Question

a principal of $2400 is invested at 7.75% interest, compounded annually. how many years will it take to accumulate $5000 or more in the account? (use the calculator provided if necessary.)
write the smallest possible whole number answer.

Explanation:

Step1: Recall Compound Interest Formula

The compound interest formula is $A = P(1 + r)^t$, where $A$ is the amount, $P$ is the principal, $r$ is the annual interest rate (decimal), and $t$ is time in years. Here, $P = 3400$, $r = 0.0775$, $A \geq 5000$. So, $5000 = 3400(1 + 0.0775)^t$.

Step2: Simplify the Equation

Divide both sides by 3400: $\frac{5000}{3400} = (1.0775)^t$. Simplify $\frac{5000}{3400} \approx 1.4706 = (1.0775)^t$.

Step3: Take Natural Logarithm

Take $\ln$ of both sides: $\ln(1.4706) = t\ln(1.0775)$.

Step4: Solve for t

Calculate $\ln(1.4706) \approx 0.3857$ and $\ln(1.0775) \approx 0.0748$. Then $t = \frac{0.3857}{0.0748} \approx 5.16$. Since we need whole number of years to accumulate $5000$ or more, we round up to $6$ (because at $t = 5$, $A = 3400(1.0775)^5 \approx 3400 \times 1.457 \approx 4953.8 < 5000$; at $t = 6$, $A = 3400(1.0775)^6 \approx 3400 \times 1.570 \approx 5338 > 5000$).

Answer:

6