Question

Insert five rational numbers between 3 and 4.

Answer 1

Hint:
Here, we need to find 5 rational numbers between 3 and 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 solution:
We have to find 5 rational numbers between 3 and 4.
Here, \[4 > 3\].
Therefore, let \[x\] be 3 and \[y\] be 4.
Since we have to find 5 rational numbers between 3 and 4, let \[n\] be 5.
Substituting \[x = 3\], \[y = 4\], and \[n = 5\] in the formula \[d = \dfrac{{y - x}}{{n + 1}}\], we get
\[ \Rightarrow d = \dfrac{{4 - 3}}{{5 + 1}}\]
Adding and subtracting the terms in the expression, we get
\[ \Rightarrow d = \dfrac{1}{6}\]
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 \[x = 3\] and \[d = \dfrac{1}{6}\] in the expression \[x + d\], we get
First rational number between 3 and 4 \[ = 3 + \dfrac{1}{6}\]
Taking the L.C.M. and simplifying the expression, we get
First rational number between 3 and 4 \[ = \dfrac{{18 + 1}}{6} = \dfrac{{19}}{6}\]
Substituting \[x = 3\] and \[d = \dfrac{1}{6}\] in the expression \[x + 2d\], we get
Second rational number between 3 and 4 \[ = 3 + 2 \times \dfrac{1}{6} = 3 + \dfrac{2}{6}\]
Taking the L.C.M. and simplifying the expression, we get
Second rational number between 3 and 4 \[ = \dfrac{{18 + 2}}{6} = \dfrac{{20}}{6} = \dfrac{{10}}{3}\]
Substituting \[x = 3\] and \[d = \dfrac{1}{6}\] in the expression \[x + 3d\], we get
Third rational number between 3 and 4 \[ = 3 + 3 \times \dfrac{1}{6} = 3 + \dfrac{3}{6}\]
Taking the L.C.M. and simplifying the expression, we get
Third rational number between 3 and 4 \[ = \dfrac{{18 + 3}}{6} = \dfrac{{21}}{6} = \dfrac{7}{2}\]
Substituting \[x = 3\] and \[d = \dfrac{1}{6}\] in the expression \[x + 4d\], we get
Fourth rational number between 3 and 4 \[ = 3 + 4 \times \dfrac{1}{6} = 3 + \dfrac{4}{6}\]
Taking the L.C.M. and simplifying the expression, we get
Fourth rational number between 3 and 4 \[ = \dfrac{{18 + 4}}{6} = \dfrac{{22}}{6} = \dfrac{{11}}{3}\]
Substituting \[x = 3\] and \[d = \dfrac{1}{6}\] in the expression \[x + 5d\], we get
Fifth rational number between 3 and 4 \[ = 3 + 5 \times \dfrac{1}{6} = 3 + \dfrac{5}{6}\]
Taking the L.C.M. and simplifying the expression, we get
Fifth rational number between 3 and 4 \[ = \dfrac{{18 + 5}}{6} = \dfrac{{23}}{6}\]

Therefore, we get the five rational numbers between 3 and 4 as \[\dfrac{{19}}{6}\], \[\dfrac{{10}}{3}\], \[\dfrac{7}{2}\], \[\dfrac{{11}}{3}\], and \[\dfrac{{23}}{6}\].

Note:
We found five rational numbers between 3 and 4. 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 3 and 4.
Rational numbers include every integer, fraction, decimal.