Question

$5,000 is deposited today into a bank account. The account earns 7.5% per annum compounded half yearly for the first 6 years, then 7.8% per annum compounded quarterly thereafter. Assuming no further deposits or withdrawals are made,
(c) Calculate the account balance 6.5 years from today.

Answer 1

Answer:

The account balance 6 years from today is $8,082.44.

Step-by-step explanation:

This is a compound interest problem

The compound interest formula is given by:

$A = P(1 + \frac{r}{n})^{nt}$

In which A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

In this problem, we have that

The total amount formula changes after 6 years, at which point each of the principal(initial money), interest rate, and n changes.

This exercise asks the account balance after 6.5 years. However, after years, some parameters change. This means that to find the balance after 6.5 years, first we have to find after 6 years, and then for the next half year, with the new parameters.

Step 1: Finding the balance after 6 years.

A is the balance, the value we have to find.

The loan is of $5,000. So $P = 5,000$.

The account earns 7.5% per annum compounded half yearly, so$r = 0.075, n = 2$.

We want to find the account balance in 6 years, so $t = 6$

$A = P(1 + \frac{r}{n})^{nt}$

$A = 5,000(1 + \frac{0.075}{2})^{12}$

$A = 7,777.27$

After 6 years, the balance is $7,777.27. Now, we compound this value for half a year, with the second definition.

Final step: Finding the balance after 6.5 years.

A is the balance.

The value that is going to be compounded is $7,777.27. So $P = 7,777.27$

7.8% per annum compounded quarterly, so $r = 0.078, n = 3$.

This compounding is only going to be valid for 6 months. However, the time is measured in years, so $t = 6$

$A = P(1 + \frac{r}{n})^{nt}$

$A = 7,777.27(1 + \frac{0.078}{3})^{1.5}$

$A = 8,082.44$

The account balance 6 years from today is $8,082.44.