Question

How many positive rational numbers less than 1 are there such that the sum of the numerator and the denominator doesn’t exceed 10.

Answer 1

Hint: Try to recall the definition of rational and try to apply the extra constraints given in the question to the values of p and q in any general rational number of the form $\dfrac{p}{q}$ .

Complete step-by-step answer:
Before moving to the options, let us talk about the definitions of rational numbers followed by irrational numbers.
So, rational numbers are those real numbers that can be written in the form of $\dfrac{p}{q}$ such that both p and q are integers and $q \ne 0$ . In other words, we can say that the numbers which are either terminating or recurring when converted to decimal form are called rational numbers. All the integers fall under this category.
Now moving to the solution. We know that a general rational number can be represented as $\dfrac{p}{q}$ such that both p and q are integers and $q \ne 0$, also it is given that the rational number must be less than 1. So, we can say that the denominator of the number must be greater than the numerator.
$\therefore q>p$
Also, it is given that the sum of p and q must not exceed the value of 10.
$\therefore p+q<10$
So, using these constraints, we can say that the possible rational numbers are:
$\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5},\dfrac{1}{6},\dfrac{1}{7},\dfrac{1}{8},\dfrac{1}{9},\dfrac{2}{3},\dfrac{2}{5},\dfrac{2}{7},\dfrac{2}{8},\dfrac{3}{4},\dfrac{3}{5},\dfrac{4}{5}$ .
So, there are a total of 15 rational numbers satisfying the conditions given in the question.

Note: See in such questions, you need to be very careful about the cases you consider, as while writing there is a very high chance of repeating the fractions because $\dfrac{1}{2}\text{ and }\dfrac{2}{4}$ are same, but would come twice while writing in an ordered manner, so you need to skip writing one of the two as I did in the above solution.