Let's write down the given equations based on the problem statement:

1. $A + B + C + D = 17$
2. $A + C = 6$
3. $P + Q + S + D = 15$
4. $P + Q + R + B = 17$
5. $P = R$ and $Q = S$

Step 1: Find the value of $B + D$ Substitute equation (2) into equation (1): $(A + C) + B + D = 17$ $6 + B + D = 17$ $B + D = 11$

Step 2: Simplify equations (3) and (4) Using the given relations $P = R$ and $Q = S$, we can rewrite equations (3) and (4): From equation (3): $P + Q + Q + D = 15$ $P + 2Q + D = 15$ --- (Equation 6)

From equation (4): $P + Q + P + B = 17$ $2P + Q + B = 17$ --- (Equation 7)

Step 3: Find the value of $P + Q$ Add Equation (6) and Equation (7) together: $(P + 2Q + D) + (2P + Q + B) = 15 + 17$ $3P + 3Q + B + D = 32$

We already know from Step 1 that $B + D = 11$. Substitute this into the equation: $3(P + Q) + 11 = 32$ $3(P + Q) = 21$ $P + Q = 7$

Step 4: Evaluate the given options

• Option A: States that $B + D < P + Q$. We found $B + D = 11$ and $P + Q = 7$. Since $11$ is not less than $7$, this statement is incorrect.
• Option B: States that $P + Q > A + C$. We found $P + Q = 7$ and we are given $A + C = 6$. Since $7 > 6$, this statement is correct.
• Options C and D: We only know that $P + Q = 7$. The individual weights of $P$ and $Q$ cannot be uniquely determined from the given information (e.g., $P$ could be $4$ and $Q$ could be $3$, or vice versa). Thus, we cannot definitively say whether $P > Q$ or $Q > P$.

Therefore, the only correct statement is Option B.