How much time will be needed for $\$ 18,000$ to grow to $\$ 20,276.87$ if deposited at $6 \%$ compounded quarterly?
For calculating use the formula $A=P\left(1+\frac{r}{k}\right)^{k t}$.
Answer
Final Answer: The time needed for $18,000 to grow to $20,276.87 if deposited at 6% compounded quarterly is approximately \(\boxed{2}\) years.
Step 1 ：We are given a problem where we need to find out how much time will be needed for $18,000 to grow to $20,276.87 if deposited at 6% compounded quarterly.
Step 2 ：We can use the formula for compound interest to solve this problem. The formula is \(A=P\left(1+\frac{r}{k}\right)^{k t}\), where A is the final amount, P is the principal amount, r is the annual interest rate, k is the number of times the interest is compounded per year, and t is the time in years.
Step 3 ：We are given the following values: A = $20,276.87, P = $18,000, r = 6% or 0.06, and k = 4 (quarterly). We need to find the value of t.
Step 4 ：We can rearrange the formula to solve for t: \(t = \frac{\log\left(\frac{A}{P}\right)}{k \log\left(1+\frac{r}{k}\right)}\)
Step 5 ：Substituting the given values into this formula, we get a value of approximately 2 for t.
Step 6 ：This means that it will take approximately 2 years for the principal amount of $18,000 to grow to $20,276.87 when compounded quarterly at an annual interest rate of 6%.
Step 7 ：Final Answer: The time needed for $18,000 to grow to $20,276.87 if deposited at 6% compounded quarterly is approximately \(\boxed{2}\) years.
