If $40\%$ of $A = 60\%$ of $B = \frac{1}{6}$ of $C$, then find the ratio of $A : B : C$.

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**Solution:**

1. Let the common value be $k$. Then,
   $$40\% 	ext{ of } A = 60\% 	ext{ of } B = \frac{1}{6} 	ext{ of } C = k$$

2. Express each variable in terms of $k$:
   - $40\%$ of $A = k \Rightarrow 0.4A = k \Rightarrow A = \frac{k}{0.4} = \frac{k}{\frac{2}{5}} = \frac{5k}{2}$
   
   - $60\%$ of $B = k \Rightarrow 0.6B = k \Rightarrow B = \frac{k}{0.6} = \frac{k}{\frac{3}{5}} = \frac{5k}{3}$

- $\frac{1}{6}$ of $C = k \Rightarrow \frac{C}{6} = k \Rightarrow C = 6k$

3. Now, the ratio $A : B : C$ is:
   $$\frac{5k}{2} : \frac{5k}{3} : 6k$$

4. To simplify, multiply all terms by the LCM of denominators, which is 6:
   $$\left(\frac{5k}{2} 	imes 6ight) : \left(\frac{5k}{3} 	imes 6ight) : (6k 	imes 6)$$
   $$= 15k : 10k : 36k$$

5. Cancel $k$ from all terms:
   $$15 : 10 : 36$$

**Answer:** $15 : 10 : 36$

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