The above example to answer questions 30 - 32.

Given: $L = 6$, $R = 9$, $B = 216$. Find $C = X$.
Let's test some common operations:

• If $C = L \times R$: $X = 6 \times 9 = 54$.
  • Then, how is $B = 216$ related to $X = 54$? $216 = 54 \times 4$.
  • So, the pattern could be: $C = L \times R$ and $B = C \times 4$.
• If $C = L \times L$: $X = 6 \times 6 = 36$.
  • Then, how is $B = 216$ related to $X = 36$? $216 = 36 \times 6$.
  • So, the pattern could be: $C = L \times L$ and $B = C \times L$. This means $B = L \times L \times L = L^3$.
  • Let's check this: $6^3 = 216$. This works perfectly for Q30!
  • If this pattern is correct, then $X = L^2 = 6^2 = 36$.

Given: $L = 9$, $R = 3$, $B = 6$. Find $C = X$.
Using the pattern $B = L^3$:
$6 = 9^3 = 729$. This is incorrect. So the pattern $B = L^3$ is not universally applicable.

Let's re-evaluate the pattern. The pattern must involve all three numbers connected to the central number.
Consider the relationship: $C = \frac{B}{L \times R}$.

• For Q30: $X = \frac{216}{6 \times 9} = \frac{216}{54} = 4$.
  • Is 4 an option for Q30? No. Options are A. 18, B. 36, C. 108, D. 54, E. 11.

Let's consider the relationship: $C = \frac{B}{L} \times R$.

• For Q30: $X = \frac{216}{6} \times 9 = 36 \times 9 = 324$. Too large.

Let's consider the relationship: $C = \frac{B}{R} \times L$.

• For Q30: $X = \frac{216}{9} \times 6 = 24 \times 6 = 144$. Too large.

Let's consider the relationship: $C = \frac{B}{L+R}$.

• For Q30: $X = \frac{216}{6+9} = \frac{216}{15} = 14.4$. Not an option.

Let's consider the relationship: $C = \frac{B}{L-R}$.

• For Q30: $X = \frac{216}{6-9} = \frac{216}{-3} = -72$. Not an option.

Let's consider the relationship: $C = \frac{B}{R-L}$.

• For Q30: $X = \frac{216}{9-6} = \frac{216}{3} = 72$. Not an option.

Let's try a different approach. What if the bottom number is the product of the central number and one of the side numbers?

• If $B = C \times L$:
  • For Q30: $216 = X \times 6 \implies X = \frac{216}{6} = 36$.
  • If $X=36$, how is it related to $L=6$ and $R=9$? $36 = 6 \times 6$ and $36 = 9 \times 4$.
  • This implies $X = L \times L$.
  • So, the pattern would be: $C = L \times L$ and $B = C \times L$. This means $B = L^3$.
  • As shown before, this works for Q30 ($6^3 = 216$), so $X = 6^2 = 36$. This is option B for Q30.
  • Let's test this on Q31: $L=9, R=3, B=6$.
    • $C = L \times L = 9 \times 9 = 81$.
    • $B = C \times L = 81 \times 9 = 729$.
    • But $B$ is given as 6. This pattern is inconsistent.

The example at the very top of the page (2-25-5, 3-18-4, 49-7, 80-7, 343) seems to be a general illustration of number relationships, not a direct operational example for the questions below. The questions 30-32 have a specific structure.

Let's assume the pattern is $C = L \times R$ and $B = C \times (R-L)$.

• For Q30: $L=6, R=9, B=216$.
  • $C = 6 \times 9 = 54$.
  • $B = 54 \times (9-6) = 54 \times 3 = 162$. This is not 216.

Let's assume the pattern is $C = L \times R$ and $B = C \times (L+R)$.

• For Q30: $L=6, R=9, B=216$.
  • $C = 6 \times 9 = 54$.
  • $B = 54 \times (6+9) = 54 \times 15 = 810$. This is not 216.

Let's assume the pattern is $C = L \times R$ and $B = C \times \frac{R}{L}$.

• For Q30: $L=6, R=9, B=216$.
  • $C = 6 \times 9 = 54$.
  • $B = 54 \times \frac{9}{6} = 54 \times 1.5 = 81$. This is not 216.

Let's assume the pattern is $C = L \times R$ and $B = C \times \frac{L}{R}$.

• For Q30: $L=6, R=9, B=216$.
  • $C = 6 \times 9 = 54$.
  • $B = 54 \times \frac{6}{9} = 54 \times \frac{2}{3} = 36$. This is not 216.

Let's try the pattern: $C = L \times R$. And $B = C \times (\text{some constant})$.

• For Q30: $X = 6 \times 9 = 54$. $216 = 54 \times 4$. So the constant is 4.
• For Q31: $X = 9 \times 3 = 27$. $B = 6$. If the constant is 4, then $27 \times 4 = 108 \neq 6$. This is not consistent.

Let's try the pattern: $C = L \times R$. And $B = C \times (R-L+k)$ or similar.

Let's reconsider the possibility that the central number is the product of the two side numbers, and the bottom number is the product of the central number and the left number.

• Pattern: $C = L \times R$ and $B = C \times L$.
  • Q30: $L=6, R=9, B=216$.
    • $C = 6 \times 9 = 54$.
    • $B = 54 \times 6 = 324$. This is not 216.

Let's try the pattern: $C = L \times R$. And $B = C \times R$.

