Question

a principal of $2600 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 (in decimal), and $t$ is the time in years. Here, $P = 2600$, $A = 5000$, and $r = 7.75\% = 0.0775$. So we need to solve for $t$ in the equation $5000 = 2600(1 + 0.0775)^t$.

Step2: Simplify the equation

Divide both sides by 2600: $\frac{5000}{2600}=(1.0775)^t$. Simplify $\frac{5000}{2600}\approx1.9231=(1.0775)^t$.

Step3: Take natural logarithm

Take $\ln$ of both sides: $\ln(1.9231)=t\ln(1.0775)$.

Step4: Solve for t

$t=\frac{\ln(1.9231)}{\ln(1.0775)}$. Calculate $\ln(1.9231)\approx0.653$ and $\ln(1.0775)\approx0.0749$. Then $t\approx\frac{0.653}{0.0749}\approx8.72$.

Step5: Round up

Since we need the smallest whole number of years to accumulate at least $5000$, we round up $8.72$ to $9$.

Answer:

9