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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You're on a roll — here are the solutions to questions 11 and 12. 11. A relief organization donated 240 kg of maize and 150 kg of beans to needy families. Each family received exactly the same quantity by mass of either maize or beans. No family received both. Determine the least possible number of needy families. Step 1: To find the least possible number of needy families, each family must receive the largest possible quantity of food. This quantity must be a common divisor of 240 kg and 150 kg. Therefore, we need to find the Greatest Common Divisor (GCD) of 240 and 150. Prime factorization of 240: 240 = 24 × 10 = (2^3 × 3) × (2 × 5) = 2^4 × 3 × 5 Prime factorization of 150: 150 = 15 × 10 = (3 × 5) × (2 × 5) = 2 × 3 × 5^2 Step 2: Calculate the GCD by taking the lowest power of common prime factors. GCD(240, 150) = 2^1 × 3^1 × 5^1 = 2 × 3 × 5 = 30 kg So, each family receives 30 kg of food. Step 3: Calculate the number of families that received maize. Number of maize families = 240 kg30 kg/family = 8 families Step 4: Calculate the number of families that received beans. Number of bean families = 150 kg30 kg/family = 5 families Step 5: Determine the total least possible number of needy families. Total families = 8 + 5 = 13 families The least possible number of needy families is 13. 12. A worker completes a job in 12 hours while another takes 8 hours. How long will it take them to finish the job all together? Step 1: Determine the work rate for each worker. Worker 1's rate (R_1) = 1 job12 hours = (1)/(12) job/hour. Worker 2's rate (R_2) = 1 job8 hours = (1)/(8) job/hour. Step 2: Calculate their combined work rate when working together. R_total = R_1 + R_2 = (1)/(12) + (1)/(8) Step 3: Find a common denominator for the fractions, which is 24. R_total = (2)/(24) + (3)/(24) = (2+3)/(24) = (5)/(24) job/hour Step 4: The time taken to complete the job together is the reciprocal of the combined rate. T_total = (1)/(R_total) = (1)/(5)24 = (24)/(5) hours Step 5: Convert the fraction to a mixed number or decimal. T_total = 4 (4)/(5) hours To express this in hours and minutes: 4 (4)/(5) hours = 4 hours + ((4)/(5) × 60) minutes = 4 hours + 48 minutes It will take them 4 (4)/(5) hours (or 4 hours and 48 minutes) to finish the job all together. Send me the next one 📸