Hello, rachael724!
stapel's method is the way to go . . .
I assume you know the Compound Interest Formula:
. . . \(\displaystyle A \;= \;P(1 + i)^n\)
where \(\displaystyle P\) = principal invested
. . . . . . \(\displaystyle i\) = periodic interest rate
. . . . . . \(\displaystyle n\) = number of periods
. . . . . \(\displaystyle A\) = final value (amount)
How many years, correct to the nearest tenth of a year, will it take for a sum of money to double
if it's invested at 8.5%, compounded quarterly?
Let \(\displaystyle y\) = number of years.
We have:
.\(\displaystyle i = \frac{8.5\%}{4} = 0.02125,\;\;n = 4y\)
We want our principal \(\displaystyle P\) to grow to \(\displaystyle 2P.\)
Applying the formula:
. \(\displaystyle 2P \;= \;P(1.02125)^{4y}\)
Then we have:
. \(\displaystyle (1.02125)^{4y} \;= \;2\)
Take logs:
. \(\displaystyle \log[(1.02125)^{4y}] \;= \;\log(2)\)
. . then:
. \(\displaystyle 4y\cdot\log(1.02125) \;= \;\log(2)\)
Finally:
. \(\displaystyle y \;= \;\frac{\log(2)}{4\cdot\log(1.02125)} \;= \;8.241012465\)
Therefore, it will take about 8.2 years.-
