If 4s/t = 6.7 and s/tn = 26.8, what is the value of n?

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\frac{4\mathrm{s}}{\mathrm{t}} = 6.7\)
  • Second equation: \(\frac{\mathrm{s}}{\mathrm{tn}} = 26.8\)
• Need to find: the value of n

2. INFER the solution strategy

• Notice both equations contain the term \(\frac{\mathrm{s}}{\mathrm{t}}\)
• Strategy: Isolate \(\frac{\mathrm{s}}{\mathrm{t}}\) from the first equation, then substitute it into the second equation
• This will allow us to solve directly for n

3. SIMPLIFY the first equation to find s/t

• Start with: \(\frac{4\mathrm{s}}{\mathrm{t}} = 6.7\)
• Divide both sides by 4: \(\frac{\mathrm{s}}{\mathrm{t}} = \frac{6.7}{4} = 1.675\) (use calculator)

4. SIMPLIFY the second equation for substitution

• Rewrite \(\frac{\mathrm{s}}{\mathrm{tn}} = 26.8\) as: \(\frac{\mathrm{s}}{\mathrm{t}} \times \frac{1}{\mathrm{n}} = 26.8\)
• This shows we can substitute our value for \(\frac{\mathrm{s}}{\mathrm{t}}\)

5. SIMPLIFY by substituting and solving

• Substitute \(\frac{\mathrm{s}}{\mathrm{t}} = 1.675\): \((1.675) \times \frac{1}{\mathrm{n}} = 26.8\)
• This gives us: \(\frac{1.675}{\mathrm{n}} = 26.8\)
• Multiply both sides by n: \(1.675 = 26.8\mathrm{n}\)
• Divide both sides by 26.8: \(\mathrm{n} = \frac{1.675}{26.8} = 0.0625\) (use calculator)

Answer: .0625 (or 1/16 or 0.062 or 0.063)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the strategic connection between the two equations through the common \(\frac{\mathrm{s}}{\mathrm{t}}\) term. Instead, they might try to solve each equation independently or attempt to isolate s, t, and n separately, leading to a much more complex system. This approach quickly becomes unworkable and causes students to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the critical division steps, particularly when calculating \(6.7 \div 4 = 1.675\) or the final step \(1.675 \div 26.8 = 0.0625\). Common calculation mistakes include getting 1.75 instead of 1.675, or miscalculating the final division, leading to incorrect decimal answers.

The Bottom Line:

This problem tests whether students can recognize structural relationships between equations and use substitution strategically. The key insight is seeing that both equations share the \(\frac{\mathrm{s}}{\mathrm{t}}\) term, making substitution the most efficient path to the solution.