Question

Suppose that any given year, the value of a certain investment is increased by 15%. If the value is now $15,000, in how many years will the value be $21,000?
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Answer 1

The answer is 2.5 years.

The compound interest formula is:
A = P(1 + r/n)ⁿˣ
A - the final value
P - the initial value
r - the interest rate
n - the number of times interest is compounded by year
t - the numbers of years of investment

We have:
A = $21,000
P = $15,000
r = 15% = 15/100 = 0.15
n = 1       (since this is per year)
t = ?

So:
$21000 = 15000(1 + 0.15/1) ^{1*t} \\ 21000=15000(1+0.15) ^{t} \\ 21000 = 15000 * 1.15 ^{t} \\ 1.15 ^{t}=21000/15000 \\ 1.15 ^{t}=1.4$

Now, logarithm both sides of the equation:
$log(1.15 ^{t})=log(1.4)$

Since $log (x^{a}) =a*log(x)$, then $log (1.15^{t}) =t*log(1.15)$
Therefore:
$t*log(1.15)=log(1.4) \\ t = \frac{log(1.4)}{log(1.15)} \\ t = \frac{0.15}{0.06} \\ t = 2.5$