1. Problem statement: - A parish agricultural officer has 450 kilograms of fertilisers and 120 litres of anti-tick vaccine to distribute equally among farmers. We need to find the number of farmers and how much each gets. - Joseph produces 180 litres of milk daily, sells 60% at 850 per litre, donates \(\frac{1}{15}\) of the remainder, and uses the leftover milk to make yoghurt and butter in ratio 1:3. We need to find how much money he collects and how much milk he uses to make butter. 2. Step (a)(i) Find the number of farmers: - Let the number of farmers be \(n\). - Since the fertiliser is distributed equally, \(n\) must divide both 450 and 120 evenly. - Find the greatest common divisor (GCD) of 450 and 120: $$\text{Factors of }450: 1,2,3,5,6,9,10,15,18,25,30,45,50,75,90,150,225,450$$ $$\text{Factors of }120: 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120$$ - Common factors: 1,2,3,5,6,10,15,30 - Greatest common divisor (GCD) is 30. - Thus, \(n=30\) farmers will receive the items. 3. Step (a)(ii) Calculate amount per farmer: - Fertiliser per farmer: $$\frac{450 \text{ kg}}{30} = 15 \text{ kg per farmer}$$ - Vaccine per farmer: $$\frac{120 \text{ litres}}{30} = 4 \text{ litres per farmer}$$ 4. Step (b)(i) Calculate money Joseph collects from milk sales: - Total milk = 180 litres - Milk sold = 60% of 180 = $$0.6 \times 180 = 108 \text{ litres}$$ - Price per litre = 850 - Money collected: $$108 \times 850 = 91800$$ 5. Step (b)(ii) Calculate milk to use for butter: - Milk left after selling = 180 - 108 = 72 litres - Milk donated to orphanage = \(\frac{1}{15}\) of 72 litres: $$\frac{1}{15} \times 72 = 4.8 \text{ litres}$$ - Milk left to make products = 72 - 4.8 = 67.2 litres - Yoghurt:Butter ratio is 1:3, total parts = 1 + 3 = 4 parts - Milk for butter: $$\frac{3}{4} \times 67.2 = 50.4 \text{ litres}$$ Final answers: - Number of farmers: 30 - Fertiliser per farmer: 15 kg - Vaccine per farmer: 4 litres - Money collected by Joseph: 91800 - Milk used for butter: 50.4 litres