Keisha’s teacher gives her the following information: m, n, p, and q are all integers and p ≠ 0 and q ≠ 0. A = m/q and B = n/p. What conclusion can Keisha make?
The Correct Answer and Explanation is:
Correct Answer:
Keisha can conclude that A and B are rational numbers.
Explanation
Keisha is given the expressions A=mqA = \frac{m}{q}A=qm and B=npB = \frac{n}{p}B=pn, where m,n,p,m, n, p,m,n,p, and qqq are all integers, and both ppp and qqq are not equal to zero. To determine what conclusion she can make, it is important to understand what these expressions represent and how the values involved affect the type of numbers AAA and BBB are.
By definition, a rational number is any number that can be expressed as the quotient of two integers, where the denominator is not zero. That is, a number rrr is rational if it can be written in the form ab\frac{a}{b}ba, where aaa and bbb are integers and b≠0b \neq 0b=0.
Looking at A=mqA = \frac{m}{q}A=qm, we know that mmm and qqq are both integers, and since q≠0q \neq 0q=0, this matches the definition of a rational number. Therefore, AAA must be a rational number.
Similarly, for B=npB = \frac{n}{p}B=pn, since both nnn and ppp are integers and p≠0p \neq 0p=0, BBB is also in the form of a rational number.
Because both expressions meet all the criteria for rational numbers — they are ratios of integers with nonzero denominators — Keisha can confidently conclude that both AAA and BBB are rational numbers.
She cannot determine whether AAA is equal to BBB, or whether one is greater than the other, without specific values. However, based on the given conditions alone, the most definite conclusion is that both AAA and BBB are rational.
