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 the questions: Question 1: Inverse Proportion A 50kg bag of rice lasts 20 people for 3 weeks. This means the total "person-weeks" of rice is 20 people × 3 weeks = 60 person-weeks. Let W be the number of weeks the rice lasts for P people. Then P × W = 60, so W = (60)/(P). i. a family of 4 Step 1: Identify the number of people. P = 4 Step 2: Calculate the number of weeks. W = (60)/(4) = 15 weeks. The rice would last for 15 weeks. ii. a family of 6 Step 1: Identify the number of people. P = 6 Step 2: Calculate the number of weeks. W = (60)/(6) = 10 weeks. The rice would last for 10 weeks. iii. a hotel that serves 60 people Step 1: Identify the number of people. P = 60 Step 2: Calculate the number of weeks. W = (60)/(60) = 1 week. The rice would last for 1 week. Question 2: Solve the equation (6P + 5)/(P + 3) = 5 Step 1: Multiply both sides by (P+3) to eliminate the denominator. 6P + 5 = 5(P + 3) Step 2: Distribute the 5 on the right side. 6P + 5 = 5P + 15 Step 3: Subtract 5P from both sides. 6P - 5P + 5 = 15 P + 5 = 15 Step 4: Subtract 5 from both sides. P = 15 - 5 P = 10 The solution is P = 10. Question 3a: Find the sizes of the lettered angles in the following diagram. The diagram shows a triangle with an altitude drawn from the top vertex to the base. The square symbol indicates that the line segment is perpendicular to the base, forming a 90^ angle. The tick marks on the base indicate that the altitude bisects the base. When an altitude bisects the base of a triangle, the triangle is an isosceles triangle. This means the two base angles are equal, and the altitude also bisects the top angle. Step 1: Determine angle U. Since the triangle is isosceles, the base angles are equal. One base angle is U, and the other is 29^. U = 29^ Step 2: Determine angles T and V. Since the triangle is isosceles, the altitude bisects the top angle, so V = T. Consider the right-angled triangle on the right side. The angles are T, 90^, and 29^. The sum of angles in a triangle is 180^. T + 90^ + 29^ = 180^ T + 119^ = 180^ T = 180^ - 119^ T = 61^ Since V = T, then V = 61^. Step 3: Determine angle S. The placement of S and 74^ is ambiguous in the diagram. However, if we assume the 74^ is an angle within the left small triangle, and S is the angle adjacent to the 90^ angle at the base, this would imply S=90^, which is redundant with the square symbol. Given the common conventions for such diagrams, the 74^ label is likely intended to be the angle U, but this contradicts the isosceles property (U=29^). Assuming the geometric properties (altitude bisects base isosceles triangle) take precedence over potentially misleading numerical labels, we proceed with U=29^. The letter S is placed at the base, to the left of the altitude, adjacent to the 90^ angle. This is the position of the angle at the foot of the altitude, which is 90^. Therefore, S = 90^. The sizes of the lettered angles are: U = 29^ V = 61^ T = 61^ S = 90^ Question 3b: Find (i) the amount and (ii) the compound interest of N500,000 for 2 years at 8% per annum. Given: Principal (P) = N500,000, Time (n) = 2 years, Rate (r) = 8% = 0.08. Step 1: Calculate the Amount (A) using the compound interest formula. A = P(1 + r)^n A = 500000(1 + 0.08)^2 A = 500000(1.08)^2 A = 500000(1.1664) A = 583200 The amount is N583,200. Step 2: Calculate the Compound Interest (CI). CI = A - P CI = 583200 - 500000 CI = 83200 The compound interest is N83,200. Question 4a: Copy and complete the table below for y = 3x + 2. Step 1: Calculate 3x for each x value. • For x = -1: 3x = 3(-1) = -3 • For x = 0: 3x = 3(0) = 0 • For x = 1: 3x = 3(1) = 3 • For x = 2: 3x = 3(2) = 6 • For x = 3: 3x = 3(3) = 9 Step 2: Calculate y = 3x + 2 for each x value. • For x = -1: y = -3 + 2 = -1 • For x = 0: y = 0 + 2 = 2 • For x = 1: y = 3 + 2 = 5 • For x = 2: y = 6 + 2 = 8 • For x = 3: y = 9 + 2 = 11 The completed table is: | x | -1 | 0 | 1 | 2 | 3 | |---|----|---|---|---|---| | 3x | -3 | 0 | 3 | 6 | 9 | | 2 | 2 | 2 | 2 | 2 | 2 | | y | -1 | 2 | 5 | 8 | 11 | That's 2 down. 3 left today — send the next one.