# you have $1000 to invest in an account with a rate 8%, compounded semi-annually. How long will it take you to double your money?

How long will it take to double an initial investment of 1,000 if you are paid 8% interest, and the interest is applied semiannually.

Use the formula `A(t)=A_0(1+r/n)^(nt) `

Here A(t) represents the amount accumulated after t years, t is in years, n is the number of compounding periods per year, and r is the annual interest rate. A0 is the initial deposit.

So we want A(t)=2000, `A_0=1000 ` , n=2, and r=.08. Note that the 8% given is assumed to be the annual rate, unless otherwise specified.

`2000=1000(1+.08/2)^(2t) ` or

`(1.04)^(2t)=2 `

We can use logarithms to solve:

`ln(1.04)^(2t)=ln(2) ` Use a property of logs:

(2t)ln(1.04)=ln(2)

`2t=(ln(2))/(ln(1.04)) `

`t~~8.83 ` years

The rule of 72 can give us an estimate -- take 72 divided by the annual percentage rate to estimate the time to double. Here, the effective annual interest rate is between 8% and 9%, so the estimate is between 8 and 9 years which agrees with our answer.

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It will take about 8.8 years to double

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