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# Insert 5 rational numbers between $4.6$ and $8.4$.

Last updated date: 20th Jun 2024
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Hint: Here, we need to find 5 rational numbers between $4.6$ and $8.4$. A rational number is a number which can be written in the form $\dfrac{p}{q}$, where the denominator $q \ne 0$. We will use the formula of rational numbers between two numbers to find the required number.

Formula Used: The $n$ rational numbers between two numbers $x$ and $y$ are given as $x + d$,
$x + 2d$, $x + 3d$, …, $x + \left( {n - 1} \right)d$,
$x + nd$ where $y > x$ and $d = \dfrac{{y - x}}{{n + 1}}$.

We have to find 5 rational numbers in between $4.6$ and $8.4$.
Here, $8.4 > 4.6$.
Therefore, let $x$ be $4.6$ and $y$ be $8.4$.
Since we have to find 5 rational numbers in between
$4.6$ and $8.4$, let $n$ be 5.
Substituting $x = 4.6$, $y = 8.4$, and $n = 5$ in the formula $d = \dfrac{{y - x}}{{n + 1}}$, we get
$\Rightarrow d = \dfrac{{8.4 - 4.6}}{{5 + 1}}$
Adding and subtracting the terms in the expression, we get
$\Rightarrow d = \dfrac{{3.8}}{6} = \dfrac{{38}}{{60}}$
Simplifying the expression, we get
$\Rightarrow d = \dfrac{{19}}{{30}}$
Now, the 5 rational numbers between two numbers $x$ and $y$ are given as $x + d$, $x + 2d$, $x + 3d$, $x + 4d$,
$x + 5d$ where
$y > x$ and $d = \dfrac{{y - x}}{{n + 1}}$.
We will substitute the value of $x$ and $d$ to find the rational numbers one by one.
Substituting $x = 4.6 = \dfrac{{46}}{{10}}$ and
$d = \dfrac{{19}}{{30}}$ in the expression $x + d$, we get
First rational number between
$4.6$ and $8.4$ $= \dfrac{{46}}{{10}} + \dfrac{{19}}{{30}}$
Taking the L.C.M. and simplifying the expression, we get
First rational number between $4.6$ and $8.4$ $= \dfrac{{138 + 19}}{{30}} = \dfrac{{157}}{{30}}$
Substituting $x = \dfrac{{46}}{{10}}$ and $d = \dfrac{{19}}{{30}}$ in the expression $x + 2d$, we get
Second rational number between $4.6$ and $8.4$ $= \dfrac{{46}}{{10}} + 2 \times \dfrac{{19}}{{30}} = \dfrac{{46}}{{10}} + \dfrac{{38}}{{30}}$
Taking the L.C.M. and simplifying the expression, we get
Second rational number between $4.6$ and $8.4$ $= \dfrac{{138 + 38}}{{30}} = \dfrac{{176}}{{30}} = \dfrac{{88}}{{15}}$
Substituting $x = \dfrac{{46}}{{10}}$ and $d = \dfrac{{19}}{{30}}$ in the expression $x + 3d$, we get
Third rational number between $4.6$ and $8.4$ $= \dfrac{{46}}{{10}} + 3 \times \dfrac{{19}}{{30}} = \dfrac{{46}}{{10}} + \dfrac{{57}}{{30}}$
Taking the L.C.M. and simplifying the expression, we get
Third rational number between $4.6$ and $8.4$ $= \dfrac{{138 + 57}}{{30}} = \dfrac{{195}}{{30}} = \dfrac{{13}}{2}$
Substituting $x = \dfrac{{46}}{{10}}$ and $d = \dfrac{{19}}{{30}}$ in the expression $x + 4d$, we get
Fourth rational number between $4.6$ and $8.4$ $= \dfrac{{46}}{{10}} + 4 \times \dfrac{{19}}{{30}} = \dfrac{{46}}{{10}} + \dfrac{{76}}{{30}}$
Taking the L.C.M. and simplifying the expression, we get
Fourth rational number between $4.6$ and $8.4$ $= \dfrac{{138 + 76}}{{30}} = \dfrac{{214}}{{30}} = \dfrac{{107}}{{15}}$
Substituting $x = \dfrac{{46}}{{10}}$ and $d = \dfrac{{19}}{{30}}$ in the expression $x + 5d$, we get
Fifth rational number between $4.6$ and $8.4$ $= \dfrac{{46}}{{10}} + 5 \times \dfrac{{19}}{{30}} = \dfrac{{46}}{{10}} + \dfrac{{95}}{{30}}$
Taking the L.C.M. and simplifying the expression, we get
Fifth rational number between $4.6$ and $8.4$ $= \dfrac{{138 + 95}}{{30}} = \dfrac{{233}}{{30}}$
Therefore, we get the 5 rational numbers between $4.6$ and $8.4$ as $\dfrac{{157}}{{30}}$, $\dfrac{{88}}{{15}}$, $\dfrac{{13}}{2}$, $\dfrac{{107}}{{15}}$, and $\dfrac{{233}}{{30}}$.

Note: Here we have found out 5 rational numbers. We can say that the number we found is a rational number because the denominator is not equal to zero. If the denominator of a fraction is zero then they are termed as infinite numbers. We could have found the answer using a number line and placing the given numbers on the number line. And then observe which numbers come in between $4.6$ and $8.4$.