Equalized multiples define shares: Some money is divided among A, B, and C such that 5 times A’s share equals 12 times B’s share, which also equals 6 times C’s share. Determine the ratio A : B : C.

Introduction / Context:
When several scaled versions of different shares are equal, set each equal to a common constant. Then express each share in terms of that constant and clear denominators to obtain an integer ratio.

Given Data / Assumptions:

• 5A = 12B = 6C = k, for some k > 0.

Concept / Approach:
From 5A = k ⇒ A = k/5; from 12B = k ⇒ B = k/12; from 6C = k ⇒ C = k/6. Combine these into A : B : C and multiply by the LCM of denominators to get whole numbers.

Step-by-Step Solution:

Verification / Alternative check:
Using A : B : C = 12 : 5 : 10, confirm 5A : 12B : 6C = 60 : 60 : 60, which are equal as required.

Why Other Options Are Wrong:
5 : 10 : 12 and 10 : 12 : 5 permute the correct order improperly; 5 : 12 : 10 misplaces A and B; 6 : 5 : 10 comes from picking an arbitrary scale rather than the exact LCM-based simplification.

Common Pitfalls:
Forgetting to use a common constant or not clearing denominators, which leads to fractional ratios or misordered terms.

Final Answer:
12 : 5 : 10