To determine the time required for an initial principal of [tex]$1,250.00 to amount to $[/tex]2,031.25 at an annual simple interest rate of 12.5%, follow these steps:
1. Understand the Simple Interest Formula:
The formula to calculate the amount in simple interest is:
[tex]\[
A = P + I
\][/tex]
where [tex]\( A \)[/tex] is the total amount, [tex]\( P \)[/tex] is the principal, and [tex]\( I \)[/tex] is the interest.
2. Interest Calculation:
Simple interest ([tex]\( I \)[/tex]) can be calculated using:
[tex]\[
I = P \times R \times T
\][/tex]
where [tex]\( R \)[/tex] is the rate per annum (expressed as a decimal) and [tex]\( T \)[/tex] is the time in years.
3. Combine Equations:
Combine the formulas to express the total amount [tex]\( A \)[/tex]:
[tex]\[
A = P + PRT
\][/tex]
[tex]\[
2031.25 = 1250 + 1250 \times 0.125 \times T
\][/tex]
4. Isolate [tex]\( T \)[/tex]:
Rearrange the equation to solve for [tex]\( T \)[/tex]:
[tex]\[
2031.25 = 1250(1 + 0.125T)
\][/tex]
[tex]\[
2031.25 = 1250 + 1250 \times 0.125T
\][/tex]
[tex]\[
2031.25 - 1250 = 1250 \times 0.125T
\][/tex]
[tex]\[
781.25 = 1250 \times 0.125T
\][/tex]
[tex]\[
781.25 = 156.25T
\][/tex]
5. Solve for [tex]\( T \)[/tex]:
[tex]\[
781.25 = 156.25T
\][/tex]
[tex]\[
T = \frac{781.25}{156.25}
\][/tex]
[tex]\[
T = 5
\][/tex]
Therefore, the time for which [tex]$1,250.00 will amount to $[/tex]2,031.25 at an annual simple interest rate of 12.5% is 5 years.
The correct answer is:
D. 5 years
