Question

Unfortunately, Diana doesn't have enough money in her account right now. She needs to make additional contributions at the end of each of the next three years to be able to pay for the repairs. Her account currently has $3,500, which, along with her additional contributions, is expected to continue earning 6% annual interest. If she makes equal contributions each year, how large must each contribution be for Diana to have $8,000 after three years

Answer 1

Diana needs to make annual contributions of $1,202.93 at the end of each year for three years, with a 6% interest rate, to reach a total savings of $8,000.

Step-by-step explanation:

The question is asking to calculate the size of equal contributions that Diana needs to make for three consecutive years at an annual interest rate of 6%, so that she ends up with $8,000 in her account at the end of this three-year period. The existing amount in Diana's account is $3,500, which will also accrue interest over this time.

First, let's use the Future Value formula for compound interest, which is FV = PV(1 + r)^n, where FV is the future value, PV is the present value, r is the interest rate and n is the number of periods.

We first calculate the future value of the current balance, which is already in the account:

$3,500(1 + 0.06)³ = $4,171.77 (rounded to two decimal places)

Next, we figure out how much additional money needs to be in the account to reach the $8,000 target, after accounting for the interest the present savings will generate:

$8,000 - $4,171.77 = $3,828.23 needed

Now, we need to calculate the size of the equal contributions, C, that Diana must make at the end of each year, which will also compound annually at the same 6% rate. We have to solve the following equation:

C[(1 + 0.06)² + (1 + 0.06) + 1] = $3,828.23

This simplifies to:

C[1.1236 + 1.06 + 1] = $3,828.23

Which further simplifies to:

C[3.1836] = $3,828.23

Dividing both sides by 3.1836 gives us:

C = $1,202.93

Therefore, Diana must make annual contributions of $1,202.93 at the end of each year to reach her goal.

Answer 2

Answer: $1,203.49

Step-by-step explanation:

The equal contributions will be an annuity. The $3,500 already there will also grow at 6% for 3 years. Expression is;

8,000 = ( 3,500 * ( 1 + 6%)^3) + Contribution * Future value interest factor of annuity, 3 years, 6%

8,000 = 4,168.56 + Contribution * 3.1836

Contribution = (8,000 - 4,168.56) / 3.1836

Contribution = $1,203.49