An oceanographer uses the equation s = 3/2p to model the speed s, in knots, of an ocean wave, where...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{s = \frac{3}{2}p}\)
  • s represents speed in knots
  • p represents period in seconds
• What we need: Express p in terms of s (solve for p)

2. INFER the solution strategy

• Since we have \(\mathrm{s = \frac{3}{2}p}\) and need p = [something with s], we need to isolate p
• The coefficient of p is \(\frac{3}{2}\), so we'll multiply both sides by its reciprocal \(\frac{2}{3}\)

3. SIMPLIFY by applying the reciprocal

• Multiply both sides by \(\frac{2}{3}\):

\(\frac{2}{3} \times \mathrm{s} = \frac{2}{3} \times \frac{3}{2}\mathrm{p}\)

• Simplify the right side:

\(\frac{2}{3}\mathrm{s} = \frac{2 \times 3}{3 \times 2} \times \mathrm{p}\)

\(= \frac{6}{6} \times \mathrm{p}\)

\(= 1\mathrm{p}\)

\(= \mathrm{p}\)

• Final result: \(\mathrm{p = \frac{2}{3}s}\)

Answer: A. \(\mathrm{p = \frac{2}{3}s}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students attempt the reciprocal approach but make computational errors with fraction multiplication. They might incorrectly calculate \(\frac{2}{3} \times \frac{3}{2}\) or get confused about which fraction is the reciprocal of \(\frac{3}{2}\).

This may lead them to select Choice B (\(\mathrm{p = \frac{3}{2}s}\)) by incorrectly keeping the original coefficient.

Second Most Common Error:

Inadequate TRANSLATE reasoning: Students misunderstand what "in terms of" means algebraically and attempt addition instead of multiplication. They think solving means adding something to both sides rather than using multiplicative inverse.

This may lead them to select Choice C (\(\mathrm{p = \frac{2}{3} + s}\)) or Choice D (\(\mathrm{p = \frac{3}{2} + s}\)) through incorrect additive reasoning.

The Bottom Line:

This problem tests fundamental algebraic manipulation skills. Success requires both recognizing the need to isolate a variable and executing fraction arithmetic correctly - two skills that many students struggle with when combined.