# Compound Interest: Time

How to find the time of the compound interest investment: formula, 1 example, and its solution.

## Formula

A compounded interest means

add the principle and the interest,

calculate the next interest,

and repeating this process.

The amount of money shows exponential growth.

To find the time

of a compound interest investment,

use the exponential growth formula:

A_{0}(1 + r)^{t} = A.

A_{0}: Initial value

r: Rate of change (per time period)

t: Number of time period

A: Final value

## Example

The initial value of the investment is $1,000.

So A_{0} = $1000.

It says

after how many years will the investment

worth more than $1,800?

So set A = $1800.

The investment is at a rate of 6% per year.

So r = 0.06/year.

A_{0} = 1000

A = 1800

r = 0.06/year

The investment is compounded yearly.

Then 1000(1 + 0.06)^{t} = 1800.

The goal is to find the time t.

Divide both sides by 1000.

1 + 0.06 = 1.06

Then 1.06^{t} = 1.8.

log 1.8 and log 1.06 are given.

So common log both sides.

log 1.06^{t} = log 1.8

log 1.06^{t} = t log 1.06

Logarithm of a Power

It says

assume log 1.8 = 0.255, log 1.06 = 0.025.

Then t⋅0.025 = 0.255.

Divide both sides by 0.025.

Move the decimal points

3 digits to the right.

0.255/0.025 = 255/25

Find the value of 255/25

to the ones.

255/25 = 10.xx

t = 10.xx

Round this up to the nearest ones:

10.xx → 11.

The unit of the time is [year].

So write

After 11 years.

t = 10.xx means

after 10.xx years,

the investment will worth exactly $1,800.

So after 11 years,

the investment will worth more than $1,800.

So

after 11 years

is the answer.