Nomonde invested R4000 at i(4) = 12% p.a. Exactly two years later the rate of interest was changed and was also accumulated monthly. Four years later the accumulated amount in the account was R9641.86.

a) Find the nominal rate of interest used during the last 4 years.

b) At what effective monthly rate of interest constant over the whole term, would Nomonde have had the same amount in the account after 6 years?

c) If Nomonde withdrew R6000 at the end of the sixth year, how many months later will she be able to withdraw R6000 again?

We will tackle the problem step-by-step.

### Given Information:

- **Initial Investment (P):** R4000
- **Initial Interest Rate:** \( i^{(4)} = 12\% \) p.a. (nominal interest rate compounded quarterly)
- **Duration of first interest period:** 2 years
- **New Accumulated Amount:** R9641.86 after 6 years (total).
- **New Interest Period:** 4 years (interest rate changed after 2 years).

---

### Part (a): Find the nominal rate of interest during the last 4 years.

#### Step 1: Calculate the accumulated value after the first 2 years.

The first 2 years follow the nominal rate of 12% compounded quarterly. The formula for compound interest is:

\[
A = P \left(1 + \frac{i}{m}\right)^{mt}
\]

Where:
- \( P \) = R4000
- \( i = 12\% = 0.12 \)
- \( m = 4 \) (quarterly compounding)
- \( t = 2 \) years

Substituting the values:

\[
A = 4000 \left(1 + \frac{0.12}{4}\right)^{4 \times 2}
\]

\[
A = 4000 \left(1 + 0.03\right)^{8}
\]
\[
A = 4000 (1.03)^8
\]

Let me calculate this value.The accumulated amount after the first 2 years is approximately R5067.08.

#### Step 2: Calculate the nominal rate of interest for the last 4 years.

Let the nominal interest rate during the last 4 years be \( i_{last} \), compounded monthly. The formula for the second period (4 years) is:

\[
A = P \left(1 + \frac{i_{last}}{12}\right)^{12 \times 4}
\]

Where:
- \( P \) is now the amount after 2 years (R5067.08),
- \( A = 9641.86 \) (the total amount after 6 years),
- \( t = 4 \) years,
- \( m = 12 \) (compounded monthly).

Substitute the values and solve for \( i_{last} \):

\[
9641.86 = 5067.08 \left(1 + \frac{i_{last}}{12}\right)^{48}
\]

Now, we will solve this equation for \( i_{last} \).It appears there was a miscalculation. Let me revise the steps for solving for the nominal interest rate \( i_{last} \). I will address the issue and calculate it correctly.It seems there is still an issue with the calculations. I will review the process to resolve this accurately, as the nominal rate should not yield a negative value.

Let me reanalyze the formula and approach for this step.It appears that solving for the nominal rate of interest using the current approach led to inconsistent results. Let's revisit the calculation approach more carefully:

Given the equation:

\[
A = P \left(1 + \frac{i_{last}}{12}\right)^{48}
\]

Where \( A = 9641.86 \), \( P = 5067.08 \), and the equation solves for \( i_{last} \). 

Allow me to try recalculating this step with a more stable method to find the correct nominal rate.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can still help guide you on how to solve this manually or answer other questions.

Would you like to continue with this, or perhaps focus on a different part of the problem?

Here are some possible directions:
1. How to isolate \( i_{last} \) for manual solving.
2. Explore how compounding interest works over multiple periods.
3. Explain why a nominal interest rate is used in different scenarios.
4. Discuss the interpretation of monthly and annual rates.
5. Tips on using financial calculators for these problems.

**Tip**: When solving for rates, using logarithms can simplify equations like \( A = P (1 + \frac{i}{m})^{mt} \).

Nomonde invested R4000 at i(4) = 12% p.a. Exactly two years later the rate of interest was changed and was also accumulated monthly. Four years later the accumulated amount in the account was R9641.86. a) Find the nominal rate of interest used during the last 4 years. b) At what effective monthly rate of interest constant over the whole term, would Nomonde have had the same amount in the account after 6 years? c) If Nomonde withdrew R6000 at the end of the sixth year, how many months later will she be able to withdraw R6000 again?

Solve for the nominal and effective interest rates for Nomonde's investment of R4000 with an initial rate of 12% p.a. compounded quarterly. Understand how to calculate the final amount of R9641.86 after 6 years. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation