Question

What are rational numbers between \[\dfrac{1}{3}\] and \[\dfrac{1}{4}\]
A. \[\dfrac{1}{3},\dfrac{7}{64},\dfrac{13}{48},\dfrac{1}{4}\]
B. \[\dfrac{1}{3},\dfrac{7}{24},\dfrac{13}{48},\dfrac{1}{4}\]
C. \[\dfrac{1}{3},\dfrac{7}{64},\dfrac{13}{68},\dfrac{1}{4}\]
D. None of the above

Answer 1

Hint: The rational number between a and b is \[\dfrac{a+b}{2}\] . Use this equation and substitute the value of a and b further simplifying we get the rational numbers between the given two numbers.

Complete step by step answer:
Before solving the problem, we need to understand the concept of rational numbers that is
Rational numbers are the numbers which can be expressed in the form of p and q where, \[q\ne 0\] Examples for rational numbers are prime and composite numbers, odd and even numbers, decimals and fractions. A number of rational numbers between two rational numbers can be located. Between any two rational numbers, countless rational numbers can be found.
Now, we come to the problem,
Here we have to find 4 rational numbers between \[\dfrac{1}{3}\] and \[\dfrac{1}{4}\].
As, we know, the rational number between a and b is given by, \[\dfrac{a+b}{2}\]
Hence, the rational number between \[\dfrac{1}{3}\] and \[\dfrac{1}{4}\] is \[\dfrac{\left( \dfrac{1}{3} \right)+\left( \dfrac{1}{4} \right)}{2}\]
Here, above step take LCM that is 12 we get:
\[\Rightarrow \dfrac{\left( \dfrac{4+3}{12} \right)}{2}\]
By simplifying this we get:
\[\Rightarrow \dfrac{7}{24}\]
Hence, Rational number between\[\dfrac{1}{3}\]and \[\dfrac{1}{4}\]is \[\dfrac{7}{24}\]
Now, we have to take another rational number between \[\dfrac{7}{24}\] and \[\dfrac{1}{4}\] is \[\dfrac{\left( \dfrac{7}{24} \right)+\left( \dfrac{1}{4} \right)}{2}\]
Here, first we need to take the LCM as 24 we get:
\[\Rightarrow \dfrac{\left( \dfrac{7+6}{24} \right)}{2}\]
By simplifying further, we get:
\[\Rightarrow \dfrac{13}{48}\]
Hence, the Rational number between \[\dfrac{7}{24}\] and \[\dfrac{1}{4}\] is \[\dfrac{13}{48}\]
Hence, the required two rational number between \[\dfrac{1}{3}\] and \[\dfrac{1}{4}\] are \[\dfrac{1}{3},\dfrac{7}{24},\dfrac{13}{48},\dfrac{1}{4}\]

So, the correct answer is “Option B”.

Note:
Whenever such a type of question appears, that is it is asked to find a rational number between any two rational numbers, suppose x and y, the simplest approach is to divide their sum by 2, as mentioned in the question. Proceed step by step, obtain the value of \[\dfrac{\text{sum}\,\,\text{of}\,\text{two number}}{2}\] then find the next rational number by dividing the sum of the first rational no. obtained in step 1 with the first number in the range, repeat it till you get all the required rational numbers.