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

Analyze the question: We are given two variation relationships and must combine them to find how x relates to z.

Concept:

• "x varies inversely as y" means x ∝ 1/y, so x = k₁/y.
• "y varies directly as z" means y ∝ z, so y = k₂z.

Solution:

1. Start with x = k₁/y.
2. Substitute y = k₂z:

x = k₁/(k₂z) = (k₁/k₂) · (1/z)

3. Since k₁/k₂ is just another constant (call it k):

x = k/z

So x ∝ 1/z.

Checking the other options:

• A (x ∝ z) would be direct variation, which is wrong because z sits in the denominator.
• C ("a ∝ z²") has a typo, uses "a" instead of x, and involves a square that never appears.
• D (x ∝ 1/z²) would need y to vary as z², but y only varies as z, so the power stays 1.

Answer: Option B.

Tip: Combine variations by substituting. Inverse of direct gives inverse: if y ∝ z, then 1/y ∝ 1/z. The powers carry through, so no new squares appear unless a relationship itself involves a square.