Let \[A = \left\{ {x:x\,{\rm{is}}\,{\rm{prime}}\,{\rm{number}}\,{\rm{and}}\,x < 30} \right\}\]. What is the number of different rational numbers whose numerator and denominator belong to A?
A. 90
B. 180
C. 91
D. 100

Hint: First we will find all prime number less than 30. Then we will find the number of ways to choose a denominator and numerator of a rational number and also we will calculate the number of rational numbers with same value. Now we will subtract the number of rational numbers with same value from the number of ways to choose a denominator and numerator of a rational number. Then add 1 with the result to find to total number of rational numbers.

Complete step by step solution: Given set is \[A = \left\{ {x:x\,{\rm{is}}\,{\rm{prime}}\,{\rm{number}}\,{\rm{and}}\,x < 30} \right\}\].
The prime numbers less 30 is 2,3,5,7,11,13,17,19,23,29.
The roster form of set A is \[\left\{ {2,3,5,7,11,13,17,19,23,29} \right\}\].
The number of element set A is 10.
We can choose the denominator and numerator from 10 elements and repetition allow.
Thus the number of numbers to choose a denominator and numerator of a rational number is \[10 \times 10\].
We know that if the denominator and numerator of a rational number are the same then the rational number is equal to 1.
We can get 10 numbers whose denominator and the numerator are the same.
Thus the number of different rational numbers whose numerator and denominator belong to A is \[\left( {10 \times 10} \right) - 10 = 90\]
Now we will add 1 to consider the rational number 1.
The total number of different rational numbers whose numerator and denominator belong to A is 90 + 1 = 91.

Option ‘C’ is correct

Note: Students often do mistakes to calculate the total number of the rational number. They did not add 1, because they thought 1 is not a rational number. But 1 is a rational number. So we have to add 1 with 90 to reach the correct answer.