Question

Insert 16 rational numbers between 2.1 and 2.2.

Answer 1

Hint: The best method of inserting rational numbers between any 2 numbers is taking the average of them. As the average always lies between 2 numbers and also, we can say the average of 2 rational numbers is always a rational number.

Complete step by step solution:
First, we find the average of given rational numbers.
$1^{st}$ number: average of 2.1, 2.2 is rational number between them
$=\ \dfrac{2.1+2.2}{2}$
By simplifying the above equation, we get the rational number as:
$=\ 2.15$.
$2^{nd}$ number: average of 2.1, 2.15 is rational number between them
$=\ \dfrac{2.1+2.15}{2}$
Simplifying, the above equation we get the rational number as:
 $=\ 2.125$.
$3^{rd}$ number: average of 2.15, 2.2 is rational number between them
$=\ \dfrac{2.15+2.2}{2}$
By simplifying the above equation, we get that:
$=\ 2.175$.
$4^{th}$ number: average of 2.2, 2.125 is rational number between them
$=\ \dfrac{2.2+2.125}{2}$
By simplifying the above equation, we get that:
$=\ 2.1125$.
$5^{th}$ number: average of 2.125, 2.15 is rational number between them
$=\ \dfrac{2.125+2.15}{2}$
By simplifying the above equation, we get the rational numbers to be
$=\ 2.1375$.
$6^{th}$ number: average of 2.15, 2.175 lies between them as rational
$=\ \dfrac{2.15+2.175}{2}$
By simplifying the above equation, we get the rational numbers to be
$=\ 2.1625$.
$7^{th}$ number: average of 2.175, 2.2 lies between them as rational
$=\ \dfrac{2.175+2.2}{2}$
By simplifying the above equation, we get the rational numbers to be
$=\ 2.1875$.
$8^{th}$ number: average of 2.1, 2.1125 lies between them as rational
$=\ \dfrac{2.1+2.1125}{2}$
By simplifying the above equation, we get the rational numbers to be
$=\ 2.10625$.
$9^{th}$ number: average of 2.1, 2.10625 lies between them as rational
$=\ \dfrac{2.1+2.10625}{2}$
By simplifying the above equation, we get the rational numbers to be
$=\ 2.103125$.
Similarly, if we do till 16 numbers, we get the following:
2.1, 2.103125, 2.10625, 2.1125, 2.125, 2.1375, 2.15, 2.15078125, 2.150625, 2.153125, 2.15625, 2.1625, 2.175, 2.1875, 2.1935, 2.196875, 2.1984375, 2.2.

Note: Take any average of numbers which are between 2.1, 2.2 they will be rational. If you do average you just add two rational numbers which will result in a rational number and then you divide with 2 which also give a rational number. We know average always lies between the numbers. So we fulfilled the given condition by this.