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$M = 2N = \frac{P}{4} = Q + 8 = \frac{R}{2} > 0$ Based on the statement above, which variable has the greatest value? $\newline$ A. $M$ $\newline$ B. $P$ $\newline$ C. $Q$ $\newline$ D. $R$

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Evaluate recursive formulas for sequences

Full solution

Q.

M = 2N = \frac{P}{4} = Q + 8 = \frac{R}{2} > 0

Based on the statement above, which variable has the greatest value?

\newline

A.

M

\newline

B.

P

\newline

C.

Q

\newline

D.

R

Analyze Relationships: Let's analyze the given relationships between the variables: $\newline$ $M = 2N = \frac{P}{4} = Q + 8 = \frac{R}{2} > 0$ $\newline$ Since all expressions are equal to each other and greater than zero, we can compare them directly to find out which variable is the greatest.
Compare $M$ and $N$ : First, let's compare $M$ and $N$ . Since $M = 2N$ , and both $M$ and $N$ are greater than zero, $N$ must be half the value of $M$ . Therefore, $M$ is greater than $N$ .
Compare M and P: Next, let's compare M and P. Since $M = \frac{P}{4}$ , multiplying both sides by $4$ gives us $P = 4M$ . This means $P$ is four times greater than $M$ . Therefore, $P$ is greater than $M$ .
Compare M and Q: Now, let's compare M and Q. Since $M = Q + 8$ , and $M$ is greater than $0$ , $Q$ must be less than $M$ because we have to subtract $8$ from $M$ to get $Q$ . Therefore, $M$ is greater than $Q$ .
Compare M and R: Finally, let's compare M and R. Since $M = \frac{R}{2}$ , multiplying both sides by $2$ gives us $R = 2M$ . This means $R$ is twice as large as $M$ . Therefore, $R$ is greater than $M$ .
Final Comparison: Comparing all the variables, we find that $P$ is four times $M$ , and $R$ is twice $M$ , while $N$ and $Q$ are less than $M$ . Since $P$ is a multiple of $4$ of $M$ and $R$ is a multiple of $M$ $1$ of $M$ , $P$ must be the greatest variable.

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