Question

as teacher gives her the following information:
n, n, p, and q are all integers and p ≠ 0 and q ≠ 0
a = \\(\frac{m}{q}\\) and b = \\(\frac{n}{p}\\)
at conclusion can keisha make?
b = \\(\frac{mp + nq}{pq}\\), so the sum of a rational number and an irrational number is an irrational number.
b = \\(\frac{mp + nq}{pq}\\), so the sum of two rational numbers is a rational number.
b = \\(\frac{mp + nq}{pq}\\), so the product of two irrational numbers is an irrational number.
a · b = \\(\frac{mn}{pq}\\), so the product of two rational numbers is a rational number.

Explanation:

Step1: Recall the definition of rational numbers

A rational number is a number that can be expressed as $\frac{a}{b}$ where $a$ and $b$ are integers and $b
eq0$. Given $A = \frac{m}{q}$ and $B=\frac{n}{p}$, with $m,n,p,q$ integers and $p
eq0,q
eq0$, so $A$ and $B$ are rational numbers.

Step2: Analyze the sum of two rational numbers

To find $A + B$, we have:
$$A + B=\frac{m}{q}+\frac{n}{p}=\frac{mp + nq}{pq}$$
Since $m,n,p,q$ are integers, $mp + nq$ and $pq$ are also integers. And since $p
eq0$ and $q
eq0$, $pq
eq0$. So $\frac{mp + nq}{pq}$ is a rational number (by the definition of rational numbers). This shows that the sum of two rational numbers is a rational number.

Step3: Eliminate other options

• Option about sum of rational and irrational: $A$ and $B$ are both rational, so this option is incorrect.
• Option about product of two irrational: $A$ and $B$ are rational, not irrational, so this option is incorrect.
• Option about product (the last option, but we are analyzing sum here, and the correct sum - related option is the one stating sum of two rationals is rational).

Answer:

The option stating " $B=\frac{mp + nq}{pq}$, so the sum of two rational numbers is a rational number" (the second option among the given sum - related options, assuming the options are presented with the sum formula and the conclusion about sum of two rationals).