The table shows the ages in years of 42 children at a birthday party. (a) Find the value of x. (b) Calculate, correct to the nearest whole number, the mean age. (c) Find the probability of selecting at random a child whose age is not less than 9 years.

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Asked on June 3, 2026Mathematics

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.

Accepted Answer · 2026-06-03T03:53:22.048Z

a) Step 1: Calculate the initial cost of labour, materials, and consultancy fee. The total cost is GHe 120,000.00. The ratio of labour : materials : consultancy is 5 : 12 : 3. The total number of ratio parts is 5 + 12 + 3 = 20. Initial Labour Cost (L_0): L_0 = (5)/(20) × 120,000 = GHe 30,000 Initial Materials Cost (M_0): M_0 = (12)/(20) × 120,000 = GHe 72,000 Initial Consultancy Fee (C_0): C_0 = (3)/(20) × 120,000 = GHe 18,000 Step 2: Express the new costs in terms of y. Labour cost increased by 2y\%: L_1 = L_0 (1 + (2y)/(100)) = 30,000 (1 + (2y)/(100)) Materials cost increased by 2.5y\%: M_1 = M_0 (1 + (2.5y)/(100)) = 72,000 (1 + (2.5y)/(100)) Consultancy fee remained unchanged: C_1 = C_0 = GHe 18,000 Step 3: Use the given condition to form an equation for y. The new cost of labour is two-fifth the new cost of materials: L_1 = (2)/(5) M_1 30,000 (1 + (2y)/(100)) = (2)/(5) × 72,000 (1 + (2.5y)/(100)) 30,000 (1 + (2y)/(100)) = 28,800 (1 + (2.5y)/(100)) Step 4: Solve the equation for y. Divide both sides by 100: 300 (1 + (2y)/(100)) = 288 (1 + (2.5y)/(100)) Divide both sides by 12: 25 (1 + (2y)/(100)) = 24 (1 + (2.5y)/(100)) 25 + (50y)/(100) = 24 + (60y)/(100) 25 + 0.5y = 24 + 0.6y 25 - 24 = 0.6y - 0.5y 1 = 0.1y y = (1)/(0.1) y = 10 The value of y is 10. b) Step 1: Calculate the new costs using y=10. New Labour Cost (L_1): L_1 = 30,000 (1 + (2 × 10)/(100)) = 30,000 (1 + (20)/(100)) = 30,000 (1.2) = GHe 36,000 New Materials Cost (M_1): M_1 = 72,000 (1 + (2.5 × 10)/(100)) = 72,000 (1 + (25)/(100)) = 72,000 (1.25) = GHe 90,000 New Consultancy Fee (C_1): C_1 = GHe 18,000 Step 2: Calculate the new total cost of building the house. New Total Cost = L_1 + M_1 + C_1 New Total Cost = 36,000 + 90,000 + 18,000 = GHe 144,000 The new cost of building the house is GHe 144,000.00. Drop the next question.

Accepted Answer · 2026-06-03T03:53:22.048Z

a) Step 1: Calculate the initial cost of labour, materials, and consultancy fee. The total cost is GHe 120,000.00. The ratio of labour : materials : consultancy is 5 : 12 : 3. The total number of ratio parts is 5 + 12 + 3 = 20. Initial Labour Cost (L_0): L_0 = (5)/(20) × 120,000 = GHe 30,000 Initial Materials Cost (M_0): M_0 = (12)/(20) × 120,000 = GHe 72,000 Initial Consultancy Fee (C_0): C_0 = (3)/(20) × 120,000 = GHe 18,000 Step 2: Express the new costs in terms of y. Labour cost increased by 2y\%: L_1 = L_0 (1 + (2y)/(100)) = 30,000 (1 + (2y)/(100)) Materials cost increased by 2.5y\%: M_1 = M_0 (1 + (2.5y)/(100)) = 72,000 (1 + (2.5y)/(100)) Consultancy fee remained unchanged: C_1 = C_0 = GHe 18,000 Step 3: Use the given condition to form an equation for y. The new cost of labour is two-fifth the new cost of materials: L_1 = (2)/(5) M_1 30,000 (1 + (2y)/(100)) = (2)/(5) × 72,000 (1 + (2.5y)/(100)) 30,000 (1 + (2y)/(100)) = 28,800 (1 + (2.5y)/(100)) Step 4: Solve the equation for y. Divide both sides by 100: 300 (1 + (2y)/(100)) = 288 (1 + (2.5y)/(100)) Divide both sides by 12: 25 (1 + (2y)/(100)) = 24 (1 + (2.5y)/(100)) 25 + (50y)/(100) = 24 + (60y)/(100) 25 + 0.5y = 24 + 0.6y 25 - 24 = 0.6y - 0.5y 1 = 0.1y y = (1)/(0.1) y = 10 The value of y is 10. b) Step 1: Calculate the new costs using y=10. New Labour Cost (L_1): L_1 = 30,000 (1 + (2 × 10)/(100)) = 30,000 (1 + (20)/(100)) = 30,000 (1.2) = GHe 36,000 New Materials Cost (M_1): M_1 = 72,000 (1 + (2.5 × 10)/(100)) = 72,000 (1 + (25)/(100)) = 72,000 (1.25) = GHe 90,000 New Consultancy Fee (C_1): C_1 = GHe 18,000 Step 2: Calculate the new total cost of building the house. New Total Cost = L_1 + M_1 + C_1 New Total Cost = 36,000 + 90,000 + 18,000 = GHe 144,000 The new cost of building the house is GHe 144,000.00. Drop the next question.

The table shows the ages in years of 42 children at a birthday party. (a) Find the value of x. (b) Calculate, correct to the nearest whole number, the mean age. (c) Find the probability of selecting at random a child whose age is not less than 9 years.

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The table shows the ages in years of 42 children at a birthday party. (a) Find the value of x. (b) Calculate, correct to the nearest whole number, the mean age. (c) Find the probability of selecting at random a child whose age is not less than 9 years.

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