Questions: 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? A · B = mp + nq/PV, so the product of two irrational numbers is an irrational number. A + B = mp + np/p, so the sum of two rational numbers is a rational number. A · B = mp + nq/N, so the product of two rational numbers is a rational number. A + B = mp + nq/pP, so the sum of a rational number and an irrational number is an irrational number.

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?
A · B = mp + nq/PV, so the product of two irrational numbers is an irrational number.
A + B = mp + np/p, so the sum of two rational numbers is a rational number.
A · B = mp + nq/N, so the product of two rational numbers is a rational number.
A + B = mp + nq/pP, so the sum of a rational number and an irrational number is an irrational number.

Transcript text: Keisha's teacher gives her the following information: - $m, n, p$, and $q$ are all integers and $p \neq 0$ and $q \neq 0$ - $A=\frac{m}{q}$ and $B=\frac{n}{p}$ What conclusion can Keisha make? $A \cdot B=\frac{m p+n q}{P V}$, so the product of two irrational numbers is an irrational number. $A+B=\frac{m p+n_{p}}{p}$, so the sum of two rational numbers is a rational number. $A \cdot B=\frac{m p+n q}{N}$, so the product of two rational numbers is a rational number. $A+B=\frac{m p+n q}{p P}$, so the sum of a rational number and an irrational number is an irrational number.

Solution

Solution Steps

Step 1: Analyze the given information

We are given:

$ m, n, p, $ and $ q $ are integers with $ p \neq 0 $ and $ q \neq 0 $.
$ A = \frac{m}{q} $ and $ B = \frac{n}{p} $.

Since $ m, n, p, $ and $ q $ are integers and $ p, q \neq 0 $, both $ A $ and $ B $ are rational numbers.

Step 2: Evaluate the sum $ A + B $

The sum of two rational numbers is: \[ A + B = \frac{m}{q} + \frac{n}{p} = \frac{mp + nq}{pq}. \] Since $ mp + nq $ and $ pq $ are integers (as $ m, n, p, q $ are integers), $ A + B $ is a rational number.

Step 3: Evaluate the product $ A \cdot B $

The product of two rational numbers is: \[ A \cdot B = \frac{m}{q} \cdot \frac{n}{p} = \frac{mn}{pq}. \] Since $ mn $ and $ pq $ are integers (as $ m, n, p, q $ are integers), $ A \cdot B $ is a rational number.

Final Answer

The correct conclusion is: \[ \boxed{A+B=\frac{mp+nq}{pq}, \text{ so the sum of two rational numbers is a rational number.}} \] and \[ \boxed{A \cdot B=\frac{mn}{pq}, \text{ so the product of two rational numbers is a rational number.}} \]

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