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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 \[d = \dfrac{{y - x}}{{n + 1}}\].

**Complete step-by-step answer:**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 \[8.4\].

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