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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Selepe, let's tackle these math problems! Question 3 3.1 Calculate: 8(1)/(3) × (3)/(8) Step 1: Convert the mixed number to an improper fraction. 8(1)/(3) = ((8 × 3) + 1)/(3) = (24 + 1)/(3) = (25)/(3) Step 2: Multiply the fractions. (25)/(3) × (3)/(8) = (25 × 3)/(3 × 8) Step 3: Simplify the expression by canceling out common factors. 25 × 33 × 8 = (25)/(8) Step 4: Convert the improper fraction to a mixed number (optional, but good practice). (25)/(8) = 3(1)/(8) (25)/(8) or 3(1)/(8) 3.2 A school had 450 learners in January. By March, the number increased to 540 learners. In June, the numbers increased by 10% from March number. What is the number of learners in June? Step 1: Identify the number of learners in March. Number of learners in March = 540 Step 2: Calculate the 10% increase from the March number. Increase = 10\% of 540 = (10)/(100) × 540 Increase = 0.10 × 540 = 54 Step 3: Add the increase to the March number to find the number of learners in June. Learners in June = Learners in March + Increase Learners in June = 540 + 54 = 594 594 learners 3.3 If 30% of a number is 45. What is 80% of that number? Step 1: Find the original number. Let the number be x. 30\% of x = 45 (30)/(100) × x = 45 0.3x = 45 x = (45)/(0.3) x = 150 Step 2: Calculate 80% of the original number. 80\% of 150 = (80)/(100) × 150 80\% of 150 = 0.8 × 150 80\% of 150 = 120 120 Question 4 4.1 Calculate: 1,250 × 16 Step 1: Perform the multiplication. 1,250 × 16 = 20,000 20,000 4.2 Divide a ribbon of 9,864m into 4 equal pieces. Step 1: Perform the division. (9,864m)/(4) = 2,466m 2,466m 4.3 A hawker buys 3750 g of apples at R12.40 per kg and 2.5kg of banana at R15.60 per kg. How much does the hawker pay all together? Step 1: Convert the mass of apples from grams to kilograms. 3750 g = (3750)/(1000) kg = 3.75 kg Step 2: Calculate the cost of the apples. Cost of apples = 3.75 kg × R12.40/kg Cost of apples = R46.50 Step 3: Calculate the cost of the bananas. Cost of bananas = 2.5 kg × R15.60/kg Cost of bananas = R39.00 Step 4: Calculate the total amount the hawker pays. Total cost = Cost of apples + Cost of bananas Total cost = R46.50 + R39.00 Total cost = R85.50 R85.50 Question 5 5.1 Use the smallest base to write 81 in exponential form. Step 1: Find the prime factors of 81. 81 = 3 × 27 27 = 3 × 9 9 = 3 × 3 So, 81 = 3 × 3 × 3 × 3. Step 2: Express 81 as a power of its smallest prime base. 81 = 3^4 3^4 5.2 Calculate: [3]27 + 3^2 Step 1: Calculate the cube root of 27. [3]27 = 3 (since 3 × 3 × 3 = 27) Step 2: Calculate the square of 3. 3^2 = 3 × 3 = 9 Step 3: Add the results. 3 + 9 = 12 12 Question 6: INTEGERS 6.1 Calculate: -4 + (-6) - 15 + (-5 + 13) Step 1: Simplify the expression inside the parentheses first. -5 + 13 = 8 Step 2: Rewrite the expression with the simplified term. -4 + (-6) - 15 + 8 -4 - 6 - 15 + 8 Step 3: Perform additions and subtractions from left to right. (-4 - 6) - 15 + 8 -10 - 15 + 8 (-10 - 15) + 8 -25 + 8 -17 -17 6.2 Complete: 6.2.1 -23 + 17 = + (-23) Step 1: Calculate the left side of the equation. -23 + 17 = -6 Step 2: Use the commutative property of addition (a + b = b + a) to find the missing number. -6 = + (-23) The missing number is 17. -23 + 17 = 17 + (-23) 17 6.2.2 (2 + 5) + = + (5 + (-3)) Step 1: Simplify the known parts of the equation. (2 + 5) = 7 5 + (-3) = 5 - 3 = 2 Step 2: Rewrite the equation with the simplified parts. 7 + = + 2 Step 3: Use the associative property of addition ((a + b) + c = a + (b + c)). If we let the missing number on the left be c and the missing number on the right be a, then: (2 + 5) + (-3) = 2 + (5 + (-3)) 7 + (-3) = 2 + 2 4 = 4 So, the missing number on the left is -3, and the missing number on the right is 2. (2 + 5) + (-3) = 2 + (5 + (-3)) 3 done, 2 left today. You're making progress.