. On the day that you were born, your grandfather opened a savings account in your name. At that time, he deposited a certain amount of money into the account, and has deposited the same amount on each of your first 20 birthdays. Today is your 21st birthday. Your grandfather has decided not to make another deposit but rather to give you a special gift. If the account has paid a consistent interest rate of 4% compounded semi-annually and has a current balance of $100,000, how much money did your grandfather deposit on each occasion?

Question

. On the day that you were born, your grandfather opened a savings account in your name. At that time, he deposited a certain amount of money into the account, and has deposited the same amount on each of your first 20 birthdays. Today is your 21st birthday. Your grandfather has decided not to make another deposit but rather to give you a special gift. If the account has paid a consistent interest rate of 4% compounded semi-annually and has a current balance of $100,000, how much money did your grandfather deposit on each occasion?

Answer 1

To find the amount of money your grandfather deposited on each occasion, we need to use the concept of compound interest.

First, let's calculate the number of compounding periods. Since the interest is compounded semi-annually, we have 2 compounding periods per year, and a total of 20 years. Therefore, the total number of compounding periods is 2 x 20 = 40.

Next, let's determine the interest rate per compounding period. Since the annual interest rate is 4%, the semi-annual interest rate is half of that, which is 4% / 2 = 2%.

Now, we can use the formula for compound interest to calculate the initial deposit made by your grandfather. The formula is:

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

Where:
A = the current balance ($100,000 in this case)
P = the initial deposit
r = interest rate per compounding period (0.02 in this case since it's 2%)
n = number of compounding periods per year (2 in this case since it's compounded semi-annually)
t = number of years (20 in this case)

Plugging in these values, we get:

100,000 = P(1 + 0.02/2)^(2*20)

Simplifying this equation, we get:

100,000 = P(1.01)^40

Now, we can solve for P by dividing both sides of the equation by (1.01)^40:

P = 100,000 / (1.01)^40

Calculating this value, we find that P is approximately $49,066.35.

Therefore, your grandfather deposited approximately $49,066.35 on each occasion.

Answer 2

To determine the amount of money your grandfather deposited on each occasion, we need to use the formula for compound interest:

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

Where:
A = the final amount (current balance) on the account ($100,000)
P = the principal amount (initial deposit on the first birthday)
r = the annual interest rate (4% or 0.04)
n = the number of times the interest is compounded per year (2, for semi-annual compounding)
t = the number of years (21 on your 21st birthday)

We can rearrange the formula to solve for the principal amount (P):

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

Now, let's plug in the values into the formula:

P = 100,000 / (1 + 0.04/2)^(2*21)

P = 100,000 / (1 + 0.02)^(42)

P = 100,000 / (1.02)^42

P ≈ $32,783.98

Therefore, your grandfather deposited approximately $32,783.98 on each occasion.

Answer 3

In order for the ordinary formulas to work, the interest period must match the payment period.

In this question, the payment period is annually, but the interest period is semiannually.
So we must first find what annual interest rate is equivalent to 4% per annum compounded semi-annually
let that rate be i
1+i = 1.02^2 = 1.0404
so i = .0404

secondly, we have to be careful not to make a mistake in counting the number of payments, let's look at the pattern
at birth -- 1st payment
1st birthday -- 2nd payment
2nd birthday -- 3rd payment
...
21st birthday -- 22nd payment

so P(1.0404)^22 - 1)/.0404 = 100,000
P(34.407256) = 100,000
P = 2906.36