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).
100.03039731 = 1.0725 = 1 + rate
rate = .0725 or 7.25%

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