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 = (log(1.4))/(log(1.15)) \\ t = (0.15)/(0.06) \\ t = 2.5`