Question

How many natural numbers are there from \[87\] to $97$ ? What fraction of them are prime numbers?
(A) $\dfrac{5}{9}$
(B) $\dfrac{4}{7}$
(C) $\dfrac{2}{{11}}$
(D) $\dfrac{5}{{13}}$

Answer 1

Hint: We shall find all the natural numbers between $87$ and $97$. Then we determine the fraction. First we will write down the natural numbers between $87$ and $97$, and count them. Then we will find which of the numbers of those are prime numbers by using a divisibility test. Finally, we will get the required fraction of the prime numbers between $87$ and $97$.

Complete step-by-step solution:
A natural number is any integer which is greater than $0$. Here we are asked to calculate the fraction of the prime numbers between $87$ and $97$.
First let us write down the natural numbers which occur between $87$ and $97$. Now the natural numbers between $87$ and $97$ are given by $87,88,89,90,91,92,93,94,95,96,97$ .
Thus we get $11$ natural numbers between $87$ and $97$.
Our next step is to find the prime numbers between $87$ and $97$.
After observing each and every natural number between $87$ and $97$, we get only $2$ prime numbers, namely $89$ and $97$.
Thus we got $2$ prime numbers out of the $11$ numbers between $87$ and $97$.
Now we find out the fraction value of the prime numbers.
Since we got only $2$ numbers as primes out of the $11$ numbers between $87$ and $97$, then the fraction value of the prime numbers must be $ = \dfrac{\text{The number of primes obtained}}{\text{The total no. of natural numbers obtained}} = \dfrac{2}{{11}}$
Therefore there are $11$ natural numbers between $87$ and $97$.
The fraction of them being prime numbers is $\dfrac{2}{{11}}$ .
Hence option $(C)$ is correct.

Note: A natural number is also called as a Positive Integer. And a prime number is defined as those numbers who are only divisible by $1$ and itself. In this problem, we obtained $2$ prime numbers $89$ and $97$. Note that, there is no other divisor of $89$ other than $1$ and $89$ itself. Same argument goes for $97$ being prime. And for the other numbers, they are mostly divisible by either $2$ or $3$or$5$ or$7$.