A, B, and C can complete a work in 15 days, 12 days, and 10 days respectively. If A and C work together for *x* days and B completes the remaining work in *2x* days, find the value of $3x$.

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**Solution:**

1. Calculate the daily work rates of A, B, and C:
   - A's rate = $\frac{1}{15}$ work/day
   - B's rate = $\frac{1}{12}$ work/day
   - C's rate = $\frac{1}{10}$ work/day

2. A and C work together for *x* days, so work done by A and C together:
   $$x 	imes \left(\frac{1}{15} + \frac{1}{10}ight) = x 	imes \frac{1}{6}$$
   (Since $\frac{1}{15} + \frac{1}{10} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6}$)

3. B works for $2x$ days, so work done by B:
   $$2x 	imes \frac{1}{12} = \frac{x}{6}$$

4. Total work done by A, B, and C together is 1 (complete work):
   $$\frac{x}{6} + \frac{x}{6} = 1$$

5. Simplify the equation:
   $$\frac{2x}{6} = 1 \implies \frac{x}{3} = 1 \implies x = 3$$

6. Find $3x$:
   $$3x = 3 	imes 3 = 9$$

**Answer:** $9$

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