Four Annual Compound Interest Formulas
Formulas and examples showing how to solve for principal, ending amount, time (years) or rate.
Total = Principal × (1 + Rate)^{years}
If you came here from the Compound Interest Calculator, then the above formula should be very familiar to you.
Anyway, how about an example?
You have $1,500 in a 5 per cent annual interest savings account.
How much money do you have after 3 years?
Total = 1,500 * (1+.05)^{3}
Total = 1,500 * 1.157625
Total = 1,736.44


Principal = Total ÷ (1 + Rate)^{years}
After 7 years in a 9% annual rate investment, you have $6,032.53.
What amount was invested originally?
Principal = 6,032.53 ÷ (1 + .09)^{7}
Principal = 6,032.53 ÷ 1.828039
Principal = 3,300.00


Years = {log(total) log(Principal)} ÷ log(1 + rate)
$11,800 invested in a 7.5 annual rate account eventually yields $28,105.
How many years did this take?
Years = {log(28,105) log(11,800)} ÷ log(1.075)
Years = {4.4487836 4.0718820} ÷ (0.03140846)
Years = {0.3769016} ÷ (0.03140846)
Years = 12


log(1 + rate) = {log(total) log(Principal)} ÷ Years
An investment of $23,000 in nine years yielded $43,182.08.
What was the annual rate of this investment?
log(1 + rate) = {log(43,182.08) log(23,000)} ÷ 9
log(1 + rate) = {4.6353036 4.3617278} ÷ 9
log(1 + rate) = {0.2735758} ÷ 9
log(1 + rate) = 0.03039731
In order to solve this equation, it's time for a quick lesson about logarithms.
The common log (or base 10 log) of 2 is .3010... and if we raise 10 to the power of .3010, we get 2.
So, basically if we raise 10 to the power of 0.03039731 it produces the value of (1 + rate).
10^{0.03039731} = 1.0725 = 1 + rate
rate = .0725 or 7.25%
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