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: 4a. Solve 2(x + 4) 5x - 1 and graph the solution. Step 1: Distribute the 2 on the left side. 2x + 8 5x - 1 Step 2: Subtract 2x from both sides. 8 5x - 2x - 1 8 3x - 1 Step 3: Add 1 to both sides. 8 + 1 3x 9 3x Step 4: Divide by 3. (9)/(3) x 3 x This can also be written as x 3. Step 5: Graph the solution. The solution x 3 means all numbers less than or equal to 3. On a number line, this is represented by a closed circle at 3 and an arrow extending to the left. ` <------------------•---------------------> -1 0 1 2 3 4 5 <----- (closed circle at 3, arrow to the left) ` The solution is x 3. 4b. Change 11101_2 to base ten numeral. To convert a binary number to base ten, multiply each digit by the corresponding power of 2. 11101_2 = (1 × 2^4) + (1 × 2^3) + (1 × 2^2) + (0 × 2^1) + (1 × 2^0) = (1 × 16) + (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1) = 16 + 8 + 4 + 0 + 1 = 29 So, 11101_2 = 29_10. 5a. Three machines in a factory produce packages. Machine A every 12 minutes, Machine B every 18 minutes, and Machine C every 24 minutes. Find the time (in minutes) when all three machines will produce a package at the same time. This problem requires finding the Least Common Multiple (LCM) of 12, 18, and 24. Step 1: Find the prime factorization of each number. 12 = 2^2 × 3^1 18 = 2^1 × 3^2 24 = 2^3 × 3^1 Step 2: Identify the highest power of each prime factor present in any of the numbers. For prime factor 2: The highest power is 2^3 (from 24). For prime factor 3: The highest power is 3^2 (from 18). Step 3: Multiply these highest powers to find the LCM. LCM(12, 18, 24) = 2^3 × 3^2 = 8 × 9 = 72 All three machines will produce a package at the same time after 72 minutes. 5b. If an item is sold at a 20% profit for GH¢60, find the cost price. Let the cost price be C. The selling price (SP) is GH¢60. The profit percentage is 20%. The selling price is the cost price plus the profit. SP = Cost Price + Profit Profit = 20\% of Cost Price = 0.20C So, 60 = C + 0.20C 60 = 1.20C To find C, divide the selling price by 1.20. C = (60)/(1.20) C = (60)/(120)100 C = (60 × 100)/(120) C = (6000)/(120) C = 50 The cost price is GH¢50. 5c. Arrange the numbers -20, 15, -2, 0, 56, 4 in ascending order. Ascending order means arranging numbers from the smallest to the largest. The given numbers are: -20, 15, -2, 0, 56, 4. Step 1: Identify the negative numbers and arrange them. -20 is smaller than -2. So, -20, -2. Step 2: Place zero. 0 comes after the negative numbers. So, -20, -2, 0. Step 3: Identify the positive numbers and arrange them. The positive numbers are 15, 56, 4. Arranging them: 4, 15, 56. Step 4: Combine all numbers in ascending order. -20, -2, 0, 4, 15, 56