x y z invest in a partnership. the amounts that x y z invest are in the ratio of x:y:z and their profits are in the ratio of z:y:x the ratio of their periods of investment is

Let's denote the amounts invested by \( X \), \( Y \), and \( Z \) as \( x \), \( y \), and \( z \), respectively. The ratio of their investments is given as \( x : y : z \).

Let's denote the periods of their investments by \( T_X \), \( T_Y \), and \( T_Z \).

Since the profits are in the ratio \( z : y : x \) (reversing the ratio of their investments), the profits will be proportional to the product of the amount invested and the time period of investment. Therefore, we have:

\[
x \times T_X : y \times T_Y : z \times T_Z = z : y : x
\]

This implies:

\[
\frac{x \times T_X}{z} = \frac{y \times T_Y}{y} = \frac{z \times T_Z}{x}
\]

Simplifying these ratios, we get:

\[
\frac{T_X}{\frac{z}{x}} = \frac{T_Y}{1} = \frac{T_Z}{\frac{x}{z}}
\]

Therefore, the ratio of the periods of investment \( T_X : T_Y : T_Z \) is:

\[
\frac{z}{x} : 1 : \frac{x}{z}
\]

Thus, the ratio of their periods of investment is:

\[
\boxed{\frac{z}{x} : 1 : \frac{x}{z}}
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. If \( x = 2 \), \( y = 3 \), and \( z = 6 \), what is the ratio of their periods of investment?
2. How would the ratio of periods change if the profit ratio was \( x : y : z \) instead of \( z : y : x \)?
3. Can this method be applied to other types of proportional investment scenarios?
4. What if \( x \), \( y \), and \( z \) are functions of time? How would the ratio of periods change?
5. How does changing the period of investment for one person affect the overall profit distribution?

**Tip:** When working with ratios, it’s important to keep track of the corresponding elements being compared to avoid confusion.

Explore how to determine the ratio of periods of investment based on investment amounts and profit ratios. This problem involves understanding proportional relationships in investments. Suitable for advanced high school students.

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