Ruby invests a certain amount of money in a bank account that pays a nominal annual interest rate of 6.8%, compounded quarterly. The amount of money in Ruby’s account at the end of each year forms a geometric sequence with a common ratio, i. Find the value of the actual interest rate p.a. represented by I.

To solve this problem, we need to determine the value of the actual interest rate per annum represented by the common ratio \( i \) of the geometric sequence, given that the nominal annual interest rate is 6.8%, compounded quarterly.

### Step 1: Understanding the Problem
- The nominal annual interest rate is 6.8%, compounded quarterly.
- Compounding quarterly means that the interest rate is divided by 4 and applied every quarter.
- The amount of money in Ruby's account at the end of each year forms a geometric sequence, where the common ratio \( i \) is the factor by which the investment grows each year.

### Step 2: Calculate the Quarterly Interest Rate
Since the nominal annual interest rate is 6.8%, the quarterly interest rate \( r \) is:
\[
r = \frac{6.8\%}{4} = \frac{6.8}{4} = 1.7\%
\]
Expressing this as a decimal:
\[
r = 0.017
\]

### Step 3: Calculate the Common Ratio \( i \)
The common ratio \( i \) of the geometric sequence is the factor by which the investment grows after one year. Since the interest is compounded quarterly, there are 4 compounding periods per year. The common ratio \( i \) is given by:
\[
i = \left(1 + r\right)^4
\]
Substituting \( r = 0.017 \) into the equation:
\[
i = \left(1 + 0.017\right)^4
\]
\[
i = (1.017)^4
\]

### Step 4: Calculate the Effective Annual Interest Rate
Now, we can compute \( i \) using the formula:
\[
i = (1.017)^4
\]

Calculating this:
\[
i \approx 1.069678
\]

### Step 5: Interpret the Result
The value \( i \approx 1.069678 \) represents the factor by which the investment grows each year. The actual annual interest rate (effective interest rate) is given by:
\[
\text{Effective Annual Interest Rate} = i - 1 = 1.069678 - 1 \approx 0.069678
\]
Expressing this as a percentage:
\[
\text{Effective Annual Interest Rate} \approx 6.9678\%
\]

### Final Answer
The actual interest rate per annum represented by \( i \) is approximately **6.9678%**.

Would you like further details on any part of this solution? Here are some related questions you might find useful:

1. How does compounding frequency affect the effective annual interest rate?
2. Can you explain how the geometric sequence relates to compound interest?
3. What would happen if the interest were compounded monthly instead of quarterly?
4. How do you calculate the nominal interest rate if the effective interest rate is known?
5. What is the difference between nominal and effective interest rates?

**Tip:** Remember, the more frequently interest is compounded, the higher the effective annual interest rate will be, even if the nominal rate remains unchanged.

Learn how to calculate the effective annual interest rate for an investment with a nominal annual rate of 6.8%, compounded quarterly. Understand key concepts such as compound interest, geometric sequences, and effective annual interest rate. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation