A carton contains a dozen (12) mirrors and is dropped. Which of the following cannot be the ratio of broken mirrors to unbroken mirrors?

Introduction / Context:
For a ratio to be feasible, broken + unbroken must equal the fixed total (12 mirrors). Thus, the sum of the ratio terms times a common multiplier must be 12. This divisibility check quickly rules out impossible options.

Given Data / Assumptions:

• Total mirrors = 12
• Broken : Unbroken must sum to 12 when scaled by some integer k.
• We test each option by checking if (a + b) * k = 12 has an integer k.

Concept / Approach:
If the ratio is a : b, then numbers are ak and bk for some integer k. Total = (a + b)k must be 12. Therefore, 12 must be divisible by (a + b).

Step-by-Step Solution:
2 : 1 ⇒ a + b = 3; 12/3 = 4 ⇒ possible3 : 1 ⇒ a + b = 4; 12/4 = 3 ⇒ possible3 : 2 ⇒ a + b = 5; 12/5 is not an integer ⇒ impossible7 : 5 ⇒ a + b = 12; 12/12 = 1 ⇒ possible

Verification / Alternative check:
Try to construct numbers for 3 : 2; you cannot reach exactly 12 total with integer counts, confirming impossibility.

Why Other Options Are Wrong:
They are feasible because their combined parts divide 12 exactly.

Common Pitfalls:
Forgetting that counts of items must be whole numbers; using non-integer k makes the ratio unrealizable.

Final Answer:
3 : 2