Rational numbers on number line - class-VII

Price$=Rs\ 10\dfrac {3}{4}=Rs.\cfrac{43}{4}$ per liter

To find a rational number between -2/3 and 1/2, we need a value satisfying -2/3 < x < 1/2. Option A: -3/6 = -1/2, which satisfies -2/3 < -1/2 < 1/2. Option B: -1/12 ≈ -0.083 also satisfies the condition, making both A and B technically correct.

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

$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
 $\displaystyle \frac{1}{2}\left ( \frac{5}{6}+\frac{6}{7} \right )=\frac{1}{2}\left ( \frac{35+36}{42} \right )=\frac{71}{84}$

The 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 $\displaystyle \frac{63}{250}=\frac{252}{1000}$

Convert the rational numbers into equivalent rational numbers with the same denominator.

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

$x=\frac { 1 }{ 3 } =\frac { 8 }{ 24 } \quad and\quad y=\frac { 1 }{ 2 } =\frac { 12 }{ 24 } ,\ $

$\dfrac{1}{4} = \dfrac{6}{24} = \dfrac{12}{48} = 0.25$

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

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 infinite rational numbers between two rational numbers.

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 $4$.

$\dfrac{1}{2}=0.5$

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

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

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

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

From the options, only $0$ can lie between $\dfrac {-5}{2}$ and $\dfrac {3}{4}.$

Clearly, $0$ and $-1$ cannot lie between $0$ and $-1$. 

When a positive integer is divided by another positive integer will yield a rational number

Required number $=$ $\dfrac{1}{2}\left(\dfrac{1}{5}+\dfrac{1}{3}\right)$

Converting the given fraction in decimal format we get $\frac { 1 }{ 3 } =0.33\quad and\quad \frac { 4 }{ 5 } =0.8$

To get the rational numbers between $\displaystyle\frac{2}{3}$ and $\displaystyle\frac{4}{5}$

Take an LCM of these two numbers: $\displaystyle\frac{10}{15}$ and $\displaystyle\frac{12}{15}$

Multiply numerator and denominator by 4: $\displaystyle\frac{40}{60}$ and $\displaystyle\frac{48}{60}$

All the numbers between $\displaystyle\frac{40}{60}$ and $\displaystyle\frac{48}{60}$ form the answer

Some of these numbers are $\displaystyle\frac{41}{60}$, $\displaystyle\frac{42}{60}$, $\displaystyle\frac{43}{60}$, $\displaystyle\frac{44}{60}$, $\displaystyle\frac{45}{60}$

2/5 = 8/20 and 3/5 = 12/20. The numbers 9/20, 10/20, and 11/20 are all strictly between 8/20 and 12/20. Thus, the statement is true.

$\dfrac{1}{3} = 0.3333.....$

Both given rational numbers are negative rational number. So, a rational number between both these rational numbers will be negative. But option B is a positive. So, option B will not lie between the given rational numbers. So, correct answer is option B.

To get the rational numbers between $\displaystyle\frac{1}{4}$ and $\displaystyle\frac{1}{2}$

Take an LCM of these two numbers: $\displaystyle\frac{1}{4}$ and $\displaystyle\frac{2}{4}$

Multiply numerator and denominator by 8: $\displaystyle\frac{8}{32}$ and $\displaystyle\frac{16}{32}$

All the numbers between $\displaystyle\frac{8}{32}$ and $\displaystyle\frac{16}{32}$ form the answer

Some of these numbers are $\displaystyle\frac{9}{32}$, $\displaystyle\frac{10}{32}$, $\displaystyle\frac{11}{32}$, $\displaystyle\frac{12}{32}$, $\displaystyle\frac{13}{32}$

To get the rational numbers between $\displaystyle\frac{-3}{2}$ and $\displaystyle\frac{5}{3}$

Take an LCM of these two numbers: $\displaystyle\frac{-9}{6}$ and $\displaystyle\frac{10}{6}$

All the numbers between $\displaystyle\frac{-9}{6}$ and $\displaystyle\frac{10}{6}$ form the answer

Some of these numbers are $\displaystyle\frac{-8}{6}$, $\displaystyle\frac{-7}{6}$, $\displaystyle{0}$, $\displaystyle\frac{1}{6}$, $\displaystyle\frac{2}{6}$

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 $\displaystyle\frac{15}{20}$

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.

$\sqrt {2  } \cong $ 1.41....and $\sqrt {3  } \cong $ 1.73..

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

-1/3 = -2/6 and 1/2 = 3/6. The numbers -1/6 and 1/6 are both between -2/6 and 3/6.

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 $\dfrac{6}{24}$.

Additive inverse of 2/3 is -2/3. Reciprocal of 9/8 is 8/9. Sum = -2/3 + 8/9 = -6/9 + 8/9 = 2/9.

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

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

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.

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

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.

Changing the fractions to denominator $=75$