$ are $\dfrac {8}{7}, \dfrac {9}{7}, \dfrac {10}{7}, \dfrac {11}{7}, \dfrac {12}{7}$.

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

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

  1. $-\dfrac {1}{4}$

  2. $-\dfrac {3}{10}$

  3. $\dfrac {3}{10}$

  4. $-\dfrac {7}{20}$

Show answer
Correct Option: C
Explanation:

Since the given rational numbers $-\dfrac {2}{5}$ and $-\dfrac {1}{5}$ are negative rational numbers, therefore, none of the positive rational number can lie between them.


Hence, the rational number $\dfrac {3}{10}$ does not lie between the rational numbers $-\dfrac {2}{5}$ and $-\dfrac {1}{5}$ 

Identity the rational number that does not lie between  $ \cfrac{3}{5}$ and $ \cfrac{2}{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{46}{75}$

  2. $ \cfrac{47}{75}$

  3. $ \cfrac{49}{75}$

  4. $ \cfrac{50}{75}$

Show answer
Correct Option: D
Explanation:

Changing the fractions to denominator $=75$


$\dfrac{3}{5} = \dfrac{45}{75}$

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

$\therefore \dfrac{50}{75}$ doesn't lie in between $\dfrac{3}{5}$ and $\dfrac{2}{3}$