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The problem states that:

- \( x \) varies inversely as \( y \).
- \( y \) varies directly as \( z \).

Let's break this down:

1. **Inverse Variation**: If \( x \) varies inversely as \( y \), this means:
   \[
   x = \frac{k_1}{y}
   \]
   where \( k_1 \) is a constant.

2. **Direct Variation**: If \( y \) varies directly as \( z \), this means:
   \[
   y = k_2 z
   \]
   where \( k_2 \) is another constant.

Now, substitute the expression for \( y \) into the equation for \( x \):

\[
x = \frac{k_1}{y} = \frac{k_1}{k_2 z}
\]

Thus, \( x \) is inversely proportional to \( z \), meaning:

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

So, the relationship between \( x \) and \( z \) is that \( x \) varies **inversely** as \( z \).

Would you like further clarification on any part of this? Here are five related questions to deepen understanding:

1. What happens to \( x \) if \( z \) increases and the constants remain the same?
2. How would the equations change if \( x \) varied directly with \( y \) instead?
3. Can you provide an example with specific values for \( k_1 \), \( k_2 \), and \( z \) to compute \( x \)?
4. How would the relationship between \( x \) and \( z \) change if \( y \) were a constant?
5. How is the inverse proportional relationship different from a direct proportional relationship?

**Tip:** Always verify the type of variation (direct or inverse) before setting up the equations for proportional relationships.

If x varies inversely as y and y varies directly as z, what is the relationship between x and z?

This page explains the relationship between x and z when x varies inversely with y and y varies directly with z. Using the laws of proportionality, the solution shows that x is inversely proportional to z, with step-by-step calculations.

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