Question

Which of the following is a rational number(s)?
(A) $\dfrac{-2}{9}$
(B) $\dfrac{4}{-7}$
(C) \[\dfrac{-3}{17}\]
(D) All the three given numbers.

Answer 1

Hint: Use the definition of rational numbers given by “Any number having the form $\dfrac{p}{q}$ where ‘p’ and ‘q’ both are integers and \[q\ne 0\] is known as Rational Number” As a boundary conditions to decide whether the numbers are rational or not.

Complete step-by-step answer:
To identify that which number is a rational number, we should know the definition of rational number so that we can verify the conditions of rational numbers and conclude whether the number is rational or not, and for that we should know the definition of rational numbers given below,
Definition of Rational Numbers:
Any number having the form $\dfrac{p}{q}$ where ‘p’ and ‘q’ both are integers and \[q\ne 0\] is known as Rational Number.
Now, we will check each option as if it is a rational number or not,
Therefore consider Option (a),
If we compare $\dfrac{-2}{9}$ with $\dfrac{p}{q}$ then we can write,
p = -2 which is an integer, q = 9 which is also an integer and \[q\ne 0\],
As $\dfrac{-2}{9}$ satisfies all the conditions of a rational number therefore we can say that $\dfrac{-2}{9}$ is a rational number.
Therefore option (a) is correct. ……………………………………………………. (1)
Now, consider Option (b),
If we compare $\dfrac{4}{-7}$ with $\dfrac{p}{q}$ then we can write,
p = 4 which is an integer, q = -7 which is also an integer and \[q\ne 0\],
As $\dfrac{4}{-7}$ satisfies all the conditions of a rational number therefore we can say that $\dfrac{4}{-7}$ is a rational number.
Therefore option (b) is correct. ……………………………………………………. (2)

Therefore consider Option (c),
If we compare $\dfrac{-3}{17}$ with $\dfrac{p}{q}$ then we can write,
p = -3 which is an integer, q = 17 which is also an integer and \[q\ne 0\],
As $\dfrac{-3}{17}$ satisfies all the conditions of a rational number therefore we can say that v is a rational number.
Therefore option (c) is correct. ……………………………………………………. (1)
From equation (1), equation (2) and equation (3) we can say that all the three options have rational numbers and therefore we can say that all the given three numbers are rational numbers.
Therefore the correct answer is option (d).

Note: As per the definition of rational numbers they should have the form $\dfrac{p}{q}$ where \[q\ne 0\] that means ‘q’ can be 1 also and if q is equal to one the all the integers are also rational numbers including zero on the numerator. So if you find an integer in the option in this type of problem, don’t get confused and write it as a rational number.