Question

If you invest $100,000 in an account earning 8% interest compounded annually, how long will it take until the account holds $300,000?

(Round to the nearest 0.1 of a year.) 



( answergoes here ) ______________________




Compound interest formulas:  

A=P(1+rn)ntA=P(1+r/n)^nt and A=Pe^rt

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Answer 1

We have been given that you invest $100,000 in an account earning 8% interest compounded annually. We are asked to find the time it will take the amount to reach $300,000.

We will use compound interest formula to solve our given problem.

$A=P(1+\frac{r}{n})^{nt}$, where,

A = Final amount after t years,

P = Principal amount,

r = Annual interest rate in decimal form,

n = Number of times interest is compounded per year,

t = Time in years.  

$8\%=\frac{8}{100}=0.08$

$300,000=100,000(1+\frac{0.08}{1})^{1\cdot t}$

$300,000=100,000(1.08)^{t}$

$\frac{300,000}{100,000}=\frac{100,000(1.08)^{t}}{100,000}$

$3=(1.08)^{t}$

$(1.08)^{t}=3$

Let us take natural log on both sides of equation.

$\text{ln}((1.08)^{t})=\text{ln}(3)$

Using natural log property $\text{ln}(a^b)=b\cdot \text{ln}(a)$, we will get:

$t\cdot \text{ln}(1.08)=\text{ln}(3)$

$\frac{t\cdot \text{ln}(1.08)}{\text{ln}(1.08)}=\frac{\text{ln}(3)}{\text{ln}(1.08)}$

$t=\frac{1.0986122886681097}{0.0769610411361283}$

$t=14.274914586$

Upon rounding to nearest tenth of year, we will get:

$t\approx 14.3$

Therefore, it will take approximately 14.3 years until the account holds $300,000.