Question

If `3000` dollars invested in a bank account for `8` years, compounded quarterly, amounts to `4571.44` dollars, what is the interest rate paid by the account?

Answer 1

About `5.3%`.

Explanation:

We'll use the compounding interest formula, `A=P(1+r/n)^(nt)`.
Here, `A` is the final amount; `P` is the principal, or original amount invested; `r` is the interest rate written in decimal form; `n` is the number of times the money is compounded per time `t`. Usually, `t` is in years and `n` is number of compoundings per year. Quarterly means `n=4`.

Plug in the information you have: `4,571.44=3,000(1+r/4)^(4*8)`
So `4,571.44=3,000(1+r/4)^32`
Divide both sides by `3,000` to get `1.52381=(1+r/4)^(32)`

Now, there are a couple of ways to solve this, with either roots or logarithms. Since you're in pre-calculus, I'll assume your teacher wants you to use logs.

Take the `log` of both sides: `log 1.52381=log (1+r/4)^(32)`
Remember the property of logarithms that says `log a^b=b log a`.
Bring the `32` in front of the expression on the right:
`log 1.52381=32log (1+r/4)`

Then divide both sides by `32`: `(log 1.52381)/32=log (1+r/4)`

Simplify: `.0057166=log (1+r/4)`

Using the inverse property of logarithms, change the expression to read: `10^.0057166 = 1+r/4`

Simplify the left: `1.02325 = 1+r/4`

Solve for `r` by subtracting `1` and multiplying by `4`.

`r=.053000`, or in percent notation, `r=5.3%`.