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 your questions: 2. WHOLE NUMBERS 2.1 Write 124 as a product of its prime factors. Step 1: Divide 124 by the smallest prime number, 2. 124 ÷ 2 = 62 Step 2: Divide 62 by 2. 62 ÷ 2 = 31 Step 3: 31 is a prime number. 124 = 2 × 2 × 31 124 = 2^2 × 31 The product of its prime factors is 2^2 × 31. 2.2 Find the HCF of 24 and 32. Step 1: List the factors of 24. F(24) = \1, 2, 3, 4, 6, 8, 12, 24\ Step 2: List the factors of 32. F(32) = \1, 2, 4, 8, 16, 32\ Step 3: Identify the common factors and the highest among them. Common factors are \1, 2, 4, 8\. The highest is 8. The HCF of 24 and 32 is 8. 2.3 Find the LCM of 12 and 18. Step 1: List the multiples of 12. M(12) = \12, 24, 36, 48, ...\ Step 2: List the multiples of 18. M(18) = \18, 36, 54, ...\ Step 3: Identify the lowest common multiple. The LCM of 12 and 18 is 36. 2.4 Divide R500 between Muliweli and Lloyd in the ratio of 6:4. Step 1: Find the total number of ratio parts. 6 + 4 = 10 parts Step 2: Determine the value of one ratio part. R50010 = R50 per part Step 3: Calculate Muliweli's share. 6 × R50 = R300 Step 4: Calculate Lloyd's share. 4 × R50 = R200 Muliweli gets R300 and Lloyd gets R200. 2.5 A packet of apples cost R22 excluding VAT. What will the price of the apples be once 14% VAT is added to the price? Step 1: Calculate the VAT amount. VAT = 14\% of R22 = 0.14 × R22 = R3.08 Step 2: Add the VAT to the original price. New Price = R22 + R3.08 = R25.08 The price of the apples with VAT will be R25.08. 3. INTEGERS 3.1 Calculate the following and show all the steps: -23 - (-18) Step 1: Simplify the double negative. -23 - (-18) = -23 + 18 Step 2: Perform the addition. -23 + 18 = -5 The result is -5. 3.2 2(8-3+4) Step 1: Perform the operations inside the parenthesis. 2(8-3+4) = 2(5+4) = 2(9) Step 2: Perform the multiplication. 2(9) = 18 The result is 18. 3.3 3^3 + 4^2 - 5^2 Step 1: Calculate the powers. 3^3 = 27 4^2 = 16 5^2 = 25 Step 2: Substitute the values back into the expression. 27 + 16 - 25 Step 3: Perform addition and subtraction from left to right. 43 - 25 = 18 The result is 18. 3.4 sqrt(9) + 16 + 7 Step 1: Calculate the square root. sqrt(9) = 3 Step 2: Perform the additions from left to right. 3 + 16 + 7 = 19 + 7 = 26 The result is 26. 3.5 (-6+3(-4)-3(4))/((2+3)(-3)) Step 1: Calculate the numerator. -6 + 3(-4) - 3(4) = -6 - 12 - 12 = -30 Step 2: Calculate the denominator. (2+3)(-3) = (5)(-3) = -15 Step 3: Divide the numerator by the denominator. (-30)/(-15) = 2 The result is 2. 4. COMMON FRACTIONS 4.1 Calculate: 10 ÷ (5)/(8) Step 1: Convert division to multiplication by the reciprocal of the fraction. 10 ÷ (5)/(8) = 10 × (8)/(5) Step 2: Multiply the numbers. = (10 × 8)/(5) = (80)/(5) Step 3: Simplify the fraction. = 16 The result is 16. 4.2 Simplify: sqrt((25)/(49)) Step 1: Take the square root of the numerator and the denominator separately. sqrt((25)/(49)) = sqrt(25)sqrt(49) Step 2: Calculate the square roots. = (5)/(7) The simplified expression is (5)/(7). 4.3 Simplify: 3/(27) + ((1)/(2))^2 Step 1: Calculate the cube root of the first term. 3/(27) = [3]8[3]27 = (2)/(3) Step 2: Calculate the square of the second term. ((1)/(2))^2 = (1^2)/(2^2) = (1)/(4) Step 3: Add the two fractions. Find a common denominator, which is 12. (2)/(3) + (1)/(4) = (2 × 4)/(3 × 4) + (1 × 3)/(4 × 3) = (8)/(12) + (3)/(12) = (8+3)/(12) = (11)/(12) The simplified expression is (11)/(12). 4.4 Determine the percentage decrease if a chocolate that cost R10 is marked down to R7. Step 1: Calculate the amount of decrease. Decrease = Original Price - New Price Decrease = R10 - R7 =