5 Rajesh deposits R3 200 into a savings account which is compounded annually at $$ 11 \% $$. At the end of two years he withdraws R1 500. 5.1 Calculate the amount of money Rajesh has in his savings account after four years. 5.2 Calculate the amount of money in the account after five years. 6 The map on the next page is part of the map of Gauteng, obtained from www.google.co.za. Tourists visiting Gauteng decide to rent a car for their tour. A car rental company charges the tourists R180 and R10 for every kilometre travelled. The car rental company gives 50 kilometres free per rental. The tourists are travelling from Rustenburg to Krugersdorp, a distance of about $$ 91,3 \mathrm{~km} $$. 6.1.1 Use the information provided and the map to determine how much money the tourists will pay for renting the car for their tour. 6.1.2 The tourists decided to visit Johannesburg from Krugersdorp. The extra distance to Johannesburg from Krugersdorp is $$ 37,0 \mathrm{~km} $$. Calculate the cost of the extra distance travelled.

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### **Problem 5**

**Rajesh** deposits **R3,200** into a savings account that is compounded annually at an **11%** interest rate. After **two years**, he withdraws **R1,500**. We are to find out how much money Rajesh has in his account after **four** and **five** years.

#### **5.1 Amount after Four Years**

1. **Calculate the amount after the first two years:**

The formula for compound interest is:
   $$
   A = P \times (1 + r)^n
   $$
   where:
   - $$ P = 3,200 $$ (principal)
   - $$ r = 0.11 $$ (annual interest rate)
   - $$ n = 2 $$ (number of years)

$$
   A = 3,200 \times (1 + 0.11)^2 = 3,200 \times 1.2321 = 3,942.72
   $$

2. **Subtract the withdrawal after two years:**

$$
   3,942.72 - 1,500 = 2,442.72
   $$

3. **Calculate the amount after the next two years:**

$$
   A = 2,442.72 \times (1 + 0.11)^2 = 2,442.72 \times 1.2321 \approx 3,009.67
   $$

**After four years, Rajesh has approximately **R3,009.67** in his savings account.**

#### **5.2 Amount after Five Years**

1. **Calculate the amount after one more year (total of five years):**

Using the amount after four years:
   
   $$
   A = 3,009.67 \times 1.11 \approx 3,340.74
   $$

**After five years, Rajesh has approximately **R3,340.74** in his savings account.**

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### **Problem 6**

**Tourists** are renting a car to travel within Gauteng. The rental company charges:
- **R180** (fixed charge)
- **R10** per kilometer traveled
- **50 kilometers** are free per rental

The tourists' initial trip is from **Rustenburg to Krugersdorp**, a distance of **91.3 km**. Later, they decide to visit **Johannesburg**, adding an extra **37.0 km**.

#### **6.1.1 Cost for Renting the Car for the Tour**

1. **Determine total kilometers traveled:**
   
   - **Initial Trip:** 91.3 km

2. **Calculate the extra kilometers beyond the free 50 km:**
   
   $$
   \text{Extra km} = 91.3 - 50 = 41.3 \text{ km}
   $$

3. **Calculate the cost for extra kilometers:**
   
   $$
   \text{Extra Cost} = 41.3 \times 10 = R413
   $$

4. **Add the fixed rental charge:**
   
   $$
   \text{Total Cost} = 180 + 413 = R593
   $$

**The tourists will pay **R593** for renting the car for their tour from Rustenburg to Krugersdorp.**

#### **6.1.2 Cost for the Extra Distance to Johannesburg**

1. **Additional distance to Johannesburg:**
   
   - **Extra Trip:** 37.0 km

2. **Since the initial free 50 km have already been utilized**, the entire extra distance is chargeable.

3. **Calculate the cost for the extra kilometers:**
   
   $$
   \text{Extra Cost} = 37.0 \times 10 = R370
   $$

**The cost for the extra distance traveled to Johannesburg is **R370**.**

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### **Summary of Answers**

- **5.1:** After four years, Rajesh has **R3,009.67** in his savings account.
- **5.2:** After five years, Rajesh has **R3,340.74** in his savings account.
- **6.1.1:** The cost for renting the car from Rustenburg to Krugersdorp is **R593**.
- **6.1.2:** The additional cost for traveling to Johannesburg is **R370**.
After four years, Rajesh has R3,009.67 in his savings account. After five years, he has R3,340.74. For the car rental, the tourists pay R593 for the trip from Rustenburg to Krugersdorp and an additional R370 for the extra distance to Johannesburg.

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