$P=\{40\%\text{ of }A)+(65\%\text{ of }B)$ and $Q=(50\%\text{ of }A]+(50\%\text{ of }B)$, where A is greater than B

In this context, which of the following statements is correct?

The correct statement is None of the above can be concluded with certainty.

Step-by-Step Explanation:

We are given the expressions for $P$ and $Q$ as follows:

$P=0.40A+0.65B$

$Q=0.50A+0.50B$

We are also given that $A>B$.

To determine the relationship between $P$ and $Q$, let us find the difference $P-Q$:

$P-Q=(0.40A+0.65B)-(0.50A+0.50B)$

$P-Q=-0.10A+0.15B$

Now, let us analyze the sign of $P-Q$ under different scenarios consistent with $A>B$:

Case 1: Both A and B are positive

Suppose $A=100$ and $B=10$ (so $A>B$ holds true):

$P-Q=-0.10\left(100\right)+0.15\left(10\right)=-10+1.5=-8.5$

Here, $P-Q<0$, which means $Q>P$.

Case 2: A and B are negative

Suppose $A=-10$ and $B=-100$ (so $A>B$ holds true):

$P-Q=-0.10(-10)+0.15(-100)=1-15=-14$

Again, $Q>P$.

However, suppose $A=2$ and $B=1.5$ (so $A>B$ holds true):

$P-Q=-0.10\left(2\right)+0.15\left(1.5\right)=-0.20+0.225=+0.025$

Here, $P-Q>0$, which means $P>Q$.

Since $P$ can be greater than, less than, or equal to $Q$ depending on the specific numerical values of $A$ and $B$, no definitive conclusion can be drawn without knowing the relative magnitudes or signs of $A$ and $B$.

Therefore, None of the above can be concluded with certainty is the correct choice.