A, B and C invested capital in the ratio of $3 : 4 : 8$. At the end of the business term, they received the profit in the ratio of $2 : 3 : 5$. What is the ratio of their invested time?

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**Solution:**

1. Let the capitals of A, B, and C be in the ratio $3 : 4 : 8$.
2. Let the times invested by A, B, and C be $x$, $y$, and $z$ respectively.
3. The profit ratio is given as $2 : 3 : 5$.

Since profit is proportional to the product of capital and time, we have:

$$	ext{Capital} 	imes 	ext{Time} = 	ext{Profit}$$

So,

- For A: $3x = 2$
- For B: $4y = 3$
- For C: $8z = 5$

4. Solve for $x$, $y$, and $z$:

$$x = \frac{2}{3}, \quad y = \frac{3}{4}, \quad z = \frac{5}{8}$$

5. To find the ratio $x : y : z$, multiply each by the LCM of denominators $3, 4, 8$ which is $24$:

$$x : y : z = \frac{2}{3} 	imes 24 : \frac{3}{4} 	imes 24 : \frac{5}{8} 	imes 24 = 16 : 18 : 15$$

**Answer:** $16 : 18 : 15$

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