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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4. a) Step 1: Find a common denominator for the fractions. The denominators are 5 and 7. The least common multiple of 5 and 7 is 5 × 7 = 35. Step 2: Convert the fractions to have the common denominator. (3)/(5) = (3 × 7)/(5 × 7) = (21)/(35) (12)/(7) = (12 × 5)/(7 × 5) = (60)/(35) Step 3: Add the fractions. (21)/(35) + (60)/(35) = (21 + 60)/(35) = (81)/(35) The fraction is in its simplest form as 81 and 35 have no common factors other than 1. It can also be expressed as a mixed number. 81 ÷ 35 = 2 with a remainder of 11 So, (81)/(35) = 2(11)/(35). The simplified expression is (81)/(35) or 2(11)/(35). 4. b) Step 1: Determine the cost per kg of meat. Given that 3 kg of meat costs Gh¢60. Cost per kg = Total CostWeight = 60 Gh¢3 kg = 20 Gh¢/kg. Step 2: Calculate the cost for each weight in the table. For 2 kg: Cost = 2 kg × 20 Gh¢/kg = 40 Gh¢. For 5 kg: Cost = 5 kg × 20 Gh¢/kg = 100 Gh¢. For 12 kg: Cost = 12 kg × 20 Gh¢/kg = 240 Gh¢. Step 3: Complete the table. | Meat (kg) | 2 | 3 | 5 | 12 | |---|---|---|---|---| | Cost Gh¢ | 40 | 60 | 100 | 240 | 4. c) Step 1: Substitute each given value of x into the relation y = 3x - 6 to find the corresponding y value. For x = -2: y = 3(-2) - 6 = -6 - 6 = -12 For x = -1: y = 3(-1) - 6 = -3 - 6 = -9 For x = 0: y = 3(0) - 6 = 0 - 6 = -6 For x = 1: y = 3(1) - 6 = 3 - 6 = -3 For x = 2: y = 3(2) - 6 = 6 - 6 = 0 Step 2: Complete the table. | x | -2 | -1 | 0 | 1 | 2 | |---|---|---|---|---|---| | y | -12 | -9 | -6 | -3 | 0 | 5. a) Step 1: Identify the length and width of the rectangle. Length L = 12.8 m Width W = 7.5 m Step 2: Use the formula for the perimeter of a rectangle, P = 2(L + W). P = 2(12.8 m + 7.5 m) P = 2(20.3 m) P = 40.6 m The perimeter of the rectangle is 40.6 m. 5. b) Step 1: Simplify the expression -30abc ÷ 60b^3c^2. (-30abc)/(60b^3c^2) Divide the numerical coefficients: (-30)/(60) = -(1)/(2) Divide the variable terms using the rule x^m/x^n = x^m-n: a^1/1 = a b^1/b^3 = b^1-3 = b^-2 = (1)/(b^2) c^1/c^2 = c^1-2 = c^-1 = (1)/(c) Combine the simplified terms: -(1)/(2) · a · (1)/(b^2) · (1)/(c) = -(a)/(2b^2c) Step 2: Substitute the given values a=1, b=-2, and c=3 into the simplified expression. -(1)/(2(-2)^2(3)) -(1)/(2(4)(3)) -(1)/(8(3)) -(1)/(24) The simplified and evaluated expression is -(1)/(24). 5. c) Step 1: Understand that supplementary angles add up to 180^. The two angles given are (3y + 5)^ and (2y + 15)^. Step 2: Set up the equation. (3y + 5) + (2y + 15) = 180 Step 3: Combine like terms. 5y + 20 = 180 Step 4: Isolate the term with y. 5y = 180 - 20 5y = 160 Step 5: Solve for y. y = (160)/(5) y = 32 The value of y is 32.