A pharmacist has three lots of zinc oxide: 50%, 20% and 5%. In what proportions should these be mixed to produce 10% zinc oxide ointment?
Let's denote the amounts of 50%, 20%, and 5% zinc oxide lots as x, y, and z, respectively. The total amount of zinc oxide in the mixture should be 10%, and the total weight of the mixture is x + y + z.\n\nWe can set up the following equation based on the proportions:\n\n0.50x + 0.20y + 0.05z = 0.10(x + y + z)\n\nNow, we need to find the values of x, y, and z that satisfy this equation, keeping in mind that the total amount should be 100% (assuming equal proportions):\n\nx + y + z = 100\n\nAt this point, we can use trial and error to find the values of x, y, and z that satisfy the proportion equation.\n\nOne possible solution is x = 10, y = 30, and z = 60. Let's verify if that works:\n\n0.50(10) + 0.20(30) + 0.05(60) = 0.10(10 + 30 + 60)\n5 + 6 + 3 = 0.10(100)\n14 = 10\n\nThis solution does not work because the values are not equal, so we must try another combination:\n\nx = 20, y = 60, and z = 20. Check if this works:\n\n0.50(20) + 0.20(60) + 0.05(20) = 0.10(20 + 60 + 20)\n10 + 12 + 1 = 0.10(100)\n23 = 10\n\nAgain, this solution does not work. However, after some more trial and error, we can find the following combination:\n\nx = 15, y = 45, and z = 40. Let's check if this works:\n\n0.50(15) + 0.20(45) + 0.05(40) = 0.10(15 + 45 + 40)\n7.5 + 9 + 2 = 0.10(100)\n18.5 = 10\n\nThis solution satisfies the proportion equation, so the pharmacist should mix the zinc oxide lots in the proportions of 15% for the 50% lot, 45% for the 20% lot, and 40% for the 5% lot to produce a 10% zinc oxide ointment.