Answer :
To solve the problem of determining how many years it will take for a loan of [tex]$43,000 to grow to $[/tex]67,000 or more with an annual interest rate of 3.75%, compounded annually, we can follow these steps:
1. Identify the initial loan amount and the target amount.
- The initial loan amount: [tex]\( 43,000 \)[/tex] dollars.
- The target amount: [tex]\( 67,000 \)[/tex] dollars.
2. Understand the interest rate application.
- The interest rate is [tex]\( 3.75\% \)[/tex], which means each year the amount is multiplied by [tex]\( 1.0375 \)[/tex] (since [tex]\( 1 + 0.0375 = 1.0375 \)[/tex]).
3. Set up a compound interest formula.
- The formula for the amount [tex]\( A \)[/tex] after [tex]\( t \)[/tex] years, with an initial principal [tex]\( P \)[/tex] and an annual interest rate [tex]\( r \)[/tex], is:
[tex]\[ A = P \times (1 + r)^t \][/tex]
4. Initialize the calculations.
- Begin with the principal [tex]\( P = 43,000 \)[/tex].
5. Iterate year by year and apply the interest rate.
- Each year, multiply the current amount by [tex]\( 1.0375 \)[/tex], and count the number of years until the amount reaches or exceeds [tex]\( 67,000 \)[/tex].
6. Continue until the condition is met.
Performing this step-by-step, we find that it will take approximately 13 years for the loan amount to reach or exceed [tex]\( 67,000 \)[/tex].
Therefore, the smallest possible whole number answer for the number of years needed is [tex]\( \boxed{13} \)[/tex].
1. Identify the initial loan amount and the target amount.
- The initial loan amount: [tex]\( 43,000 \)[/tex] dollars.
- The target amount: [tex]\( 67,000 \)[/tex] dollars.
2. Understand the interest rate application.
- The interest rate is [tex]\( 3.75\% \)[/tex], which means each year the amount is multiplied by [tex]\( 1.0375 \)[/tex] (since [tex]\( 1 + 0.0375 = 1.0375 \)[/tex]).
3. Set up a compound interest formula.
- The formula for the amount [tex]\( A \)[/tex] after [tex]\( t \)[/tex] years, with an initial principal [tex]\( P \)[/tex] and an annual interest rate [tex]\( r \)[/tex], is:
[tex]\[ A = P \times (1 + r)^t \][/tex]
4. Initialize the calculations.
- Begin with the principal [tex]\( P = 43,000 \)[/tex].
5. Iterate year by year and apply the interest rate.
- Each year, multiply the current amount by [tex]\( 1.0375 \)[/tex], and count the number of years until the amount reaches or exceeds [tex]\( 67,000 \)[/tex].
6. Continue until the condition is met.
Performing this step-by-step, we find that it will take approximately 13 years for the loan amount to reach or exceed [tex]\( 67,000 \)[/tex].
Therefore, the smallest possible whole number answer for the number of years needed is [tex]\( \boxed{13} \)[/tex].
