Question

Find 3 rational numbers between 0 and 1.
A. $ \dfrac{3}{2} $ , $ \dfrac{1}{4} $ and $ \dfrac{3}{4} $
B. $ \dfrac{1}{2} $ , $ \dfrac{1}{4} $ and $ \dfrac{3}{4} $
C. $ \dfrac{1}{2} $ , $ \dfrac{5}{4} $ and $ \dfrac{3}{4} $
D. $ \dfrac{1}{2} $ , $ \dfrac{1}{4} $ and $ \dfrac{7}{4} $

Answer 1

Hint: Here, we are asked to find rational numbers between two certain numbers. For this, we need to use the mean value method to find the desired number of rational numbers. First, we will find the mean of two given numbers which will be our first rational number. Then, in order to find more rational numbers, we will repeat the same process with the newly obtained and old rational numbers.
 Formula used:
 $ \overline x = \dfrac{{{x_1} + {x_2} + {x_3} + ... + {x_n}}}{n} $
Where, $ \overline x $ is the mean value, $ {x_1},{x_2},{x_3},...,{x_n} $ are the given terms and $ n $ is the total number of terms.

Complete step-by-step answer:
We will start by finding the first rational number between 0 and 1 which will be the mean of both these numbers 0 and 1.
Using the formula $ \overline x = \dfrac{{{x_1} + {x_2} + {x_3} + ... + {x_n}}}{n} $ , the mean of 0 and 1 is given by
 $ \dfrac{{0 + 1}}{2} = \dfrac{1}{2} $
Thus, $ \dfrac{1}{2} $ is our first rational number between 0 and 1.
Now, in order to find the second rational number we need to find the mean of 0 and $ \dfrac{1}{2} $ , which is
 $ \dfrac{{0 + \dfrac{1}{2}}}{2} = \dfrac{1}{{2 \times 2}} = \dfrac{1}{4} $
Thus, $ \dfrac{1}{4} $ is our second rational number.
We will now find the third rational number by taking mean of $ \dfrac{1}{2} $ and 1, which is
 $ \dfrac{{\dfrac{1}{2} + 1}}{2} = \dfrac{{1 + 2}}{{2 \times 2}} = \dfrac{3}{4} $
Thus, $ \dfrac{3}{4} $ is our third rational number.
Therefore, the 3 rational number between 0 and 1 are $ \dfrac{1}{2} $ , $ \dfrac{1}{4} $ and $ \dfrac{3}{4} $
So, the correct answer is “ $ \dfrac{1}{2} $ , $ \dfrac{1}{4} $ and $ \dfrac{3}{4} $ ”.

Note: In this problem, we have determined only three rational numbers between 0 and 1. However, it is important to know that there are an infinite number of rational numbers between any two numbers. Therefore, by repeating the same process again and again, we can find any required number of rational numbers between two given numbers.