A quantity y varies inversely with w, so y = k/w for a constant k (where w neq 0).The constant...

1. TRANSLATE the problem information

• Given information:
  • y varies inversely with w: \(\mathrm{y = k/w}\)
  • The constant \(\mathrm{k = -54}\)
  • When \(\mathrm{y = 6}\), find w

• This means we substitute these values into our inverse variation equation

2. TRANSLATE into a solvable equation

• Substitute the known values into \(\mathrm{y = k/w}\):

\(\mathrm{6 = -54/w}\)

• Now we have a simple algebraic equation to solve

3. SIMPLIFY to isolate w

• Multiply both sides by w to eliminate the fraction:

\(\mathrm{6w = -54}\)

• Divide both sides by 6:

\(\mathrm{w = -54/6 = -9}\)

4. Verify the answer

• Check: If \(\mathrm{w = -9}\), then \(\mathrm{y = -54/(-9) = 6}\) ✓

Answer: -9

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution with negative numbers: Students often struggle with the sign when dividing -54 by 6, or they might incorrectly handle the negative signs throughout the algebraic manipulation. For example, they might get \(\mathrm{w = 9}\) instead of \(\mathrm{w = -9}\), forgetting that \(\mathrm{-54 ÷ 6 = -9}\).

This confusion with negative number operations can lead to selecting the wrong answer or getting stuck.

Second Most Common Error:

Weak TRANSLATE reasoning: Some students set up the equation incorrectly, such as writing \(\mathrm{6 = w/(-54)}\) instead of \(\mathrm{6 = -54/w}\), because they misunderstand how to substitute values into the inverse variation formula.

This incorrect setup leads to solving the wrong equation entirely and getting an incorrect answer.

The Bottom Line:

Success on this problem requires careful attention to negative signs and accurate substitution into the inverse variation formula. The algebraic manipulation is straightforward once the equation is set up correctly.