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Insert 5 rational numbers between \[4.6\] and \[8.4\].

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Last updated date: 20th Jun 2024
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Answer
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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 \[8.4\].