The principal amount, P, that must be invested at a simple annual interest rate, r, to have a total amount,...

1. INFER the approach needed

• Given: \(\mathrm{P = \frac{A}{(1 + rt)}}\)
• Goal: Solve for r in terms of P, A, and t
• Strategy: Use algebraic manipulation to isolate r

2. SIMPLIFY by clearing the fraction first

• Multiply both sides by (1 + rt):

\(\mathrm{P \times (1 + rt) = A}\)

• This eliminates the fraction and gives us: \(\mathrm{P(1 + rt) = A}\)

3. SIMPLIFY by distributing

• Apply distributive property on the left side:

\(\mathrm{P + Prt = A}\)

• Now we have r in a term we can isolate

4. SIMPLIFY by isolating the r term

• Subtract P from both sides:

\(\mathrm{Prt = A - P}\)

• Now the r term is by itself on one side

5. SIMPLIFY by solving for r

• Divide both sides by Pt:

\(\mathrm{r = \frac{(A - P)}{(Pt)}}\)

Answer: B. r = (A - P) / (Pt)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make algebraic mistakes during the multi-step process, particularly when distributing or rearranging terms.

For example, they might incorrectly distribute \(\mathrm{P(1 + rt)}\) as \(\mathrm{P + rt}\) instead of \(\mathrm{P + Prt}\), or they might subtract incorrectly when isolating terms. These algebraic errors lead to wrong expressions that might match other answer choices like A or D.

Second Most Common Error:

Poor INFER reasoning about strategy: Some students attempt to work directly with the fraction form without clearing it first, leading to more complex manipulations that increase error likelihood.

They might try to rearrange \(\mathrm{P = \frac{A}{(1 + rt)}}\) without multiplying through by the denominator, making the algebraic steps unnecessarily complicated and error-prone.

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires methodical step-by-step work with careful attention to distributive property and variable isolation techniques.