Question

Number of rational numbers between 15 and 18 is:
(A) infinite
(B) finite
(C) zero
(D) one

Answer 1

Hint: Rational number is a product of two integers. Use this definition to think about how many pairs of integers you can find to write them in a ratio, such that the ratio is between 15 and 18. Then compare the result you get with the options to select the correct option.

Complete step-by-step answer:
By definition. Rational numbers are defined as the numbers which are in a ratio of any two integers. We can represent a set of rational numbers as,
 $ \mathbb{Q} = \left\{ {\dfrac{p}{q}|p,q \in \mathbb{Z},q \ne 0,HCF(p,q) = 1} \right\} $
If $ q = 1 $ , then $ \mathbb{Q} = \mathbb{Z} $ . That means, every integer is also a rational number.
Now, by using above definition of rational numbers, we can write the following
 $ \dfrac{{15}}{1},\dfrac{{15}}{2},\dfrac{{15}}{3},\dfrac{{15}}{4},\dfrac{{15}}{5},\dfrac{{15}}{6},\dfrac{{15}}{7}.... $
The important thing to observe here is that we can keep on increasing the denominator by 1 every time and we will get a new rational number.
And since, there are infinite integers, we can keep on adding 1 infinite times. That way, we will get an infinite number of rational numbers without ever reaching 16.
Therefore, we can conclude that there are an infinite number of rational numbers between 15 and 18.
Therefore, from the above explanation, the correct answer is, option (A) infinite
So, the correct answer is “Option A”.

Note: This is a logical reasoning based question than a calculation based question. To solve this question, you need to understand what is a rational number. How it is defined and how can we prove that there are infinite rational numbers between every two integers.