Question

Olivia deposited $2,136.70 into a saving account with an interest rate of 2.7% compounded quarterly. About how long will it take for the account to be worth $4,000?

Answer 1

23.26 years

Explanation:

Use the compound interest formula:

A = P
`( 1 + (r)/(100n) )^(nt)`

A = final amount

P = initial principal balance

r = interest rate

n = number of times interest applied per time period

t = number of time periods elapsed

P = $2,136.70

r = 2.7

n = 4

A = $4,000

Now we plug everything in and solve for t.

$4,000 = $2,136.70
`(1 + (2.7)/(100 * 4) )^(4t)`

$4,000 = $2,136.70
`(1 + .00675 ) ^(4t)`

$4,000 = $2,136.70
`(1.00675)^(4t)` Divide each side by $2,136.70

$4,000/$2,136.70 =
`(1.00675)^(4t)`

1.87 =
`(1.00675)^(4t)`

log (1.87) = log
`(1.00675)^(4t)`

log 1.87 = (4t) (log 1.00675) Divide each side by (log 1.00675)

`(log 1.87)/(log 1.00675)` = 4t

93.044 = 4t Divide each side by 4.

93.044/4 = t

23.26 = t

23.26 years

Answer 2

23.3 years

Explanation:

4000 = 2136.7(1 + .027/4)^(4t)

4000 / 2136.7 = 1.00675^4t

ln (4000 / 2136.7) / ln 1.00675^4 = t

23.301690290541729158877031400229 = t

23.3 years