If x/y
is rational, must x and y each be rational?
a) yes; because the quotient of rationals is rational.
b) no; because x and y could have a common irrational factor.
c) no; because the quotient of irrationals is always irrational.
d) yes; because x and y must be integers, and integers are rational.

The answer is B.

Consider the fraction .

Both the numerator and denominator, when by themselves, are irrational.

However, when they are divided, they result in the rational number .

When have the same irrational factor, the irrational factor will be cancelled from the fraction and (possibly) leave a rational number.

The right answer is option d.