Carrie put $125 in an account for two years at 15% annual interest. How much money did Carrie have after two years?

`color(blue)("Preamble")`

The symbol % is a bit like a unit of measurement but 1 that is worth `1/100`

So `15%->15xx%->15xx1/100=15/100`

Thus we have `15/100xx$125` for 1 year

What follows is one way of working it out in your head (for `15/100`)

`15` is the same as `10+5` and `5` is `1/2" of "10`

So we have `10/100+5/100`

but `10/100 -=1/10`

so `5/100-=1/2xx1/10`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question using the above 'trick'")`

`1/10xx125=12.5`
`1/2xx12.5=ul(color(white)("d")6.25 larr" Add"`
`color(white)("DDDDDDD")18.75 color(white)("dddddddd")larr" this is "15/100xx125`

but the above is for 1 year. So for 2 years we have:

`2xx$18.75= $37.50` Interest
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question the traditional way")`

for 2 years we have 2 lots of `15/100xx$125` giving:

`2xx15/100xx$125`

`(3cancel(0))/(10cancel(0))xx$125`

`3xx1/10xx$125`

`3xx$12.50`

`$37.50` Interest

For simple interest use the formula

`color(white)(aaaaa)``I=Prt`

Where `I` is the interest accumulated
and `P` is the principal (amount invested)
`r` is the annual rate of interest
`t` is the time (in years if the interest rate is per year)

Substituting into the formula,
`I=Prt`
`I=(125)(0.15)(2)`
`I=37.50`

So with simple interest she has the amount she invested, $125, plus the interest she earned, $37.50, giving her a total amount of $162.50 in her account.

If the interest is compounded, use the compound interest formula

`color(white)(aaaaa)``A=P(1+i)^n`

`P``color(white)(aaaaa)`represents the principal (the money put in the account)
`A``color(white)(aaaaa)` represents the amount that has accumulated in the account at the end of the time period
`i` `color(white)(aaaaa)`represents the rate of interest at each compound
`n` `color(white)(aaaaa)`represents the number of times it is compounded (here once each year.)

So `P=125`
`A=?`
`i=15%``color(white)(aaa)``(15%=0.15)`
`n=2`

`A=P(1+i)^n`
`A=125(1+0.15)^2`
`A=125(1.3225)`
`A=165.3125`

Therefore she has $165.31 after 2 years using compound interest.

(We call this the PAIN formula)