Solve the Arithmetic Puzzle: a×b×8c : (2a : (72b×c))

To solve the problem, follow these detailed steps:

• Step 1: Simplify the innermost expression:
  Calculate $2a : (72b \cdot c)$. This equates to dividing $2a$ by $72b \cdot c$:
  $2a : (72b \cdot c) = \frac{2a}{72b \cdot c}$
• Step 2: Simplification
  $= \frac{2a}{72b \cdot c} = \frac{2a}{72bc} = \frac{a}{36bc}$
• Step 3: Complete the outer expression:
  Now, substitute this back into the outer expression:
  $a \cdot b \cdot 8c : \left(\frac{a}{36bc}\right)$
• Step 4: Simplify this result:
  This means multiplying $a \cdot b \cdot 8c$ by the reciprocal of $\frac{a}{36bc}$):
  $= a \cdot b \cdot 8c \cdot \frac{36bc}{a}$
• Step 5: Cancel and compute:
  Notice $a$ cancels out:
  $= b \cdot 8c \cdot 36bc$
• Step 6: Final simplification:
  $= 8 \cdot 36 \cdot b^2 \cdot c^2$
  $= 288b^2c^2$

Therefore, the solution to the given expression is $288b^2c^2$, which corresponds to choice ID 4.