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Let's go through the questions step by step.

### **Question 1.1: Nominal interest rate compounded monthly**

Mr. Ngobe is offered a 12% effective annual interest rate, and he wants to withdraw monthly. We need to convert this into a nominal interest rate compounded monthly.

The formula to convert an effective annual interest rate to a nominal monthly interest rate is:

\[
(1 + i_{\text{effective}}) = \left(1 + \frac{i_{\text{nominal}}}{m}\right)^m
\]

Where:
- \(i_{\text{effective}}\) = 12% or 0.12 annually
- \(m\) = 12 (since it's compounded monthly)

We need to solve for \(i_{\text{nominal}}\).

### **Question 1.2.1: Future price of a new taxi after 5 years with inflation**

The formula to calculate the future value of a price under inflation is:

\[
\text{Future Price} = \text{Present Price} \times (1 + \text{inflation rate})^n
\]

Where:
- Present price = R600,000
- Inflation rate = 5.2% or 0.052
- \(n\) = 5 years

### **Question 1.2.2: Book value after 5 years using reducing balance method**

The reducing balance depreciation method formula is:

\[
\text{Book Value} = \text{Initial Value} \times (1 - \text{depreciation rate})^n
\]

Where:
- Initial value = R600,000
- Depreciation rate = 9.5% or 0.095
- \(n\) = 5 years

### **Question 1.2.3: Sinking fund calculation**

The amount needed in the sinking fund is the future cost of the new taxi minus the scrap value (book value of the old taxi). This is:

\[
\text{Amount needed} = \text{Future Price} - \text{Book Value}
\]

Let me calculate each of these step by step.Here are the results for the questions:

### **Question 1.1:** 
The nominal interest rate compounded monthly is **11.39%** per annum.

### **Question 1.2.1:**
The price of a new taxi after 5 years, considering an average inflation rate of 5.2%, will be **R773,089.81**.

### **Question 1.2.2:**
The book value (scrap value) of the taxi after 5 years, using the reducing balance depreciation method at 9.5% per annum, will be **R364,245.46**.

### **Question 1.2.3:**
The amount that must be saved in a sinking fund after 5 years, if the book value of the old taxi is used as a deposit, will be **R408,844.35**.

Do you have any questions or want more details?

Here are 5 related questions for practice:
1. How would the nominal interest rate change if compounded quarterly instead of monthly?
2. What would be the future price of the taxi if the inflation rate was 6% instead of 5.2%?
3. How much would the taxi depreciate after 3 years instead of 5?
4. What if the depreciation rate was higher, say 12% per year, how would the book value change?
5. How can you calculate the monthly payment to a sinking fund if payments are made at the end of each month?

**Tip:** Always ensure to distinguish between nominal and effective interest rates when dealing with different compounding periods.

1.1 A bank offers Mr. Ngobe 12% p.a compounded annually as an effective interest rate, but Mr. Ngobe wants to withdraw the interest only monthly. Calculate the nominal interest rate that will be compounded monthly.
1.2 A taxi owner buys a taxi for R600,000. The taxi owner plans to replace the taxi in five years' time.
1.2.1 Calculate the price of a new taxi in five years' time, if the average rate of inflation is 5.2% p.a.
1.2.2 Calculate the book value (Scrap value) if the taxi is depreciating at 9.5% p.a on a reducing balance method for five years.
1.2.3 Calculate the amount of money that must be in a sinking fund after five years, if they use the book value of the old taxi as a deposit.

Solve Mr. Ngobe's bank interest rate problem, calculate the future price of a taxi considering inflation, and determine the depreciated value of the taxi. This problem covers nominal interest rates, future value under inflation, and reducing balance depreciation.

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