Write all ratios in their simplest form. 1. There are three children in the Skosana family. Each week the eldest earns R270, the middle child earns $$ R 180 $$ and the youngest earns R90. Express these amounts as a ratio. 2. To make a certain colour of household paint, you need to mix $$ \frac{3}{4} \ell $$ of white paint with 15 ml of red paint and 10 ml of yellow paint. Express the paint mixture as a ratio. If you need a total volume of 31 litres of this paint, how much of each colour must you mix? 3. On Wednesday evening Sonto spent $$ \frac{1}{2} $$ hour doing homework, 45 minutes doing housework, 1 hour visiting friends and $$ 1 \frac{1}{2} $$ hours watching television. Write this information as a ratio. 4. In a country there are 48000 people in the defence services. They work in the army, navy and air force in the ratio $$ 5: 2: 3 $$. How many people are in each service

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Sure, let's tackle each problem step by step to express the ratios in their simplest forms and perform the necessary calculations.

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### **Problem 1**

**Earnings of the Skosana Family Children:**

- **Eldest Child:** R270
- **Middle Child:** R180
- **Youngest Child:** R90

**Step 1: Express as a Ratio**

The earnings can be expressed as:
$$ 270 : 180 : 90 $$

**Step 2: Simplify the Ratio**

Find the greatest common divisor (GCD) of the numbers to simplify the ratio.

- **GCD of 270, 180, and 90 is 90.**

Divide each term by 90:
$$
\frac{270}{90} : \frac{180}{90} : \frac{90}{90} = 3 : 2 : 1
$$

**Simplified Ratio:**  
$$ 3 : 2 : 1 $$

---

### **Problem 2**

**Paint Mixture Requirements:**

- **White Paint:** $$ \frac{3}{4} $$ litre (750 millilitres)
- **Red Paint:** 15 ml
- **Yellow Paint:** 10 ml

**Step 1: Convert All Measurements to the Same Unit**

Convert litres to millilitres:
$$
\frac{3}{4} \text{ litre} = 750 \text{ ml}
$$

**Step 2: Express as a Ratio**

$$
750 \text{ ml} : 15 \text{ ml} : 10 \text{ ml}
$$

**Step 3: Simplify the Ratio**

Find the GCD of 750, 15, and 10, which is 5.

Divide each term by 5:
$$
\frac{750}{5} : \frac{15}{5} : \frac{10}{5} = 150 : 3 : 2
$$

**Simplified Ratio:**  
$$ 150 : 3 : 2 $$

**Step 4: Determine Amounts for 31 Litres Total Volume**

Total parts in the ratio:
$$
150 + 3 + 2 = 155 \text{ parts}
$$

Calculate the value of each part:
$$
\text{Total volume} = 31 \text{ litres} = 31,000 \text{ ml} \
\text{Value of one part} = \frac{31,000}{155} = 200 \text{ ml per part}
$$

**Amounts to Mix:**
$$
\text{White Paint} = 150 \times 200 = 30,000 \text{ ml} = 30 \text{ litres} \
\text{Red Paint} = 3 \times 200 = 600 \text{ ml} \
\text{Yellow Paint} = 2 \times 200 = 400 \text{ ml}
$$

---

### **Problem 3**

**Sonto's Activities on Wednesday Evening:**

- **Homework:** $$ \frac{1}{2} $$ hour
- **Housework:** 45 minutes
- **Visiting Friends:** 1 hour
- **Watching Television:** $$ 1 \frac{1}{2} $$ hours

**Step 1: Convert All Times to the Same Unit (Minutes)**

$$
\frac{1}{2} \text{ hour} = 30 \text{ minutes} \
45 \text{ minutes} = 45 \text{ minutes} \
1 \text{ hour} = 60 \text{ minutes} \
1 \frac{1}{2} \text{ hours} = 90 \text{ minutes}
$$

**Step 2: Express as a Ratio**

$$
30 : 45 : 60 : 90
$$

**Step 3: Simplify the Ratio**

Find the GCD of 30, 45, 60, and 90, which is 15.

Divide each term by 15:
$$
\frac{30}{15} : \frac{45}{15} : \frac{60}{15} : \frac{90}{15} = 2 : 3 : 4 : 6
$$

**Simplified Ratio:**  
$$ 2 : 3 : 4 : 6 $$

---

### **Problem 4**

**Distribution of People in Defence Services:**

- **Total People:** 48,000
- **Ratio (Army : Navy : Air Force) = 5 : 2 : 3**

**Step 1: Determine Total Parts in the Ratio**

$$
5 + 2 + 3 = 10 \text{ parts}
$$

**Step 2: Calculate the Number of People per Part**

$$
\text{Each part} = \frac{48,000}{10} = 4,800 \text{ people}
$$

**Step 3: Calculate People in Each Service**

$$
\text{Army} = 5 \times 4,800 = 24,000 \text{ people} \
\text{Navy} = 2 \times 4,800 = 9,600 \text{ people} \
\text{Air Force} = 3 \times 4,800 = 14,400 \text{ people}
$$

**Number of People in Each Service:**
- **Army:** 24,000
- **Navy:** 9,600
- **Air Force:** 14,400

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These solutions provide the simplest forms of the required ratios and perform the necessary calculations based on the given information.
**Simplified Answers:** 1. **Earnings Ratio:** $$ 3 : 2 : 1 $$ 2. **Paint Mixture Ratio:** $$ 150 : 3 : 2 $$ - **White Paint:** 30 litres - **Red Paint:** 600 ml - **Yellow Paint:** 400 ml 3. **Sonto's Activities Ratio:** $$ 2 : 3 : 4 : 6 $$ 4. **Defence Services Distribution:** - **Army:** 24,000 - **Navy:** 9,600 - **Air Force:** 14,400

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