Question

You’re prepared to make monthly payments of $270, beginning at the end of this month, into an account that pays 6.8 percent interest compounded monthly.

  

How many payments will you have made when your account balance reaches $18,000? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Answer 1

Here, the payments will be same every month, so it is an annuity. We will use the following future value of annuity formula to calculate the required number of payments:

FVA = P * ((1 + r)^-n - 1 / r)

where, FVA is future value of annuity = $18000, P is the periodical amount = $270, r is the rate of interest = 6.8% compounded monthly, so monthly rate of interest = 6.8% / 12 = 0.57%  and n is the time period.

Now, putting these values in the above formula, we get,

$18000 = $270 * ((1 + 0.57%)ⁿ - 1 / 0.57%)

$18000 = $270 * ((1 + 0.0057)ⁿ - 1 / 0.0057)

$18000 / $270 =  (1.0057)ⁿ - 1  / 0.0057

$66.6667 =  (1.0057)ⁿ - 1  / 0.0057

$66.6667 * 0.0057 =  (1.0057)ⁿ - 1

0.38 =  (1.0057)ⁿ - 1

0.38 + 1 =  (1.0057)ⁿ

1.38 =  (1.0057)ⁿ

(1.0057)^56.7 = (1.0057)ⁿ

so, n = 56.7 or 57

So, a total of 57 payments will have to be made when account balance reaches $18000.