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 are the solutions to questions 3, 4, and 5: 3. Engineer and Assistant Wages Step 1: Define variables and set up equations. Let E be the daily wage of the engineer and A be the daily wage of the assistant. From the first week's work: 6E + 7A = 1,350,000 (1) From the second week's work: 5E + 3A = 870,000 (2) Step 2: Solve the system of equations using elimination. Multiply equation (1) by 3 and equation (2) by 7 to eliminate A: 3 × (6E + 7A) = 3 × 1,350,000 18E + 21A = 4,050,000 (3) 7 × (5E + 3A) = 7 × 870,000 35E + 21A = 6,090,000 (4) Step 3: Subtract equation (3) from equation (4) to find E. (35E + 21A) - (18E + 21A) = 6,090,000 - 4,050,000 17E = 2,040,000 E = (2,040,000)/(17) E = 120,000 Step 4: Substitute the value of E into equation (2) to find A. 5(120,000) + 3A = 870,000 600,000 + 3A = 870,000 3A = 870,000 - 600,000 3A = 270,000 A = (270,000)/(3) A = 90,000 The daily wage of the engineer is ugx 120,000 and the daily wage of the assistant is ugx 90,000. 4. Age Problem and School Admissions a) Find the son's present age. Step 1: Define variables and set up equations. Let F be the father's present age and S be the son's present age. From the first statement: F = S + 30 (1) From the second statement (5 years ago): F - 5 = 4(S - 5) (2) Step 2: Substitute equation (1) into equation (2). (S + 30) - 5 = 4(S - 5) S + 25 = 4S - 20 Step 3: Solve for S. 25 + 20 = 4S - S 45 = 3S S = (45)/(3) S = 15 The son's present age is 15 years. b) Help the head teacher to find the number of students to admit for S.5. Step 1: Express the number of students in each class in terms of S_1. Let S_1, S_2, S_3, S_4, S_5 be the number of students in each class. S_2 = S_1 - 5 S_3 = S_1 - 15 S_4 = (1)/(2)S_1 S_5 = (1)/(4)S_1 Step 2: Use the total enrolment from S.1 to S.4 to find S_1. The total enrolment from S.1 to S.4 is 680 students: S_1 + S_2 + S_3 + S_4 = 680 Substitute the expressions from Step 1: S_1 + (S_1 - 5) + (S_1 - 15) + (1)/(2)S_1 = 680 3S_1 + (1)/(2)S_1 - 20 = 680 3.5S_1 = 680 + 20 3.5S_1 = 700 S_1 = (700)/(3.5) S_1 = 200 Step 3: Calculate the number of students for S.5. S_5 = (1)/(4)S_1 S_5 = (1)/(4) × 200 S_5 = 50 The number of students to admit for S.5 is 50. 5. Probability *A) Two dice whose faces are marked from 1-to-6 inclusive are rolled together. Find the probability that the sum of