• Pattern: $C = L \times R$ and $B = C \times R$.
  • Q30: $L=6, R=9, B=216$.
    • $C = 6 \times 9 = 54$.
    • $B = 54 \times 9 = 486$. This is not 216.

Let's try the pattern: $C = L \times R$. And $B = C \times (L+R)/k$.

What if the bottom number is the product of the central number and the difference of the side numbers?

• Pattern: $C = L \times R$ and $B = C \times (R-L)$.
  • Q30: $L=6, R=9, B=216$.
    • $C = 6 \times 9 = 54$.
    • $B = 54 \times (9-6) = 54 \times 3 = 162$. Not 216.

What if the bottom number is the product of the central number and the sum of the side numbers?

• Pattern: $C = L \times R$ and $B = C \times (L+R)$.
  • Q30: $L=6, R=9, B=216$.
    • $C = 6 \times 9 = 54$.
    • $B = 54 \times (6+9) = 54 \times 15 = 810$. Not 216.

Let's try a pattern where the bottom number is the product of the three numbers: $L \times R \times C = B$.

• For Q30: $6 \times 9 \times X = 216 \implies 54X = 216 \implies X = \frac{216}{54} = 4$.
  • This is not among the options for Q30.

Let's try the pattern: $C = \frac{B}{L \times R}$.

• For Q30: $X = \frac{216}{6 \times 9} = \frac{216}{54} = 4$. Not in options.

Let's try the pattern: $C = \frac{B}{L} - R$.

• For Q30: $X = \frac{216}{6} - 9 = 36 - 9 = 27$. Not in options.

Let's try the pattern: $C = \frac{B}{R} - L$.

• For Q30: $X = \frac{216}{9} - 6 = 24 - 6 = 18$.
  • This is option A for Q30. Let's test this pattern on Q31.

Given: $L = 9$, $R = 3$, $B = 6$. Find $C = X$.
Using the pattern $C = \frac{B}{R} - L$:

$$X = \frac{6}{3} - 9$$ $$X = 2 - 9$$ $$X = -7$$

This is not among the options for Q31 (A. 18, B. 27, C. 36, D. 54, E. 108). So this pattern is also inconsistent.

This implies that the pattern is not straightforward or there might be an error in the question/options. Given the constraints, I must find a consistent pattern that leads to one of the options.

Let's reconsider the pattern $B = L^3$ for Q30. It worked perfectly.
$L=6 \implies C=L^2=36$, $B=L^3=216$.
If this is the pattern, then $X=36$ for Q30.

Let's assume the pattern is $C = L \times R$.

• Q30: $X = 6 \times 9 = 54$. (Option D)
• Q31: $X = 9 \times 3 = 27$. (Option B)
• Q32: The central number is -3.6. The left number is X, the right number is 4.
  • So, $-3.6 = X \times 4 \implies X = \frac{-3.6}{4} = -0.9$.
  • This is not among the options for Q32 (A. -12.96, B. -12.40, C. 12.40, D. 12.96, E. 14.40).
  • So, $C = L \times R$ is not the pattern for Q32.

Let's assume the pattern is $C = L \times R$ and $B = C \times (L+R)/k$ or $B = C \times (L \times R)/k$.

Let's try to find a pattern that works for Q32 first, as it has negative numbers.
Q32: $L=X, R=4, C=-3.6, B=-57.6$.
If $C = L \times R$: $-3.6 = X \times 4 \implies X = -0.9$.
If $B = C \times R$: $-57.6 = -3.6 \times 4 = -14.4$. This is not -57.6.
If $B = C \times L$: $-57.6 = -3.6 \times X \implies X = \frac{-57.6}{-3.6} = 16$.
If $X=16$, then $C = L \times R = 16 \times 4 = 64$. But $C=-3.6$. So this is not consistent.

Let's try the pattern: $B = C \times L \times R$.

• For Q32: $-57.6 = -3.6 \times X \times 4 \implies -57.6 = -14.4X \implies X = \frac{-57.6}{-14.4} = 4$.
  • Is 4 an option for Q32? No.

Let's try the pattern: $B = C \times (L+R)$.

• For Q32: $-57.6 = -3.6 \times (X+4)$.
  • $\frac{-57.6}{-3.6} = X+4$
  • $16 = X+4$
  • $X = 12$.
  • Is 12 an option for Q32? No.

Let's try the pattern: $B = C \times (L \times R)$.

• For Q32: $-57.6 = -3.6 \times (X \times 4)$.
  • $-57.6 = -14.4X$.
  • $X = \frac{-57.6}{-14.4} = 4$. Not an option.

Let's try the pattern: $B = C \times L \times R$.

• For Q32: $-57.6 = -3.6 \times X \times 4 \implies -57.6 = -14.4X \implies X = 4$. Not an option.

Let's try the pattern: $B = C \times (L+R)$.

• For Q32: $-57.6 = -3.6 \times (X+4) \implies 16 = X+4 \implies X = 12$. Not an option.

Let's try the pattern: $B = C \times (L-R)$.

• For Q32: $-57.6 = -3.6 \times (X-4) \implies 16 = X-4 \implies X = 20$. Not an option.

Let's try the pattern: $B = C \times (R-L)$.

• For Q32: $-57.6 = -3.6 \times (4-X) \implies 16 = 4-X \implies X = 4-16 = -12$.
  • Is -12 an option for Q32? No, but -