Question

Suppose Brianna invests $1,300 each year for 5 years, in an annuity that pays 6% annual interest, compounded annually. What percentage of the balance, after 5 years, is from Brianna’s contributions? Enter your answer as a percentage, rounded to two decimal places, but omit the percent sign.

Answer 1

88.70

Explanation:

$88.70$

To answer this question, we will calculate the total deposits into the account and then use the savings annuity formula to calculate the balance in the account after 5 years. The sum of all deposits D5 is given by

D5=5×$1,300=$6,500.

We will use the formula

PN=d((1+r/k)N⋅k−1)r/k

to find the value of P5. The question tells us that r=0.06, d=$1,300, k=1 compounding period per year, and N=5 years. Substituting these values into the formula gives

P5=$1,300((1+0.06/1)5×1−1)0.06/1.

We now simplify and calculate the value of P5 to give

P5=$1,300×((1.06)5−1)/(0.06),

thus giving

P5=$7,328.22.

Now we find the ratio D5/P5, which gives

D5P5=$6,500$7,328.22=0.88698

(to five decimals). Converting to a percentage gives 88.70%, which we enter as 88.70.

Answer 2

74.73%

Explanation:

First, we're gonna find out her total amount

We're gonna use the count interest formula: P = A(1 + r)ⁿ

P = final amount

A = starting amount (1300)

r = rate (0.06)

n = years (5)

P = 1300(1 + 0.06)⁵

P = 1739.693251

Now divide the starting amount by the total amount

1300 ÷ 1739.693251 = 0.7472582