X, Y, and Z can complete a work in 20 days, 16 days, and 8 days respectively. If X and Z work together for $D$ days, and Y completes the remaining work in $2D$ days, then find the value of $3D$.

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**Solution:**

1. Calculate the daily work rates of X, Y, and Z:
   - X's rate = $\frac{1}{20}$ work/day
   - Y's rate = $\frac{1}{16}$ work/day
   - Z's rate = $\frac{1}{8}$ work/day

2. X and Z work together for $D$ days, so the work done by them is:
   $$D 	imes \left( \frac{1}{20} + \frac{1}{8} ight) = D 	imes \left( \frac{1}{20} + \frac{5}{40} ight) = D 	imes \frac{7}{40}$$

3. The remaining work is completed by Y in $2D$ days:
   $$	ext{Work done by Y} = 2D 	imes \frac{1}{16} = \frac{2D}{16} = \frac{D}{8}$$

4. Total work done is 1, so:
   $$D 	imes \frac{7}{40} + \frac{D}{8} = 1$$

5. Find a common denominator and solve for $D$:
   $$\frac{7D}{40} + \frac{5D}{40} = 1$$
   $$\frac{12D}{40} = 1$$
   $$12D = 40$$
   $$D = \frac{40}{12} = \frac{10}{3}$$

6. Calculate $3D$:
   $$3D = 3 	imes \frac{10}{3} = 10$$

**Answer:** $10$

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