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

​

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+(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\%=(8)/(100)=0.08`

`300,000=100,000(1+(0.08)/(1))^(1\cdot t)`

`300,000=100,000(1.08)^(t)`

`(300,000)/(100,000)=(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=(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.