Give step-by-step solution with explanation and final answer:QUESTION 2.

21 An investor invests R100 000 at a bank with an interes ate of 6% per annum that
‘compounds monthly. Determine the future

Question

Give step-by-step solution with explanation and final answer:QUESTION 2.

21 An investor invests R100 000 at a bank with an interes ate of 6% per annum that
‘compounds monthly. Determine the future value (FV) of this investment afer 1 year
and ator 8 years ©

22 An investment of R20 000 will become R31 737.49 if an interest rate of 8% per
‘annum is apple. Af how many years, wil the R31 737.49 be colcted? @

23 You are given an opportunity of investing R50 000 today with a promised future
Value of RED 525.50 in 5 years’ ime. Al what rate is the Investment accruing
interest? @

24 An investor invests R10 000 at the end of ach year nto savings account yielding
an interes ate of 5% por annum thats compounded annually. Determine the furs
Value of the investment at the end of 12 years. @

25 An investor has the opportunity o receive annuity that pays RED 000 per year for
the next 10 years, wih an 8% discount rate. How much should be invested today
for:

25.1 Ordinary annul. @
252 Annuity due @

26 Staden Manufacturing manufaciures cast ron products. The ota fied cost is RZ
1000 000 por year, the variable cost i R100 per unt. The company sels is cast on
product f

VIPSolutions · Accepted Answer

Let's solve **Question 2** step-by-step with explanations and final answers.

---

### **2.1 Compound Interest Future Value**

**Given:**
- Principal ($P$) = R100,000  
- Interest rate ($r$) = 6% per annum, compounded annually
- Time ($n$) = 1 year and 8 years

**Formula:**  
$$ FV = P(1 + r)^n $$
where $ r $ is in decimal form.

#### **After 1 year:**
$$
FV_1 = 100,000 \times (1 + 0.06)^1 = 100,000 \times 1.06 = \boxed{R106,000}
$$

#### **After 8 years:**
$$
FV_8 = 100,000 \times (1.06)^8
$$

First, calculate $ (1.06)^8 $:

$$
(1.06)^8 \approx 1.59385
$$

$$
FV_8 = 100,000 \times 1.59385 = \boxed{R159,385}
$$

---

### **2.2 Find Number of Years Given FV, PV, and Rate**

**Given:**
- $PV = R20,000$
- $FV = R31,737.49$
- $r = 8\% = 0.08$
- Find $n$.

**Formula:**  
$$ FV = PV (1 + r)^n $$
$$ \frac{FV}{PV} = (1 + r)^n $$
$$ n = \frac{\ln(FV/PV)}{\ln(1 + r)} $$

$$
n = \frac{\ln(31,737.49 / 20,000)}{\ln(1.08)}
$$
$$
n = \frac{\ln(1.5868745)}{\ln(1.08)}
$$
$$
n = \frac{0.46115}{0.07696} \approx 5.99
$$

**Final Answer:**  
$$
\boxed{6 \text{ years (rounded up)}}
$$

---

### **2.3 Find the Rate Given FV, PV, and Time**

**Given:**
- $PV = R50,000$
- $FV = R80,525.50$
- $n = 5$ years
- Find $r$.

**Formula:**  
$$ FV = PV(1 + r)^n $$
$$ \frac{FV}{PV} = (1 + r)^n $$
$$ 1 + r = \left(\frac{FV}{PV}\right)^{1/n} $$
$$ r = \left(\frac{FV}{PV}\right)^{1/n} - 1 $$

$$
r = \left(\frac{80,525.50}{50,000}\right)^{1/5} - 1
$$
$$
r = (1.61051)^{0.2} - 1
$$
$$
r \approx 1.10 - 1 = 0.10 = \boxed{10\% \text{ per annum}}
$$

---

### **2.4 FV of Ordinary Annuity (Equal Deposits at End of Each Year)**

**Given:**
- $PMT = R10,000$
- $r = 9\% = 0.09$
- $n = 12$ years

**Formula:**  
$$
FV = PMT \times \frac{(1 + r)^n - 1}{r}
$$

$$
FV = 10,000 \times \frac{(1.09)^{12} - 1}{0.09}
$$

First, calculate $(1.09)^{12}$:

$$
(1.09)^{12} \approx 2.8138
$$
$$
FV = 10,000 \times \frac{2.8138 - 1}{0.09}
$$
$$
FV = 10,000 \times \frac{1.8138}{0.09}
$$
$$
FV = 10,000 \times 20.153
$$
$$
FV = \boxed{R201,530}
$$

---

### **2.5 Present Value of Annuity**

#### **2.5.1 Ordinary Annuity (Payments at End of Each Year)**

**Given:**
- $PMT = R60,000$
- $r = 8\% = 0.08$
- $n = 10$

**Formula:**  
$$
PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}
$$

$$
PV = 60,000 \times \frac{1 - (1.08)^{-10}}{0.08}
$$

First, calculate $(1.08)^{-10}$:

$$
(1.08)^{10} \approx 2.1589
$$
$$
(1.08)^{-10} = \frac{1}{2.1589} \approx 0.4632
$$
$$
PV = 60,000 \times \frac{1 - 0.4632}{0.08}
$$
$$
PV = 60,000 \times \frac{0.5368}{0.08}
$$
$$
PV = 60,000 \times 6.71
$$
$$
PV = \boxed{R402,480}
$$

---

#### **2.5.2 Annuity Due (Payments at Beginning of Each Year)**

**PV of annuity due = PV ordinary annuity × (1 + r)**

$$
PV_{due} = 402,480 \times 1.08 = \boxed{R434,678}
$$

---

## **Summary Table of Final Answers**

| Qn | Final Answer |
|----|-------------|
|2.1 (1yr) | R106,000 |
|2.1 (8yr) | R159,385 |
|2.2 | 6 years |
|2.3 | 10% per annum |
|2.4 | R201,530 |
|2.5.1 | R402,480 |
|2.5.2 | R434,678 |

---

Let me know if you need help with **Section 2.6** (Staden Manufacturing)!

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