Question

Just won $30,000 and deposited into an account that pays 7.5 percent interest, compounded annually. How long will you have to wait until your winnings are worth $70,000?

Answer 1

1) In this problem, let's consider that there was no other investment after the initial investment of $30,000. Since the interest rate of this investment will be compounded annually, we can write the following below:

`\begin{gathered} F=P(1+(r)/(n))^(nt) \\ \\ 70,000=30,000(1+(0.075)/(1))^(1t) \\ \\ 70000=30000(1.075)^t \end{gathered}`

2) Now, we can solve for "t" applying logarithms:

`\begin{gathered} 30000\cdot \:1.075^t=70000 \\ \\ (30000\cdot \:1.075^t)/(30000)=(70000)/(30000) \\ \\ 1.075^t=(7)/(3) \\ \\ t\ln \left(1.075\right)=\ln \left((7)/(3)\right) \\ \\ (t\ln \left(1.075\right))/(\ln \left(1.075\right))=(\ln \left((7)/(3)\right))/(\ln \left(1.075\right)) \\ \\ t=(\ln\left((7)/(3)\right))/(\ln\left(1.075\right))\approx11.71 \end{gathered}`

3) So, you have to wait almost 12 years (11.7) so that the investmente reaches