The table below shows the weekly timetable for a sports tournament. We’ve got the time the game begins, the end of period one, the end of period two, when the game ends, and the time that the medals are awarded for each day of the week. So on Monday, the game begins at 8:45, period one ends at 9:19, period two ends at 10 o’clock, the game ends at 11:55, and no medals are awarded. Meanwhile, on Thursday, for example, the game begins at 9:01, end of period one is at 9:40, end of the period two is at 10:15, the game ends at 11:58, and the medals are awarded at 12:40. Part a) How long does the game take on Monday?
Well, if we imagine a timeline, we can see that the game on Monday begins at 8:45 and ends at 11:55. Now, we can see that the time interval between nine o’clock and 11 o’clock is a span of two hours, then between 11 and 11:55 is 55 minutes, and between 8:45 and nine o’clock is 15 minutes.
So if we add all those times together, well, let’s add the minutes together first: 15 minutes plus 55 minutes. Five and five is 10. So that’s zero units, carry one 10. And one and five is six plus the one that we just carried is 70. So that’s 70 minutes. And 70 minutes is an hour and 10 minutes.
So the 15 minutes and the 55 minutes add up to one hour and 10 minutes. And if we add in the two hours in the middle, that gives us a total time of three hours and 10 minutes.
But before we move on, there’s another way of looking at this and this might be a bit quicker. You might notice that we could go from 8:45 to 11:45; that’s a time difference of three hours. And then from 11:45 to 11:55 is just a further 10 minutes.
So we get the same answer either way. But I think you’ll agree that using this timeline to visualize what’s going on helps us to work out the times more easily.
Now, Gunter says: “All the games this week take longer than three hours.” Is Gunter correct? Give reasons for your answer.
So we’ve got to justify our answer. We can’t just say “yes” or “no.” We have to actually explain why it’s yes or why it’s no. So if Gunter is correct, then we’re gonna have to go through every day and prove that the total length of the game is longer than three hours. If he’s wrong, then we just need to give one example of one day, where the length of the game is less than three hours.
We already know from part a that on Monday the game is longer than three hours; it’s three hours and 10 minutes. On Tuesday, the game starts at 8:59, ends at 12:14. Well, if it started at nine and ended at 12, that would be three hours. But it starts before nine o’clock and ends after 12 o’clock. So yes, that is greater than three hours. We don’t need to know the exact time, but we have shown that it’s longer than three hours.
On Wednesday, it starts at 8:15 and finishes at 11:26. Well, if it went from 8:15 to 11:15, that would be three hours. But it finishes at 11:26, which is after 11:15. So yes, again, that is greater than three hours. But on Thursday, the game begins at 9:01 and ends at 11:58. If it began at nine and ended at 12, then that would be three hours. But in fact, it begins after nine and ends before 12.
So by beginning one minute after nine and finishing two minutes before 12, it’s three minutes short of being three hours. And three minutes short of three hours is two hours and 57 minutes. So we can say, “No, Gunter is wrong because Thursday’s game only lasts for two hours and 57 minutes.”
And remember when you’re disproving something, you only have to give one counterexample. As soon as we found one day on which the game is less than three hours, we could stop and just write down our answer. We didn’t even get to look at the length of Friday’s game because we didn’t need to.