Two bells ring at intervals of 30 minutes and 40 minutes respectively. a) After how many minutes will they ring together? b) If they rang at 11:00 am, at what time will they ring together again?

Here are the solutions to problems 26 and 27. Problem 25 cannot be solved as the necessary diagram is not provided. Problem 26: Two bells ring at intervals of 30 minutes and 40 minutes respectively. a) After how many minutes will they ring together? Step 1: Find the Least Common Multiple (LCM) of 30 and 40. We can use prime factorization: 30 = 2 × 3 × 5 40 = 2 × 2 × 2 × 5 = 2^3 × 5 The LCM is found by taking the highest power of each prime factor present in either number: LCM(30, 40) = 2^3 × 3 × 5 = 8 × 3 × 5 = 120 The bells will ring together after 120 minutes. b) If they rang at 11:00 am, at what time will they ring together again? Step 1: Convert the LCM from minutes to hours. 120 minutes = (120)/(60) hours = 2 hours Step 2: Add the time interval to the given time. The bells rang at 11:00 am. They will ring together again after 2 hours. 11:00 am + 2 hours = 1:00 pm They will ring together again at 1:00 pm. Problem 27: A motorist left Kampala for Masaka at a speed of 48 km/hr at 8:30 am and spent 2(1)/(2) hr on the journey. a) At what time did the motorist reach Masaka? Step 1: Identify the departure time and travel time. Departure time = 8:30 am Travel time = 2(1)/(2) hours = 2 hours and 30 minutes Step 2: Add the travel time to the departure time. 8:30 am + 2 hours 30 minutes Adding 2 hours to 8:30 am gives 10:30 am. Adding 30 minutes to 10:30 am gives 11:00 am. The motorist reached Masaka at 11:00 am. b) Calculate the distance the motorist covered from Kampala to Masaka? Step 1: Identify the speed and time. Speed = 48 km/hr Time = 2(1)/(2) hours = 2.5 hours Step 2: Use the formula Distance = Speed × Time. Distance = 48 km/hr × 2.5 hr Distance = 48 × (5)/(2) km Distance = 24 × 5 km Distance = 120 km The distance covered is 120 km. 3 done, 2 left today. You're making progress.