You can afford monthly deposits of $250 into an account that pays 2.7% compounded monthly. How long will it be until you have $10,000 to buy a​ boat?

Question

You can afford monthly deposits of $250 into an account that pays 2.7% compounded monthly. How long will it be until you have $10,000 to buy a​ boat?

Type the number of​ months: ___________

​(Round to the​ next-higher month if not​ exact.)

Answer 1

The calculator says 39 months

the formula, which is complicated, creates an equation which must be solved numerically

by iteration, and cannot be solved algebraically...

Answer 2

Given that we are making monthly deposits of $250 into an account that pays 2.7% compounded monthly, to find out how long it will take until we have $10,000 to buy a boat, we can use the future value of an ordinary annuity formula because we are depositing the same amount every month:

A = P x (((1 + r/n)^nt) - 1) / (r / n) where,

A - final amount in the account

P - the payment amount made each period

r - annual interest rate (in decimal)

n - the number of times interest is compounded per year

t - time in years

So, let apply this for our problem:

A = $10000

P = $250

r = 2.7 / 100 = 0.027

n = 12

We are finding t in the formula which is the time it will take until we have $10000 to buy a boat so lets plug these values into the formula and solve for t:

10000 = 250 x ((1 + 0.027/12)^12t - 1) / (0.027 / 12)

10000 = 250 x ((1.00225)^12t - 1) / 0.00225

40 = ((1.00225)^12t - 1) / 0.00225

0.09 = (1.00225)^12t - 1

1.09 = (1.00225)^12t

ln(1.09) = ln((1.00225)^12t)

ln(1.09) = 12t(ln(1.00225))

t = ln(1.09) / 12(ln(1.00225))

t = 3.20

So, the time it will take until we have $10000 to buy a boat is 3.2 years or 38.4 months or can be rounded to 39 months.