QUESTION 1 1.1 Given the list of numbers: $$ \sqrt{1296} ; \frac{33}{3} ; 0.21 ; \pi ; 36 ; \sqrt{14} $$. write down at least two of the following. 1.1.1 Rational numbers $$ \qquad $$ (1) 1.1.2 Integers $$ \qquad $$ (1) 1.1.3 Multiples of 6 $$ \qquad $$ (1) 1.1.4 Irrational numbers. (1) 1.2 Find the LCM and HCF of the following values by using prime factorization: $$ \mathbf{7 2} $$ and $$ \mathbf{1 8 0} $$. (4) 1.3 The ratio of boys to girls at a party is $$ 4: 5 $$. If there are 20 boys, how many girls are there? (2) 1.4 Thato runs 10 km in 30 minutes. how long (in minutes) will it take him to run 50 km ? (2) $$ \square $$ [12]

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### 1.1 Classifying Numbers

Given the list of numbers: 
$$
\sqrt{1296}, \frac{33}{3}, 0.21, \pi, 36, \sqrt{14}
$$

#### 1.1.1 Rational Numbers
A rational number can be expressed as the quotient of two integers.

- $$ \sqrt{1296} = 36 $$ (which is rational)
- $$ \frac{33}{3} = 11 $$ (which is rational)
- $$ 0.21 $$ (which is rational)

**Rational Numbers:** $$ 36, 11 $$

#### 1.1.2 Integers
An integer is a whole number that can be positive, negative, or zero.

- $$ 36 $$ (which is an integer)
- $$ 11 $$ (which is an integer)

**Integers:** $$ 36, 11 $$

#### 1.1.3 Multiples of 6
A multiple of 6 is any number that can be expressed as $$ 6n $$ where $$ n $$ is an integer.

- $$ 36 $$ (since $$ 36 = 6 \times 6 $$)
- $$ 0 $$ (since $$ 0 = 6 \times 0 $$)

**Multiples of 6:** $$ 36, 0 $$

#### 1.1.4 Irrational Numbers
An irrational number cannot be expressed as a simple fraction.

- $$ \pi $$ (which is irrational)
- $$ \sqrt{14} $$ (which is irrational)

**Irrational Numbers:** $$ \pi, \sqrt{14} $$

### 1.2 Finding LCM and HCF using Prime Factorization

To find the LCM and HCF of $$ 72 $$ and $$ 180 $$, we first perform prime factorization.

- **Prime Factorization of $$ 72 $$:**
  $$
  72 = 2^3 \times 3^2
  $$

- **Prime Factorization of $$ 180 $$:**
  $$
  180 = 2^2 \times 3^2 \times 5^1
  $$

#### HCF (Highest Common Factor)
The HCF is found by taking the lowest power of all prime factors present in both numbers.

- For $$ 2 $$: $$ \min(3, 2) = 2 $$
- For $$ 3 $$: $$ \min(2, 2) = 2 $$
- For $$ 5 $$: Not present in $$ 72 $$

Thus, 
$$
\text{HCF} = 2^2 \times 3^2 = 4 \times 9 = 36
$$

#### LCM (Lowest Common Multiple)
The LCM is found by taking the highest power of all prime factors present in either number.

- For $$ 2 $$: $$ \max(3, 2) = 3 $$
- For $$ 3 $$: $$ \max(2, 2) = 2 $$
- For $$ 5 $$: $$ \max(0, 1) = 1 $$

Thus, 
$$
\text{LCM} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360
$$

**HCF:** $$ 36 $$  
**LCM:** $$ 360 $$

### 1.3 Ratio of Boys to Girls

The ratio of boys to girls is $$ 4:5 $$. If there are $$ 20 $$ boys, we can set up the equation:

Let $$ g $$ be the number of girls. Then:
$$
\frac{4}{5} = \frac{20}{g}
$$

Cross-multiplying gives:
$$
4g = 100 \implies g = \frac{100}{4} = 25
$$

**Number of Girls:** $$ 25 $$

### 1.4 Time to Run 50 km

Thato runs $$ 10 $$ km in $$ 30 $$ minutes. To find the time to run $$ 50 $$ km, we first find the time per km:

$$
\text{Time per km} = \frac{30 \text{ minutes}}{10 \text{ km}} = 3 \text{ minutes/km}
$$

Now, for $$ 50 $$ km:
$$
\text{Time for 50 km} = 50 \text{ km} \times 3 \text{ minutes/km} = 150 \text{ minutes}
$$

**Time to run 50 km:** $$ 150 $$ minutes

### Summary of Answers
1.1.1 Rational Numbers: $$ 36, 11 $$  
1.1.2 Integers: $$ 36, 11 $$  
1.1.3 Multiples of 6: $$ 36, 0 $$  
1.1.4 Irrational Numbers: $$ \pi, \sqrt{14} $$  
1.2 HCF: $$ 36 $$, LCM: $$ 360 $$  
1.3 Number of Girls: $$ 25 $$  
1.4 Time to run 50 km: $$ 150 $$ minutes  
**1.1 Classifying Numbers:** - **Rational Numbers:** 36, 11 - **Integers:** 36, 11 - **Multiples of 6:** 36, 0 - **Irrational Numbers:** π, √14 **1.2 Prime Factorization:** - **HCF of 72 and 180:** 36 - **LCM of 72 and 180:** 360 **1.3 Ratio of Boys to Girls:** - Number of Girls: 25 **1.4 Time to Run 50 km:** - Time: 150 minutes

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