Questions: The amount of 5000 is invested at 4% for 3 years. Write your answers to the nearest cent, if necessary. (a) Compute the ending balance if the bank calculates simple interest. The total amount in the account is 5600 (b) Compute the ending balance if the bank calculates interest compounded annually. In year 1, the interest earned will be 200. In year 1, the total amount in the account will be 5200

The amount of 5000 is invested at 4% for 3 years. Write your answers to the nearest cent, if necessary.

(a) Compute the ending balance if the bank calculates simple interest.

The total amount in the account is 5600

(b) Compute the ending balance if the bank calculates interest compounded annually.

In year 1, the interest earned will be 200.

In year 1, the total amount in the account will be 5200

Transcript text: The amount of $\$ 5000$ is invested at $4 \%$ for 3 years. Write your answers to the nearest cent, if necessary. (a) Compute the ending balance if the bank calculates simple interest. The total amount in the account is $\$ 56$ $\square$ 5600 (b) Compute the ending balance if the bank calculates interest compounded annually. In year 1 , the interest earned will be $\$ 200$. In year 1 , the total amount in the account will be $\$ 5200$

Solution

Solution Steps

Solution Approach

For simple interest, use the formula $ A = P(1 + rt) $ where $ P $ is the principal amount, $ r $ is the annual interest rate, and $ t $ is the time in years.
For compound interest, use the formula $ A = P(1 + \frac{r}{n})^{nt} $ where $ n $ is the number of times interest is compounded per year. In this case, $ n = 1 $ since the interest is compounded annually.

Step 1: Calculate the Ending Balance with Simple Interest

To calculate the ending balance with simple interest, we use the formula: \[ A = P(1 + rt) \] where:

$ P = 5000 $ (principal amount)
$ r = 0.04 $ (annual interest rate)
$ t = 3 $ (time in years)

Substituting the values, we get: \[ A = 5000(1 + 0.04 \times 3) = 5000(1 + 0.12) = 5000 \times 1.12 = 5600.00 \]

Step 2: Calculate the Ending Balance with Compound Interest

To calculate the ending balance with compound interest, we use the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where:

$ P = 5000 $ (principal amount)
$ r = 0.04 $ (annual interest rate)
$ t = 3 $ (time in years)
$ n = 1 $ (compounded annually)

Substituting the values, we get: \[ A = 5000 \left(1 + \frac{0.04}{1}\right)^{1 \times 3} = 5000 \left(1 + 0.04\right)^3 = 5000 \times 1.124864 = 5624.32 \]

Step 3: Calculate the Total Amount in the Account After Year 1

To calculate the total amount in the account after year 1 with compound interest, we use the same formula as above but with $ t = 1 $: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where:

$ P = 5000 $ (principal amount)
$ r = 0.04 $ (annual interest rate)
$ t = 1 $ (time in years)
$ n = 1 $ (compounded annually)

Substituting the values, we get: \[ A = 5000 \left(1 + \frac{0.04}{1}\right)^{1 \times 1} = 5000 \left(1 + 0.04\right) = 5000 \times 1.04 = 5200.00 \]