The equation 8s + p = d relates distance d (in miles), initial position p (in miles), and speed s...

1. TRANSLATE the problem requirements

• Given: The equation \(\mathrm{8s + p = d}\)
• Find: The value of \(\mathrm{d - p}\) expressed in terms of \(\mathrm{s}\)
• This means we need to write \(\mathrm{d - p}\) using only the variable \(\mathrm{s}\) (no \(\mathrm{d}\) or \(\mathrm{p}\) in our final answer)

2. INFER the solution strategy

• Since we need \(\mathrm{d - p}\) but only have an equation with all three variables, we should substitute the expression for \(\mathrm{d}\)
• From \(\mathrm{8s + p = d}\), we can write \(\mathrm{d = 8s + p}\)
• Then substitute this into \(\mathrm{d - p}\)

3. SIMPLIFY through algebraic manipulation

\(\mathrm{d - p = (8s + p) - p}\)
\(\mathrm{d - p = 8s + p - p}\)
\(\mathrm{d - p = 8s}\)

Answer: \(\mathrm{(D) \ 8s}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might misunderstand what "express in terms of \(\mathrm{s}\)" means and try to solve for individual variables instead of finding the value of the expression \(\mathrm{d - p}\).

For example, they might try to solve for \(\mathrm{s}\) first, leading to confusion about how to use the result. This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors during the substitution and simplification process.

They might write \(\mathrm{d - p = (8s + p) - p}\) but then incorrectly simplify to get \(\mathrm{8s + p}\) or \(\mathrm{8s - p}\) instead of \(\mathrm{8s}\). This may lead them to select Choice \(\mathrm{(B) \ (s)}\) if they somehow lose the coefficient, or get confused between similar-looking expressions.

The Bottom Line:

This problem tests whether students can work with expressions rather than solving for individual variables, and whether they can perform clean algebraic substitution without getting lost in the manipulation steps.