Question

Insert three rational numbers between \[ - 2\] and 5.

Answer 1

Hint:Here, we need to find 5 rational numbers between \[ - 2\] and 5. We will use the formula of \[n\] rational numbers between two numbers to find the required rational numbers. A rational number is a number which can be written in the form \[\dfrac{p}{q}\], where the denominator \[q \ne 0\].

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 3 rational numbers between \[ - 2\] and 5.
Here, \[5 > - 2\].
Therefore, let \[x\] be \[ - 2\] and \[y\] be 5.
Since we have to find 3 rational numbers between \[ - 2\] and 5, let \[n\] be 3.
Substituting \[x = - 2\], \[y = 5\], and \[n = 3\] in the formula \[d = \dfrac{{y - x}}{{n + 1}}\], we get
\[ \Rightarrow d = \dfrac{{5 - \left( { - 2} \right)}}{{3 + 1}}\]
Simplifying the fraction, we get
\[ \Rightarrow d = \dfrac{{5 + 2}}{{3 + 1}}\]
Adding and subtracting the terms in the expression, we get
\[ \Rightarrow d = \dfrac{7}{4}\]
Now, the 3 rational numbers between two numbers \[x\] and \[y\] are given as \[x + d\], \[x + 2d\], \[x + 3d\] 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 \[x = - 2\] and \[d = \dfrac{7}{4}\] in the expression \[x + d\], we get
First rational number between \[ - 2\] and 5 \[ = - 2 + \dfrac{7}{4}\]
Taking the L.C.M. and simplifying the expression, we get
First rational number between \[ - 2\] and 5 \[ = \dfrac{{ - 8 + 7}}{4} = \dfrac{{ - 1}}{4} = - \dfrac{1}{4}\]
Substituting \[x = - 2\] and \[d = \dfrac{7}{4}\] in the expression \[x + 2d\], we get
Second rational number between \[ - 2\] and 5 \[ = - 2 + 2 \times \dfrac{7}{4} = - 2 + \dfrac{7}{2}\]
Taking the L.C.M. and simplifying the expression, we get
Second rational number between \[ - 2\] and 5 \[ = \dfrac{{ - 4 + 7}}{2} = \dfrac{3}{2}\]
Substituting \[x = - 2\] and \[d = \dfrac{7}{4}\] in the expression \[x + 3d\], we get
Third rational number between \[ - 2\] and 5 \[ = - 2 + 3 \times \dfrac{7}{4} = - 2 + \dfrac{{21}}{4}\]
Taking the L.C.M. and simplifying the expression, we get
Third rational number between \[ - 2\] and 5 \[ = \dfrac{{ - 8 + 21}}{4} = \dfrac{{13}}{4}\]
Therefore, we get the 3 rational numbers between \[ - 2\] and 5 as \[ - \dfrac{1}{4}\], \[\dfrac{3}{2}\], and \[\dfrac{{13}}{4}\], or \[ - 0.25\], \[1.5\], and \[3.25\].

Note: Here we have found out 3 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 2 and 5.