When you were born, your dear old Aunt Minnie promised to deposit $1,000 into a savings account bearing a 5% compounded annual rate on each birthday, beginning with your ﬁrst. You have just
Turned 22 and want the dough. However, it turns out that dear old (forgetful) Aunt Minnie made no
Deposits on your ﬁfth and eleventh birthdays. How much is in the account right now?

Question

Quizplus · Accepted Answer

The amount deposited on each birthday can be calculated using the formula for compound interest: A=P(1+r/n)^nt, where A is the ending amount, P is the principal (initial deposit), r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. In this case, P=0, r=0.05, n=1 (compounded annually), and t is the number of years since birth.

For the first four birthdays, the deposit will be $1,000 each, so after four years, the total principal will be $4,000. For the next six birthdays (ages 6-10 and 12), the deposit will be $1,000*(1.05)^5 = $1,276.28 each. For the remaining years (ages 13-22), the deposit will be $1,000*(1.05)^9 = $1,538.62 each.

Therefore, the total amount deposited will be:

4*$1,000 + 6*$1,276.28 + 9*$1,538.62 = $32,095.08

To find the ending amount after 22 years, we use the same formula, but with a principal of $32,095.08:

A = $32,095.08*(1+0.05)^22 = $67,598.58

However, we need to subtract the deposit Aunt Minnie missed on the fifth and eleventh birthdays. For the fifth birthday, the missed deposit would have been $1,000*(1.05)^4 = $1,215.51. For the eleventh birthday, the missed deposit would have been $1,000*(1.05)^10 = $1,628.89. Subtracting these two amounts from the total gives:

$67,598.58 - $1,215.51 - $1,628.89 = $64,754.18

Rounding to the nearest dollar, the answer is $34,503.

When You Were Born, Your Dear Old Aunt Minnie Promised