Question

You deposit $10,000 into an account that pays 3% interest compounded quarterly. Approximately how long will it take for your money to double?

Answer 1

Approximately 23.1914 years.

Explanation:

Compound interest can be calculated as:

`A=A_0*(1+r/n)^(nt)`, where `A_0` is your starting amount, `n` is the number of times compounded per year, `r` is the interest rate as a decimal, and `t` is time in years. So...

`A_0=10000`, `r=0.03`, `n=4`, and we want to find `t` when `A=20000`, twice the starting amount.

`10000(1+0.03/4)^(4t)=20000`.

Since this was asked in Algebra, I used a graphing calculator to find where `y=10000(1+0.03/4)^(4t)` and `y=20000` intersect and got the ordered pair `(23.1914, 20000)`. The ordered pair is of the form `(t, A)`, so the time is approximately 23.1914 years.

If you're looking for an exact answer, that goes beyond algebra, maybe:

Start with:
`10000(1+0.03/4)^(4t)=20000`.

Divide through by 10000:
`(1+0.03/4)^(4t)=2`

Take natural log of both sides:

`ln((1+0.03/4)^(4t))=ln(2)`

Use the property that `ln(a^b) = bln(a)`:

`(4t)ln((1+0.03/4)=ln(2)`

divide both sides by `4ln(1+0.03/4)`:

`t=ln(2)/(4ln(1+0.03/4))`

which is the exact value.