Question

Insert four rational numbers between 4 and 4.5.

Answer 1

Hint:
Here, we need to find 4 rational numbers between 4 and \[4.5\]. We will use the formula of \[n\] rational numbers between two numbers to find the value of the common difference. Substituting the value of common difference, we will find the 4 rational numbers between 4 and \[4.5\].

Formula Used:
We will the formula of 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 solution:
We have to find 4 rational numbers between 4 and \[4.5\].
Here, \[4.5 > 4\].
Therefore, let \[x\] be 4 and \[y\] be \[4.5\].
Since we have to find 4 rational numbers between 4 and \[4.5\], let \[n\] be 4.
Substituting \[x = 4\], \[y = 4.5\], and \[n = 4\] in the formula \[d = \dfrac{{y - x}}{{n + 1}}\], we get
\[ \Rightarrow d = \dfrac{{4.5 - 4}}{{4 + 1}}\]
Adding and subtracting the terms in the expression, we get
\[ \Rightarrow d = \dfrac{{0.5}}{5}\]
Simplifying the expression, we get
\[ \Rightarrow d = 0.1 = \dfrac{1}{{10}}\]
Now, the 4 rational numbers between two numbers \[x\] and \[y\] are given as \[x + d\], \[x + 2d\], \[x + 3d\], \[x + 4d\] 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 \[x = 4\] and \[d = \dfrac{1}{{10}}\] in the expression \[x + d\], we get
First rational number between 4 and \[4.5\] \[ = 4 + \dfrac{1}{{10}}\]
Taking the L.C.M. and simplifying the expression, we get
First rational number between 4 and \[4.5\] \[ = \dfrac{{40 + 1}}{{10}} = \dfrac{{41}}{{10}}\]
Substituting \[x = 4\] and \[d = \dfrac{1}{{10}}\] in the expression \[x + 2d\], we get
Second rational number between 4 and \[4.5\] \[ = 4 + 2 \times \dfrac{1}{{10}} = 4 + \dfrac{1}{5}\]
Taking the L.C.M. and simplifying the expression, we get
Second rational number between 4 and \[4.5\] \[ = \dfrac{{20 + 1}}{5} = \dfrac{{21}}{5}\]
Substituting \[x = 4\] and \[d = \dfrac{1}{{10}}\] in the expression \[x + 3d\], we get
Third rational number between 4 and \[4.5\] \[ = 4 + 3 \times \dfrac{1}{{10}} = 4 + \dfrac{3}{{10}}\]
Taking the L.C.M. and simplifying the expression, we get
Third rational number between 4 and \[4.5\] \[ = \dfrac{{40 + 3}}{{10}} = \dfrac{{43}}{{10}}\]
Substituting \[x = 4\] and \[d = \dfrac{1}{{10}}\] in the expression \[x + 4d\], we get
Fourth rational number between 4 and \[4.5\] \[ = 4 + 4 \times \dfrac{1}{{10}} = 4 + \dfrac{2}{5}\]
Taking the L.C.M. and simplifying the expression, we get
Fourth rational number between 4 and \[4.5\] \[ = \dfrac{{20 + 2}}{5} = \dfrac{{22}}{5}\]

Therefore, we get the four rational numbers between 4 and \[4.5\] as \[\dfrac{{41}}{{10}}\], \[\dfrac{{21}}{5}\], \[\dfrac{{43}}{{10}}\], and \[\dfrac{{22}}{5}\], or \[4.1\], \[4.2\], \[4.3\], and \[4.4\] in decimal form.

Note:
We found five rational numbers between 4 and \[4.5\]. A rational number is a number which can be written in the form \[\dfrac{p}{q}\], where the denominator \[q \ne 0\]. For example, \[5,\dfrac{7}{2}, - \dfrac{{15}}{7},5.6\], etc. are rational numbers. Rational numbers include every integer, fraction, decimal. Also, an integer can be a rational number but a rational number cannot always be an integer.