Tim is considering a different savings account that pays [tex]1\%[/tex], compounded monthly.

(When interest is compounded monthly, the bank pays interest every month instead of every year. The function representing compounded interest is [tex]S(t)=P\left(1+\frac{r}{n}\right)^{n t}[/tex], where [tex]P[/tex] is the principal, or initial deposit in the account, [tex]r[/tex] is the interest rate, [tex]n[/tex] is the number of times the interest is compounded per year, [tex]t[/tex] is the number of years, and [tex]S(t)[/tex] is the savings after [tex]t[/tex] years.)

How many years will it take Tim to earn [tex]\$200[/tex] at this bank? Should he switch?



Answer :

To determine the number of years, [tex]\( t \)[/tex], it takes for Tim to earn \[tex]$200 in the given bank account, we follow these steps: 1. Identify the known variables: - \( P \): Initial deposit, which is \$[/tex]200.
- [tex]\( r \)[/tex]: Annual interest rate, which is [tex]\(0.01\)[/tex] (or [tex]\(1\%\)[/tex]).
- [tex]\( n \)[/tex]: Number of times the interest is compounded per year, which is monthly, so [tex]\(n = 12\)[/tex].
- [tex]\( A \)[/tex]: Desired amount, which is \[tex]$200. 2. Write the formula for compound interest: The formula for compound interest is: \[ S(t) = P \left(1 + \frac{r}{n}\right)^{nt} \] 3. Set up the equation with the known amounts: We need to solve for \( t \) when the final amount \( S(t) \) is \$[/tex]200:
[tex]\[ 200 = 200 \left(1 + \frac{0.01}{12}\right)^{12t} \][/tex]

4. Simplify the equation:
Dividing both sides by 200, we get:
[tex]\[ 1 = \left(1 + \frac{0.01}{12}\right)^{12t} \][/tex]

Simplifying further:
[tex]\[ 1 = \left(1 + \frac{1}{1200}\right)^{12t} \][/tex]
Note that [tex]\(1 + \frac{1}{1200} \approx 1.000833\)[/tex].

5. Solve for [tex]\( t \)[/tex]:
Since the base of the exponent on both sides of the equation is 1, we can conclude that the exponential part [tex]\( (1.000833)^{12t} \)[/tex] must be equal to 1. For any positive base different from 1, the exponent must be 0 for the result to be 1. Therefore:
[tex]\[ 12t = 0 \][/tex]
Solving for [tex]\( t \)[/tex], we have:
[tex]\[ t = 0 \][/tex]

6. Determine if Tim should switch banks:
We found that it takes [tex]\( t = 0 \)[/tex] years for Tim to earn \[tex]$200 at this bank account with the given conditions. This implies there's no additional time needed for his savings to reach \$[/tex]200 from his initial deposit of \[tex]$200. Hence, the time required is less than any desired number of years, and Tim should switch to this bank account, as evidenced by the stated conditions. So, the detailed, step-by-step solution is: - It will take Tim \( 0 \) years to earn \$[/tex]200 at this bank (essentially, he already has the desired amount without any waiting period).
- Tim should switch to this bank with the compounded monthly interest rate.