2.6. Determine the HCF and LCM 28 and 72 using product of prime factor Question 3 [18] 3.1. The temperature in Austria one morning was \( -5^{\circ} \mathrm{C} \) at \( 08: 00 \) and increased by \( 2^{\circ} \mathrm{C} \) every hour until \( 12: 00 \). What will the temperature be at 11:30? (1) 3.2. Jannie receives R150 pocket money per month. In the new year his mother decided to increase his pocket money in the ratio \( 6: 5 \). Calculate Jannie's adjusted monthly pocket money. (2) 3.3. The Grade 8 learners decided to start living more healthily. They will either jog or cycle. There are 125 Grade 8 learners, and they jog and cycle in the ratio 3:2. Calculate how many learners participate in each sport? (3) \( \qquad \) \( \qquad \) 3.4. Michael, a farmer, wants to buy a new tractor. The price of the tractor is R160 000, VAT excluded. He can afford a deposit of R20 000. He decides to buy the tractor on hire purchase over a period of 60 months and simple interest of \( 10 \% \). What would he pay in total after 60 months? (4) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \)

Below are the detailed solutions: ───────────────────────────── Problem 2.6: Determine the HCF and LCM of 28 and 72 using the product of prime factors. Step 1. Write each number as a product of its prime factors. • 28 = 2 × 2 × 7 = 2² × 7 • 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3² Step 2. Find the Highest Common Factor (HCF) by taking the common primes with the lowest exponents. • Both numbers have the prime 2 in common. The lower power is 2². Thus, HCF = 2² = 4 Step 3. Find the Least Common Multiple (LCM) by taking every prime factor with the highest exponent from either number. • For 2: the highest exponent is 2³ • For 3: only present in 72 to the power 3² • For 7: only present in 28 to the power 7¹ Thus, LCM = 2³ × 3² × 7 = 8 × 9 × 7 = 504 ───────────────────────────── Problem 3.1: Temperature in Austria The temperature was –5°C at 08:00 and increased by 2°C every hour. • At 08:00: –5°C • At 09:00: –5 + 2 = –3°C • At 10:00: –3 + 2 = –1°C • At 11:00: –1 + 2 = 1°C • At 12:00: 1 + 2 = 3°C Since 11:30 is halfway between 11:00 and 12:00, the increase in that half-hour is half of 2°C, which is 1°C. Therefore, at 11:30 the temperature = 1°C + 1°C = 2°C ───────────────────────────── Problem 3.2: Adjusted Monthly Pocket Money Jannie originally receives R150. His mother increases his money in the ratio 6:5. This means: New Amount : Old Amount = 6 : 5 New Amount = (6/5) × R150 = R180 ───────────────────────────── Problem 3.3: Grade 8 Learners' Sporting Choices There are 125 learners, and the ratio of those who jog to cycle is 3:2. Total parts = 3 + 2 = 5 • Number jogging = (3/5) × 125 = 75 • Number cycling = (2/5) × 125 = 50 ───────────────────────────── Problem 3.4: Tractor Purchase on Hire Purchase Michael’s tractor costs R160,000 (VAT excluded). He puts down a deposit of R20,000. The amount to be financed is: Principal = R160,000 – R20,000 = R140,000 The hire purchase plan is over 60 months (which is 5 years) with a simple interest rate of 10% per annum. Calculate the interest using simple interest formula: Interest = Principal × Rate × Time Interest = R140,000 × 0.10 × 5 = R70,000 Total amount to be paid via hire purchase = Principal + Interest = R140,000 + R70,000 = R210,000 Including the deposit, the total paid over the period is: Total Payment = Deposit + Hire Purchase Amount = R20,000 + R210,000 = R230,000 ───────────────────────────── Summary of Answers: • HCF of 28 and 72: 4 • LCM of 28 and 72: 504 • Temperature at 11:30: 2°C • Jannie’s adjusted monthly pocket money: R180 • Learners who jog: 75 and who cycle: 50 • Michael’s total payment after 60 months: R230,000