The present value P of an investment can be determined using the equation \(\mathrm{P = \frac{F}{(1 + r)^n}}\), where F...

1. TRANSLATE the problem requirements

• Given: \(\mathrm{P = F/(1 + r)^n}\) for any number of periods
• Need: Expression for \(\mathrm{r}\) when \(\mathrm{n = 1}\) (single period)
• Find: Which answer choice correctly expresses \(\mathrm{r}\) in terms of \(\mathrm{P}\) and \(\mathrm{F}\)

2. INFER the solution approach

• Since we want the single period case, substitute \(\mathrm{n = 1}\) into the original formula
• Then use algebraic manipulation to isolate \(\mathrm{r}\) on one side of the equation

3. SIMPLIFY by substituting n = 1

• Original: \(\mathrm{P = F/(1 + r)^n}\)
• With \(\mathrm{n = 1}\): \(\mathrm{P = F/(1 + r)^1 = F/(1 + r)}\)

4. SIMPLIFY by solving for r systematically

• Start with: \(\mathrm{P = F/(1 + r)}\)
• Multiply both sides by \(\mathrm{(1 + r)}\): \(\mathrm{P(1 + r) = F}\)
• Divide both sides by \(\mathrm{P}\): \(\mathrm{1 + r = F/P}\)
• Subtract 1 from both sides: \(\mathrm{r = F/P - 1}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic manipulation errors during the isolation process.

Common mistakes include flipping \(\mathrm{P}\) and \(\mathrm{F}\) when dividing (getting \(\mathrm{P/F}\) instead of \(\mathrm{F/P}\)), or incorrectly handling the subtraction of 1. For example, they might write \(\mathrm{r = 1 - F/P}\) instead of \(\mathrm{r = F/P - 1}\), confusing the order of operations.

This may lead them to select Choice B (\(\mathrm{r = P/F - 1}\)) or Choice C (\(\mathrm{r = 1 - F/P}\)).

Second Most Common Error:

Poor INFER reasoning: Students try to work with the general formula \(\mathrm{P = F/(1 + r)^n}\) without first substituting \(\mathrm{n = 1}\).

This makes the algebraic manipulation much more complex and often leads to confusion about how to handle the exponent. Students may get overwhelmed and either guess randomly or attempt incorrect algebraic steps.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

The key insight is recognizing that substituting \(\mathrm{n = 1}\) first dramatically simplifies the problem, turning it into a straightforward algebraic manipulation. Students who jump straight into complex algebraic work with the general formula often get lost in unnecessary complications.