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]

### 1.1 Classification of Numbers **1.1.1 Rational Numbers** A number is rational if it can be expressed as a fraction of two integers. From the list, two examples are: - \( \frac{33}{3} \) (which simplifies to \(11\)) - \( 0.21 \) (which can be written as \(\frac{21}{100}\)) --- **1.1.2 Integers** An integer is a whole number (positive, negative, or zero) without fractional parts. Two examples from the list are: - \( \sqrt{1296} \) (since \(\sqrt{1296} = 36\)) - \( 36 \) --- **1.1.3 Multiples of 6** A number is a multiple of 6 if it is divisible by 6. Two entries from the list, noting that \( \sqrt{1296} \) and \( 36 \) represent the same value, are: - \( \sqrt{1296} \) (which is \(36\) and \(36 \div 6 = 6\)) - \( 36 \) *(They both represent the same multiple, but are both given in the list.)* --- **1.1.4 Irrational Numbers** A number is irrational if it cannot be expressed as a fraction of two integers. Two examples are: - \( \pi \) - \( \sqrt{14} \) --- ### 1.2 LCM and HCF of \(72\) and \(180\) **Step 1: Prime Factorization** Factor each number into primes: \[ 72 = 2^3 \times 3^2 \] \[ 180 = 2^2 \times 3^2 \times 5 \] **Step 2: Highest Common Factor (HCF)** For the HCF, take the lowest power of the common primes: - For \(2\): lowest power is \(2^2\) - For \(3\): lowest power is \(3^2\) \[ \text{HCF} = 2^2 \times 3^2 = 4 \times 9 = 36 \] **Step 3: Least Common Multiple (LCM)** For the LCM, take the highest power of each prime present: - For \(2\): highest power is \(2^3\) - For \(3\): highest power is \(3^2\) - For \(5\): appears as \(5^1\) \[ \text{LCM} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360 \] --- ### 1.3 Ratio of Boys to Girls The ratio of boys to girls is given as \(4:5\). If there are 20 boys, set up the proportion: \[ 4k = 20 \quad \Rightarrow \quad k = \frac{20}{4} = 5 \] Then the number of girls is: \[ 5k = 5 \times 5 = 25 \] --- ### 1.4 Time for 50 km Run Thato runs \(10 \text{ km}\) in \(30 \text{ minutes}\). To find the time taken to run \(50 \text{ km}\), note that: \[ \text{Time} \propto \text{Distance} \] Using the ratio: \[ \frac{50 \text{ km}}{10 \text{ km}} = 5 \quad \Rightarrow \quad \text{Time} = 30 \times 5 = 150 \text{ minutes} \]