M varies directly as n and inversely as the square of p. If M= 3 when n = 2 and p = 1, find M in terms of n and p.

Analyze the question: M varies directly as n and inversely as p², so we write a variation equation with a constant k and then find k from the given values.

Concept:

• Direct variation: M ∝ n
• Inverse variation: M ∝ 1/p²
• Combined: M = kn/p²

Solution:

1. Write the equation: M = kn/p²

2. Substitute M = 3, n = 2, p = 1: 3 = k(2)/(1)² 3 = 2k k = 3/2

3. Put k back into the equation: M = (3/2)(n/p²) = 3n/(2p²)

Check: n = 2, p = 1 gives M = 3(2)/(2·1) = 3 ✓

Answer: A. M = 3n/(2p²)

Tip: In variation questions, always follow the same three steps: write the equation with k, find k from the given values, then rewrite the equation with k substituted. "Directly" puts the variable on top, and "inversely" puts it at the bottom. Squares and other powers go with the variable they describe.