Tag: maths

Questions Related to maths

Choose the rational number which does not lie between rational numbers $\dfrac{3}{5}$ and $\dfrac{2}{3}$.

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

  1. $\dfrac{46}{75}$

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

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

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

Show answer
Correct Option: D
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.

Rationalising the denominator of $\dfrac {5}{\sqrt 3-\sqrt 5}$ is -

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

  1. $(\frac {5}{2}(\sqrt 3+\sqrt 5)$

  2. $(-\frac {5}{2}(\sqrt 3+\sqrt 5)$

  3. $(\frac {5}{2}(\sqrt 3-\sqrt 5)$

  4. $(-\frac {5}{2}(\sqrt 3-\sqrt 5)$

Show answer
Correct Option: B
Explanation:

here, $\dfrac {5}{\sqrt 3-\sqrt 5}$

$=\dfrac {5}{\sqrt 3-\sqrt 5}\times \dfrac {\sqrt 3+\sqrt 5}{\sqrt 3+\sqrt 5}$

$=\dfrac {5(\sqrt 3+\sqrt 5)}{3-5}$


$=-\dfrac {5}{2}(\sqrt 3+\sqrt 5)$

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

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

  1. $\dfrac{2}{5}$

  2. $\dfrac{4}{7}$

  3. $\dfrac{3}{7}$

  4. $\dfrac{3}{3}$

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Correct Option: A

A train of length 180 m crosses a man standing on a platform in 12 seconds and cross another train coming from opposite direction in 12 sec. If the second train running at 2/3 rd speed of the firstthen find the length of the second train?

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

  1. 56

  2. 120

  3. 20

  4. 44

Show answer
Correct Option: B
Explanation:

Length of the first train$=180m$


 Time taken by  the train to cross the man standing on the platform$=12s$


Speed of the first train$=\dfrac{180}{12}$

                                      $=15m/s$

Speed of the second train$=\dfrac{2}{3}\times15$

                                            $=10m/s$

Relative speed$=15+10$

                          $=25m/s$
 
Let the length of the train be $y$ metres.

$Distance =Speed\times time$

$y+180=25\times12$

$y+180=300$

$y=300-180$
$y=120$
So, the length of the second train$=120m$

 Rational numbers between $\displaystyle \frac{3}{8}$ and $\displaystyle \frac{7}{12}$ are

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

  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

Show answer
Correct Option: A
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 $\dfrac {3}{8} ,\dfrac {41}{96}, \dfrac {23}{48}, \dfrac {7}{12}$

__________ are rational numbers between between 5 and -2.

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

  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

Show answer
Correct Option: B
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 $

________ are rational numbers between $\displaystyle \frac{1}{3}$ and $\displaystyle \frac{1}{4}$

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

  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

Show answer
Correct Option: B
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 $\frac {1}{3} ,\frac {7}{24}, \frac {13}{48}, \frac {1}{4}$

 ________ are rational numbers between $\displaystyle -\dfrac{3}{4}$ and $\displaystyle \dfrac{1}{2}.$

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

  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

Show answer
Correct Option: C
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} $

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

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

  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. $\displaystyle \frac{117}{300}$,$\displaystyle \frac{287}{400}$

Show answer
Correct Option: D
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.

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?

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

  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. $\frac {a^2+b^2+c^2}{a+c-b}$

Show answer
Correct Option: D