1. **Find the HCF of 540 and 1350** The Highest Common Factor (HCF) is the greatest number that divides both numbers exactly. Use prime factorization: $$540 = 2^2 \times 3^3 \times 5$$ $$1350 = 2 \times 3^3 \times 5^2$$ Common factors: $2^1$, $3^3$, $5^1$ $$\text{HCF} = 2 \times 3^3 \times 5 = 2 \times 27 \times 5 = 270$$ 2. **Find the HCF of 1440 and 15120** Prime factorization: $$1440 = 2^5 \times 3^2 \times 5$$ $$15120 = 2^4 \times 3^3 \times 5 \times 7$$ Common factors: $2^4$, $3^2$, $5^1$ $$\text{HCF} = 2^4 \times 3^2 \times 5 = 16 \times 9 \times 5 = 720$$ 3. **Find the LCM of 180 and 540** Prime factorization: $$180 = 2^2 \times 3^2 \times 5$$ $$540 = 2^2 \times 3^3 \times 5$$ LCM takes highest powers: $$\text{LCM} = 2^2 \times 3^3 \times 5 = 4 \times 27 \times 5 = 540$$ 4. **Find the LCM of 810 and 720** Prime factorization: $$810 = 2 \times 3^4 \times 5$$ $$720 = 2^4 \times 3^2 \times 5$$ LCM takes highest powers: $$\text{LCM} = 2^4 \times 3^4 \times 5 = 16 \times 81 \times 5 = 6480$$ 5. **Simplify ratios:** (1) $48 : 84$ Find HCF of 48 and 84: $$48 = 2^4 \times 3$$ $$84 = 2^2 \times 3 \times 7$$ HCF = $2^2 \times 3 = 12$ Divide both by 12: $$\frac{48}{12} : \frac{84}{12} = 4 : 7$$ (2) $7 m : 3800 cm$ Convert 3800 cm to meters: $$3800 \text{ cm} = \frac{3800}{100} = 38 \text{ m}$$ Ratio: $$7 : 38$$ (3) $480 g : 40 kg$ Convert 40 kg to grams: $$40 \text{ kg} = 40 \times 1000 = 40000 \text{ g}$$ Ratio: $$480 : 40000$$ Find HCF: $$480 = 2^5 \times 3 \times 5$$ $$40000 = 2^6 \times 5^4$$ HCF = $2^5 \times 5 = 32 \times 5 = 160$ Divide both by 160: $$\frac{480}{160} : \frac{40000}{160} = 3 : 250$$ 6. **Foot-to-height ratio for mother and daughter** Mother: $$\frac{24}{152} = \frac{24}{152} = \frac{3}{19}$$ Daughter: $$\frac{21}{136} = \frac{21}{136} = \frac{3}{19}$$ Both ratios are equal, so foot-to-height ratio is the same. 7. **Scale of the map** Map distance = 5 cm Real distance = 300 km = $300 \times 100000 = 30000000$ cm Scale = map distance : real distance $$5 : 30000000 = 1 : \cancel{5}6000000$$ Scale is 1 : 6000000 8. **Ratio problems** (1) Ratio 4 : 7, larger number 56 Find smaller number: $$\frac{7}{4} = \frac{56}{x} \Rightarrow x = \frac{4 \times 56}{7} = 32$$ (2) Red to blue balls 3 : 5, red balls 27 Find blue balls: $$\frac{3}{5} = \frac{27}{x} \Rightarrow x = \frac{5 \times 27}{3} = 45$$ (3) Divide 400 in ratio 2 : 3 : 5 Sum of parts = 2 + 3 + 5 = 10 Parts: $$\frac{2}{10} \times 400 = 80$$ $$\frac{3}{10} \times 400 = 120$$ $$\frac{5}{10} \times 400 = 200$$ (4) Increase 180 in ratio 3 : 7 Sum = 3 + 7 = 10 Parts: $$\frac{3}{10} \times 180 = 54$$ $$\frac{7}{10} \times 180 = 126$$ (5) Decrease 180 in ratio 5 : 2 Sum = 5 + 2 = 7 Parts: $$\frac{5}{7} \times 180 = \frac{900}{7} \approx 128.57$$ $$\frac{2}{7} \times 180 = \frac{360}{7} \approx 51.43$$ 9. **Cell phone store profit and price** Cost price = 24000 (1) Selling price = cost price + 25% of cost price $$\text{Selling price} = 24000 + 0.25 \times 24000 = 24000 + 6000 = 30000$$ Profit = 30000 - 24000 = 6000 (2) Percentage profit: $$\frac{6000}{24000} \times 100 = 25\%$$ (3) Ratio of profit to selling price: $$\frac{6000}{30000} = \frac{1}{5}$$ (4) Selling price including 15% VAT: $$30000 + 0.15 \times 30000 = 30000 + 4500 = 34500$$ (5) VAT charged per phone = 4500 (6) Discounted selling price for phones costing 2500 with 10% discount: Original selling price = 2500 + 0.25 \times 2500 = 2500 + 625 = 3125 Discount = 10% of 3125 = 312.5 Discounted price = 3125 - 312.5 = 2812.5 10. **Currency conversion** (1) Cost in rands for $1200 phone at R16.17 per dollar: $$1200 \times 16.17 = 19404$$ (2) Cost in pounds for R600000 at R21 per pound: $$\frac{600000}{21} \approx 28571.43$$ 11. **Hire purchase loan** Cost = 250000 Deposit = 10% of 250000 = 25000 Balance = 250000 - 25000 = 225000 Interest for 24 months (2 years) at 16% per annum: $$\text{Interest} = 225000 \times 0.16 \times 2 = 72000$$ Total amount paid = 225000 + 72000 = 297000 12. **Water and car problems** (1) Cost of 12 litres at R2 per litre: $$12 \times 2 = 24$$ (2) Car uses 15 litres for 450 km Rate in km/litre: $$\frac{450}{15} = 30 \text{ km/litre}$$ Rate in litres/km: $$\frac{15}{450} = \frac{1}{30} \text{ litres/km}$$ (3) Car travels 585 km in 6.5 hours Speed in km/h: $$\frac{585}{6.5} = 90 \text{ km/h}$$ Speed in km/min: $$\frac{90}{60} = 1.5 \text{ km/min}$$ (4) Distance cycled in 45 minutes at 20 km/h Convert 45 minutes to hours: $$\frac{45}{60} = 0.75 \text{ hours}$$ Distance: $$20 \times 0.75 = 15 \text{ km}$$ (5) Time to travel 400 km at 80 km/h $$\frac{400}{80} = 5 \text{ hours}$$ (6) Compare speeds: 400 m in 48 s: Convert 400 m to km: 0.4 km Speed: $$\frac{0.4}{\frac{48}{3600}} = \frac{0.4}{0.01333} = 30 \text{ km/h}$$ 56 km in 2 hours: Speed = 28 km/h Fastest is 400 m in 48 s (30 km/h)