For a sum of money invested at 8.5​% compounded quarterly for 9 years state the following values.
​(a) the number of compounding periods
​(b) the periodic rate of interest
​(c) the compounding factor left parenthesis 1 plus i right parenthesis Superscript n
​(d) the numerical value of the compounding factor
Question content area bottom
Part 1
​(a) The number of compounding periods is 
  
enter your response here.
​(Type an integer or a​ decimal.)
Part 2
​(b) The periodic rate of interest is 
  
enter your response here​%.
​(Round to six decimal places as​ needed.)
Part 3
​(c) The compounding factor is left parenthesis 1 plus nothing right parenthesis Superscript nothing
  
  
.
​(Type integers or​ decimals.)
Part 4
​(d) The numerical value of the compounding factor is 
  
enter your response here.
​(Round to six decimal places as​ needed.)

Let's break down the problem into the required parts:

### Given:
- Annual interest rate \( r = 8.5\% \)
- Compounding frequency: Quarterly (4 times a year)
- Time \( t = 9 \) years

### Part 1: The Number of Compounding Periods
The number of compounding periods \( n \) is calculated as:

\[
n = m \times t = 4 \times 9 = 36
\]

So, the number of compounding periods is \( 36 \).

### Part 2: The Periodic Rate of Interest
The periodic rate of interest \( i \) is calculated by dividing the annual interest rate by the number of compounding periods per year:

\[
i = \frac{r}{m} = \frac{8.5\%}{4} = 2.125\% \text{ per period}
\]

Expressing \( i \) as a decimal and rounding to six decimal places:

\[
i = \frac{2.125}{100} = 0.021250
\]

So, the periodic rate of interest is \( 0.021250 \).

### Part 3: The Compounding Factor \((1+i)^n\)
The compounding factor is given by the expression \((1 + i)^n\):

\[
\text{Compounding Factor} = (1 + i)^n = (1 + 0.021250)^{36}
\]

### Part 4: The Numerical Value of the Compounding Factor
Now, let's calculate the numerical value of the compounding factor:

\[
(1 + 0.021250)^{36} = 1.021250^{36}
\]

Calculating this expression:

\[
\text{Numerical Value} \approx 2.122704
\]

### Final Answers:
1. **(a) The number of compounding periods** is \( 36 \).
2. **(b) The periodic rate of interest** is \( 2.125\% \), or \( 0.021250 \) as a decimal.
3. **(c) The compounding factor** is \( (1 + 0.021250)^{36} \).
4. **(d) The numerical value of the compounding factor** is approximately \( 2.122704 \).

Would you like to explore any of these steps in more detail?

Here are some related questions you might consider:
1. How would the compounding factor change if the investment period was extended to 10 years?
2. What would be the periodic rate of interest if the annual interest rate was 10%?
3. How does the compounding factor change with different compounding frequencies, like monthly or semi-annually?
4. Can you explain how to calculate the future value of an investment using the compounding factor?
5. What is the significance of the compounding factor in financial planning?
6. How would you calculate the present value of an amount using the compounding factor?
7. What happens to the periodic rate of interest if the compounding is done monthly?
8. How can the rule of 72 be used to estimate the doubling time of an investment at a given interest rate?

**Tip:** When calculating the compounding factor, ensure that your calculator settings are accurate, especially when working with exponents.

Learn how to calculate compound interest with quarterly compounding at 8.5% annual rate over 9 years. Explore the number of compounding periods, periodic interest rate, and the compounding factor (1 + i)^n. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation