Rob and Paula work in a dance school. Rob starts a new lesson every 40 minutes. Paula starts a new lesson every 50 minutes. At 11 am, both Rob and Paula start a new lesson. Part a) At which of these times could Rob’s first lesson have started? Circle your answer. Is it 8:10 am, 8:20 am, 8:30 am, or 8:40 am.
There is also a part b and c to this question that we will look at later. We’re told in the question that Rob starts a new lesson every 40 minutes. And one of these lessons started at 11 am. In order to work out when Rob’s first lesson may have started, we need to count backwards in 40-minute jumps from 11 am. There are 60 minutes in an hour. Therefore, 40 minutes before 11 am is 10:20 am. 40 minutes before this is 9:40 am. Going back 20 minutes takes us to 10 o’clock. And another 20 minutes takes us to 9:40. 40 minutes prior to this is nine o’clock or nine am. As this is still not one of our options, we need to go back another 40 minutes. 40 minutes before nine am is 8:20 am. This means that Rob’s first lesson could have started at 8:20 am. We need to circle this answer as the only one of the four options that Rob’s first lesson could have started is 8:20 am.
The second part of the question says the following. b) Neither Rob nor Paula takes a lunch break. When is the next time that Rob and Paula will start a new lesson at the same time?
We know that both Rob and Paula start a new lesson at 11 am. Rob starts a new lesson every 40 minutes and Paula starts a new lesson every 50 minutes. As neither of them take a lunch break, we know that Rob’s lessons start at 11 am, 11:40, 12:20, 13:00 hours or one pm, 13:40, 14:20, 15:00 hours or three pm, and so on. We have worked out these times by adding 40 minutes each time. Paula will start her lessons at 11 o’clock, 11:50, 12:40, 13:30, 14:20, and so on. To work out Paula’s times, we have added 50 minutes to the previous time.
We can see that 14:20 is in both lines. This means that both Rob and Paula will start a lesson at 14:20. The next time after 11 am that Rob and Paula will start a new lesson at the same time is 14:20.
An alternative method to save writing out all the times would be to find the lowest common multiple of 40 and 50. This is equal to 200 as 200 is the lowest number that is a multiple of both 40 and 50. The lowest common multiple tells us that Rob and Paula will start a new lesson at the same time, 200 minutes after 11 am. 200 minutes is the same as three hours and 20 minutes. There is 60 minutes in an hour. Therefore, three hours is equal to 180 minutes. 200 minus 180 gives us 20 minutes. We need to work out the time that is three hours and 20 minutes after 11 am. This is equal to 14:20 or 2:20 pm, which is the same answer as we got using the first method.
The final part of the question says the following. c) A third teacher starts a new lesson every hour. Her first lesson starts at 8:15 am. Will she ever start a lesson at the same time as Rob or Paula? Give a reason for your answer.
As this third teacher starts a lesson every hour, she will start a lesson at 8:15, 9:15, 10:15, 11:15, 12:15, and so on. Every lesson will start at 15 minutes past the hour. We know that every one of Rob and Paula’s lessons start at a time ending in zero. This means that the correct answer is no. The third teacher will never start a lesson at the same time as Rob or Paula as both Rob’s and Paula’s lessons always start at a time ending in zero, whereas the third teacher will always start their lessons at quarter past or 15 minutes past the hour.