A contract can be fulfilled by making an immediate payment of ​$8838​, or equal payments at the end of every three months for 4 years. What is the size of the quarterly payments at 8 % per annum compounded annually question mark
Question content area bottom
Part 1
The quarterly payments are ​$

enter your response here.
​(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as​ needed.)

To find the size of the quarterly payments, we need to determine the equivalent value of these payments, given that they are equivalent to an immediate payment of $8,838. Since the payments are made quarterly, and the interest rate is 8% per annum compounded annually, we'll use the present value of an annuity formula.

### Step 1: Calculate the number of payments
The payments are made quarterly over 4 years, so:
\[
n = 4 \times 4 = 16 \text{ payments}
\]

### Step 2: Adjust the interest rate
Since the interest rate is given as 8% per annum compounded annually, the effective quarterly interest rate $i$ can be calculated as:
\[
i = \left(1 + \frac{0.08}{1}\right)^{\frac{1}{4}} - 1
\]
Let's calculate $i$.

\[
i = \left(1.08\right)^{\frac{1}{4}} - 1
\]

### Step 3: Use the present value of an annuity formula
The present value $PV$ of an annuity can be calculated using the formula:
\[
PV = R \times \frac{1 - (1 + i)^{-n}}{i}
\]
where $R$ is the amount of each quarterly payment.

Rearranging to solve for $R$:
\[
R = \frac{PV \times i}{1 - (1 + i)^{-n}}
\]

### Step 4: Substitute the values and compute $R$
Given:
- $PV = 8838$
- $n = 16$
- We’ll substitute the value of $i$ we calculated earlier.

Let's calculate this step-by-step.The size of the quarterly payments is approximately **$647.97**.

Would you like more details on how this was calculated or have any questions?

Here are 8 related questions:
1. How is the effective interest rate calculated for different compounding periods?
2. What is the difference between simple interest and compound interest?
3. How would the payment size change if the interest rate were compounded quarterly instead of annually?
4. How do you calculate the future value of an annuity?
5. What is the difference between the present value of an annuity and the future value of an annuity?
6. How does the length of the payment period affect the size of the payments?
7. Can the annuity formula be applied to non-fixed payments?
8. How would inflation affect the real value of these quarterly payments over time?

**Tip:** Always double-check whether the interest rate is compounded annually, quarterly, or monthly to use the correct formulas.

Learn how to calculate quarterly payments using the present value of annuity formula. This problem involves determining quarterly payments equivalent to an immediate payment of $8,838 over 4 years at 8% per annum compounded annually. Detailed step-by-step solution included.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation