Math Problem Statement

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.

Solution

To solve this problem, we need to determine the value of the actual interest rate per annum represented by the common ratio ii 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 ii 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 rr is: r=6.8%4=6.84=1.7%r = \frac{6.8\%}{4} = \frac{6.8}{4} = 1.7\% Expressing this as a decimal: r=0.017r = 0.017

Step 3: Calculate the Common Ratio ii

The common ratio ii 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 ii is given by: i=(1+r)4i = \left(1 + r\right)^4 Substituting r=0.017r = 0.017 into the equation: i=(1+0.017)4i = \left(1 + 0.017\right)^4 i=(1.017)4i = (1.017)^4

Step 4: Calculate the Effective Annual Interest Rate

Now, we can compute ii using the formula: i=(1.017)4i = (1.017)^4

Calculating this: i1.069678i \approx 1.069678

Step 5: Interpret the Result

The value i1.069678i \approx 1.069678 represents the factor by which the investment grows each year. The actual annual interest rate (effective interest rate) is given by: Effective Annual Interest Rate=i1=1.06967810.069678\text{Effective Annual Interest Rate} = i - 1 = 1.069678 - 1 \approx 0.069678 Expressing this as a percentage: Effective Annual Interest Rate6.9678%\text{Effective Annual Interest Rate} \approx 6.9678\%

Final Answer

The actual interest rate per annum represented by ii 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.

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Math Problem Analysis

Mathematical Concepts

Compound Interest
Geometric Sequences
Effective Annual Interest Rate

Formulas

Quarterly Interest Rate: r = Nominal Annual Rate / 4
Common Ratio: i = (1 + r)^4
Effective Annual Interest Rate: i - 1

Theorems

Concept of Compounding Interest

Suitable Grade Level

Grades 10-12