Compound interest is calculated on both the initial payment and the interest earned in previous periods.
problem
If you deposit $4300 into an account paying 9% annual interest compounded quarterly, how long until there is $2720 in the account?
solution
It wil take approximately 10 months and 8 days for the account to go from $4300 to $2720.
explanation
To find time we use formula:
$$ A = P \left( 1 + \frac{r}{n} \right)^{\Large{n \cdot t}} $$ |
A = total amount P = principal (amount of money deposited) r = annual interest rate n = number of times compounded per year t = time in years |
In this example we have
$$ A = \$2720 ~,~ P = \$4300 , r = 9\% ~~ \text{and} ~ n = 4$$After plugging the given information we have
$$ \begin{aligned} 2720 &= 4300 \left( 1 + \frac{ 0.09 }{ 4 } \right)^{\Large{ 4 \cdot t }} \\ 2720 &= 4300 \cdot {( 1.0225 )} ^ { \Large{ 4 \cdot t } } \\ { (1.0225) } ^ { \Large{ 4 \cdot t } } &= \frac{ 2720 }{ 4300 } \\ { (1.0225) } ^ { \Large{ 4 \cdot t } } &= 0.63256 ~~ \text{Take the natural logarithm of both sides} \\ \ln \left({ 1.0225 } ^ { \Large{ 4 \cdot t } } \right) &= ln \left( 0.63256 \right) \\ 4 \cdot t \cdot ln \left( 1.0225 \right) &= ln \left( 0.63256 \right) \\ 4 \cdot t &= \frac{ ln \left( 0.63256 \right) }{ ln \left( 1.0225 \right) } \\ 4 \cdot t &= -20.58282 \\ t &= -5.1457 ~ \text{years}= \text{ 10 months and 8 days } \end{aligned}$$Please tell me how can I make this better.