To determine the time for which [tex]$1250.00 will accumulate to $[/tex]2031.25 at an annual simple interest rate of 12.5%, we can follow these steps:
1. Identify the given values:
- Principal ([tex]\(P\)[/tex]) = [tex]$1250.00
- Final Amount (\(A\)) = $[/tex]2031.25
- Annual Interest Rate ([tex]\(R\)[/tex]) = 12.5%
2. Understand the formula for simple interest:
The amount ([tex]\(A\)[/tex]) after time ([tex]\(T\)[/tex]) years can be calculated using the formula:
[tex]\[
A = P + PRT
\][/tex]
which can be rearranged to:
[tex]\[
A = P(1 + RT)
\][/tex]
3. Rearrange to solve for [tex]\(T\)[/tex]:
Starting from:
[tex]\[
A = P(1 + RT)
\][/tex]
Isolating [tex]\(T\)[/tex], we get:
[tex]\[
1 + RT = \frac{A}{P}
\][/tex]
[tex]\[
RT = \frac{A}{P} - 1
\][/tex]
[tex]\[
T = \frac{\frac{A}{P} - 1}{R}
\][/tex]
4. Substitute the known values into the equation:
- [tex]\(A = 2031.25\)[/tex]
- [tex]\(P = 1250.00\)[/tex]
- [tex]\(R = \frac{12.5}{100} = 0.125\)[/tex]
[tex]\[
T = \frac{\frac{2031.25}{1250.00} - 1}{0.125}
\][/tex]
5. Perform the division and subtraction inside the fraction:
[tex]\[
\frac{2031.25}{1250.00} = 1.625
\][/tex]
[tex]\[
T = \frac{1.625 - 1}{0.125}
\][/tex]
[tex]\[
T = \frac{0.625}{0.125}
\][/tex]
6. Complete the division to find [tex]\(T\)[/tex]:
[tex]\[
T = 5
\][/tex]
Hence, the time for which [tex]$1250.00 will amount to $[/tex]2031.25 at a 12.5% per annum simple interest rate is [tex]\( \boxed{5} \)[/tex] years. Therefore, the correct answer is:
D. 5 years
