Question

Kristen invests $5,745 in a bank. The bank pays 6.5% interest compounded monthly. How long must she leave the money in the bank for it to double? Round to nearest tenth of a year.

Answer 1

To solve this we are going to use the compounded interest formula:
`A=P(1+ (r)/(n))^(nt)`
where

`A` is the final amount

`P` is the initial investment

`r` is the interest rate in decimal form

`t` is the time in years

We know that initial investment is $5,745, so
`P=5745`. We also know that Kristen wants his investment to double, so
`A=2*5745=11490`. Now, to convert the interest rate to decimal form, we are going to divide the rate by 100%:
`r= (6.5)/(100)=0.065`. Since the interest is compounded monthly, it is compounded 12 times per year: therefore,
`n=12`. Now that we have all the information we need, lets replace the values in our formula and solve for
`t`:

`A=P(1+ (r)/(n))^(nt)`

`11490=5745(1+ (0.065)/(12))^(12t)`

`(11490)/(5745) =(1+ (0.065)/(12))^(12t)`

`2=( (2413)/(2400) )^(12t)`

`ln(2)=ln((2413)/(2400) )^(12t)`

`ln(2)=12tln((2413)/(2400) )`

`(ln(2))/(12ln((2413)/(2400) )) =t`

`t=(ln(2))/(12ln((2413)/(2400) ))`

`t=10.7`

We can conclude that she must leave the money for 10.7 years in the bank for it to double.