This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here's the breakdown of the millet mixture problem: Part a) Calculate the selling price of one bag of the mixture. Step 1: Calculate the total cost of Type A millet. Cost of Type A = 30 bags × Ksh 400/bag = Ksh 12000 Step 2: Calculate the total cost of Type B millet. Cost of Type B = 50 bags × Ksh 380/bag = Ksh 19000 Step 3: Calculate the total cost of the mixture. Total Cost = Ksh 12000 + Ksh 19000 = Ksh 31000 Step 4: Calculate the total number of bags in the mixture. Total Bags = 30 bags + 50 bags = 80 bags Step 5: Calculate the cost price per bag of the mixture. Cost Price per Bag = Total CostTotal Bags = Ksh 3100080 = Ksh 387.50 Step 6: Calculate the selling price per bag with a 20% profit. Selling Price = Cost Price × (1 + Profit Percentage) Selling Price = Ksh 387.50 × (1 + 0.20) Selling Price = Ksh 387.50 × 1.20 Selling Price = Ksh 465 The selling price of one bag of the mixture is Ksh 465. Part b) Find the ratio x:y. Step 1: Calculate the cost price per bag of the mixture. Given Selling Price (SP) = Ksh 498.55 and Profit = 30%. Let Cost Price (CP) be the cost per bag. SP = CP × (1 + Profit Percentage) Ksh 498.55 = CP × (1 + 0.30) Ksh 498.55 = CP × 1.30 CP = Ksh 498.551.30 = Ksh 383.50 Step 2: Set up an equation for the cost price per bag based on the ratio x:y. Let x be the number of bags of Type A and y be the number of bags of Type B. Total cost of mixture = x × Ksh 400 + y × Ksh 380 = 400x + 380y Total number of bags = x+y Cost price per bag = (400x + 380y)/(x+y) We equate this to the calculated cost price per bag: (400x + 380y)/(x+y) = 383.50 Step 3: Solve for the ratio x:y. 400x + 380y = 383.50(x+y) 400x + 380y = 383.50x + 383.50y 400x - 383.50x = 383.50y - 380y 16.50x = 3.50y To find the ratio x:y: (x)/(y) = (3.50)/(16.50) Multiply numerator and denominator by 100 to remove decimals: (x)/(y) = (350)/(1650) Simplify the fraction by dividing both by 50: (x)/(y) = (7)/(33) The ratio x:y is 7:33. Part c) Calculate the ratio of Type A millet to Type B millet in the final mixture. Step 1: Determine the proportion of Type A and Type B in one bag of mixture from part (a). From part (a), the mixture has 30 bags of Type A and 50 bags of Type B, totaling 80 bags. In one bag of mixture (a): Amount of Type A = (30)/(80) = (3)/(8) Amount of Type B = (50)/(80) = (5)/(8) Step 2: Determine the proportion of Type A and Type B in one bag of mixture from part (b). From part (b), the ratio of Type A to Type B is 7:33. The total parts are 7+33=40. In one bag of mixture (b): Amount of Type A = (7)/(40) Amount of Type B = (33)/(40) Step 3: Calculate the total amount of Type A and Type B when one bag of each mixture is combined. Total Type A = (Amount of Type A in bag from a) + (Amount of Type A in bag from b) Total Type A = (3)/(8) + (7)/(40) To add, find a common denominator (40): Total Type A = (3 × 5)/(8 × 5) + (7)/(40) = (15)/(40) + (7)/(40) = (22)/(40) Total Type B = (Amount of Type B in bag from a) + (Amount of Type B in bag from b) Total Type B = (5)/(8) + (33)/(40) To add, find a common denominator (40): Total Type B = (5 × 5)/(8 × 5) + (33)/(40) = (25)/(40) + (33)/(40) = (58)/(40) Step 4: Calculate the ratio of Type A millet to Type B millet in the final mixture. Ratio = Total Type A : Total Type B Ratio = (22)/(40) : (58)/(40) This simplifies to 22:58. Divide both sides by their greatest common divisor, which is 2: Ratio = 11:29 The ratio of Type A millet to Type B millet in the final mixture is 11:29. Last free one today — make it count tomorrow, or type /upgrade for unlimited.