Question

Marianna finds an annuity that pays 8% annual interest, compounded quarterly. She invests in this annuity and contributes $10,000 each quarter for 6 years. How much money will be in her annuity after 6 years? Enter your answer rounded to the nearest hundred dollars.

Answer 1

$304,200

Explanation:

To find the value of A(6), use the savings annuity formula

A(t)=d((1+r/n)n⋅t−1)r/n.

From the question, we know that r=0.08, d=$10,000, n=4 compounding periods per year, and t=6 years. Substitute these values into the formula gives

A(6)=$10,000 ((1+0.08/4)6⋅4−1)/(0.08/4).

Simplifying further gives A(6)=$10,000 ((1.02)24−1)/(0.02) and thus A(6)=$304,218.62.

Rounding as requested, our answer is $304,200.

Answer 2

Marianna will have $19,700 in her annuity after 6 years.

Explanation:

To find the amount of money in the annuity after 6 years, we need to calculate the compound interest earned over that time period.

First, let's find the interest rate per quarter: 8% annual interest divided by 4 quarters per year = 2% interest per quarter.

Next, we'll calculate the number of quarters that the money is invested: 6 years x 4 quarters per year = 24 quarters.

Now we can use the formula for compound interest to find the total amount of money in the annuity after 6 years:

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

Where:

A = final amount

P = initial investment ($10,000 in this case)

r = interest rate per quarter (2% or 0.02)

n = number of times the interest is compounded in a year (4)

t = number of years invested (6)

Plugging in the values:

A = $10,000 * (1 + 0.02)^(24)

A = $10,000 * 1.96634438

A = $19,663.44

Rounding to the nearest hundred dollars: $19,700

So, Marianna will have $19,700 in her annuity after 6 years.