Question

Find 5 rational numbers between 1 and 2. We can approach this problem in at least two ways.

Answer 1

Hint: A rational number is a number which can be expressed in the form ${\displaystyle \frac{a}{b}}$ where a and b are integers and $b\ne 0$ .
Rational numbers are either terminating or non-terminating and repeating in decimal form.
e.g. 3.4, 5.7777... = $5.\overline{7}$, 8.13 etc.
A number Z, which is greater than X and smaller than Y, is said to be between X and Y.

Complete step-by-step answer:
Method 1:
The numbers 1 and 2 have a denominator 1 and can be written as ${\displaystyle \frac{1}{1}}$ and ${\displaystyle \frac{2}{1}}$ respectively.
The numerators differ by 1 here. Since we have to find 5 rational numbers between them, let us change their denominators to such numbers so that the difference between the numerators becomes at least 6.
${\displaystyle \frac{1}{1}}={\displaystyle \frac{1\times 10}{1\times 10}}={\displaystyle \frac{10}{10}}$ and ${\displaystyle \frac{2}{1}}={\displaystyle \frac{2\times 10}{1\times 10}}={\displaystyle \frac{20}{10}}$ .
Now, using the properties of numbers, we find that ${\displaystyle \frac{11}{10}},{\displaystyle \frac{12}{10}},{\displaystyle \frac{13}{10}},{\displaystyle \frac{14}{10}},{\displaystyle \frac{15}{10}}$ etc. are all between ${\displaystyle \frac{10}{10}}$ and ${\displaystyle \frac{20}{10}}$ .
Method 2:
The numbers 1 and 2 can also be written as 1.0 and 2.0. Now a number which is greater than 1.0 and less than 2.0 will be between them.
In other words, a number whose units place is 1 and the decimal part is more than 0, will be more than 1.0.
Therefore, 1.1, 1.2, 1.3, 1.4, 1.5 etc. are all between 1.0 and 2.0.

Note: There are infinitely many numbers between any two distinct real numbers.
 $0.\overline{9}$ and 1 are not distinct numbers, they are equal.
The denominators can be changed to any number but it is easier to convert into multiples of 10.