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.
ScanSolve AI Answer
You're on a roll — Step 1: Calculate the initial costs for labour, materials, and contractor's charges. The total estimated cost is \64,000.00. The ratio of labour : materials : contractor's charges is 12 : 15 : 5. The sum of the ratio parts is 12 + 15 + 5 = 32. Initial cost of labour (L): L = (12)/(32) × \64,000 = 12 × \2,000 = \24,000 Initial cost of materials (M): M = (15)/(32) × \64,000 = 15 × \2,000 = \30,000 Initial contractor's charges (C): C = (5)/(32) × \64,000 = 5 × \2,000 = \10,000 Step 2: Express the new costs after the increases. The cost of labour increased by x \%. New labour cost (L'): L' = L (1 + (x)/(100)) = \24,000 (1 + (x)/(100)) The cost of materials increased by 2x \%. New materials cost (M'): M' = M (1 + (2x)/(100)) = \30,000 (1 + (2x)/(100)) The contractor's charges remained the same. New contractor's charges (C'): C' = C = \10,000 Step 3: Use the given condition to find the value of x. After the increase, the cost of labour was two-thirds the cost of materials: L' = (2)/(3) M' Substitute the expressions for L' and M': 24,000 (1 + (x)/(100)) = (2)/(3) × 30,000 (1 + (2x)/(100)) 24,000 (1 + (x)/(100)) = 20,000 (1 + (2x)/(100)) Divide both sides by 1,000: 24 (1 + (x)/(100)) = 20 (1 + (2x)/(100)) Divide both sides by 4: 6 (1 + (x)/(100)) = 5 (1 + (2x)/(100)) Expand the equation: 6 + (6x)/(100) = 5 + (10x)/(100) Multiply by 100 to clear the denominators: 600 + 6x = 500 + 10x Rearrange and solve for x: 600 - 500 = 10x - 6x 100 = 4x x = (100)/(4) x = 25 a) The value of x: x = 25 Step 4: Calculate the new cost of building the restaurant. First, calculate the new individual costs using x=25. New labour cost (L'): L' = \24,000 (1 + (25)/(100)) = \24,000 × (1 + 0.25) = \24,000 × 1.25 = \30,000 New materials cost (M'): M' = \30,000 (1 + (2 × 25)/(100)) = \30,000 × (1 + 0.50) = \30,000 × 1.50 = \45,000 New contractor's charges (C'): C' = \10,000 The new total cost of building the restaurant is the sum of the new labour, materials, and contractor's charges: New Total Cost = L' + M' + C' New Total Cost = \30,000 + \45,000 + \10,000 New Total Cost = \85,000 b) The new cost of building the restaurant: \$85,000.00 What's next?