Tag: operations on rational numbers

Questions Related to operations on rational numbers

State true or false:

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

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

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

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

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 $\dfrac{3}{5}$

  2. $\dfrac{2}{5}$ and $\dfrac{3}{5}$

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

  4. $\dfrac{3}{5}$ and $\dfrac{6}{5}$

Show answer
Correct Option: B
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.

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

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

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

  2. $\sqrt{6}$

  3. 1.6

  4. 1.9

Show answer
Correct Option: C
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.

The Rational Number $\dfrac { -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$-2$

  2. $-3$ and $-4$

  3. $-4$ and $-5$

  4. $-5$ and$-6$

Show answer
Correct Option: B
Explanation:

$-\cfrac { 18 }{ 5 } =-3.6\Rightarrow $ lies between $-4$ and $-3$

Find the real numbers between the following

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

  1. $8$ and $9$

  2. $\dfrac{1}{9}$ and $\dfrac{2}{9}$

  3. $-4$ and $-3$

  4. $0.75$ and $1.2$

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Correct Option: A

Find two rational numbers lying between $\dfrac{-1}{3}$ and $\dfrac{1}{2}$.

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

  1. -1/6,1 /6

  2. -2/3, 2/3

  3. none

  4. both

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Correct Option: A

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

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

  1. $ {\dfrac{ - 5} {24}}$

  2. ${\dfrac {25} {12}}$

  3. ${\dfrac {51}  {24}}$

  4. ${\dfrac {5} {12}}$

Show answer
Correct Option: A
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.

What is the sum of the addictive inverse of $\frac{2}{3}$ and the reciprocal of $\frac{9}{8}$?

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

  1. $\frac{3}{8}$

  2. -$\frac{3}{8}$

  3. $\frac{2}{9}$

  4. -$\frac{2}{9}$

Show answer
Correct Option: C

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

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

  1. $\displaystyle\frac{-8}{6},\,\displaystyle\frac{-13}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$ 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 $\displaystyle\frac{2}{6}$

Show answer
Correct Option: D
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 $\cfrac{5}{3}$ 
$\Rightarrow \cfrac{-8}{6},\,\cfrac{-7}{6},\,\cfrac{0}{6},\,\cfrac{1}{6}and\cfrac{2}{6}$

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
$\frac{19}{30},\frac{20}{30},\frac{21}{30},\frac{22}{30},\frac{23}{30},$