Question

Find two rational numbers between \[ - 3\] and \[ - 2\].

Answer 1

Hint: Here, we will find the average of the given numbers by using the average. Then, again using the formula of average, we will find the average between the obtained number and any one of the given numbers. Hence, these will be the two required rational numbers between the given numbers.

Formula Used:
\[x = \dfrac{1}{2}\left( {a + b} \right)\], where the average of the two given numbers is \[x\] and the two given numbers are \[a,b\] respectively.

Complete step-by-step answer:
According to the question, we have to find two rational numbers between the given numbers.
We know that a rational number is a number that can be written in the form of \[\dfrac{p}{q}\] where the denominator \[q \ne 0\]
Also, these numbers can be both positive and negative.
Now, we know that the average of two rational numbers will also be a rational number and it will lie between those two numbers.
Therefore, if the average of the two given numbers is \[x\] and, if the two given numbers are \[a,b\] respectively, then we can use the formula is \[x = \dfrac{1}{2}\left( {a + b} \right)\].
In other words, the average is equal to the sum of observations divided by the total number of observations.
Hence, substituting the known values, we get,
\[{x_1} = \dfrac{1}{2}\left( { - 3 - 2} \right) = \dfrac{{ - 5}}{2}\]
Now, for finding the second rational number, we will find a rational number between \[\dfrac{{ - 5}}{2}\] and \[ - 2\].
This is because \[\dfrac{{ - 5}}{2}\] also lies between \[ - 3\] and \[ - 2\].
Hence, using the same formula, we get
The average of \[\dfrac{{ - 5}}{2}\] and \[ - 2\] as
\[{x_2} = \dfrac{1}{2}\left[ {\dfrac{{ - 5}}{2} - 2} \right]\]
Taking LCM inside the fraction, we get
\[ \Rightarrow {x_2} = \dfrac{1}{2}\left[ {\dfrac{{ - 5 - 4}}{2}} \right]\]
Simplifying the equation, we get
\[ \Rightarrow {x_2} = \dfrac{{ - 9}}{4}\]

Therefore, the required two rational numbers between \[ - 3\] and \[ - 2\] are \[\dfrac{{ - 5}}{2}\] and \[\dfrac{{ - 9}}{4}\].

Hence, this is the required answer.

Note: In this question, we are given two integers and integers are those numbers that can either be positive, negative, or zero. Integers are complete or whole numbers and cannot be in fractions or decimals. The difference between integers and rational numbers is that all integers are rational numbers but all the rational numbers cannot be integers but their numerator and denominator are respectively integers.