The given equation relates the positive numbers P, N, and C. Which equation correctly expresses C in terms of P...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{P = N(19 - C)}\)
• Goal: Express \(\mathrm{C}\) in terms of \(\mathrm{P}\) and \(\mathrm{N}\)
• All variables represent positive numbers

2. INFER the approach

• We need to isolate \(\mathrm{C}\) by undoing the operations affecting it
• \(\mathrm{C}\) is currently inside parentheses, multiplied by \(\mathrm{N}\)
• Strategy: Work backwards - first undo the multiplication by \(\mathrm{N}\), then undo the subtraction

3. SIMPLIFY by dividing both sides by N

• \(\mathrm{P = N(19 - C)}\)
• \(\mathrm{P/N = (19 - C)}\)
• Now we have: \(\mathrm{P/N = 19 - C}\)

4. SIMPLIFY by isolating the C term

• \(\mathrm{P/N = 19 - C}\)
• Subtract 19 from both sides: \(\mathrm{P/N - 19 = -C}\)
• We now have \(\mathrm{C}\) with a negative coefficient

5. SIMPLIFY by eliminating the negative sign

• \(\mathrm{P/N - 19 = -C}\)
• Multiply both sides by -1: \(\mathrm{-(P/N - 19) = C}\)
• This gives us: \(\mathrm{19 - P/N = C}\)

Answer: D. \(\mathrm{C = 19 - P/N}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when working with the negative coefficient of \(\mathrm{C}\).

Many students correctly get to \(\mathrm{P/N - 19 = -C}\) but then incorrectly think that \(\mathrm{C = P/N - 19}\), forgetting that \(\mathrm{C}\) has a negative coefficient. This sign confusion leads them to an expression that's off by a negative sign.

This may lead them to select Choice C (\(\mathrm{C = 19 + P/N}\)) or cause confusion and guessing.

Second Most Common Error:

Poor INFER reasoning about operation order: Students attempt to "move terms around" without systematic algebraic steps.

Some students try to rearrange terms intuitively rather than following proper algebraic procedures. They might incorrectly try to get \(\mathrm{C}\) alone by manipulating the original equation \(\mathrm{P = N(19 - C)}\) without first eliminating the multiplication by \(\mathrm{N}\).

This leads to confusion and incorrect algebraic manipulations, potentially leading them to select Choice A (\(\mathrm{C = (19+P)/N}\)) or Choice B (\(\mathrm{C = (19-P)/N}\)).

The Bottom Line:

This problem tests whether students can execute multi-step algebraic manipulations while carefully tracking signs and following the proper sequence of operations to isolate a variable.