Pre-algebra word problems | Worked example

- (Instructor) At a mock trial competition, the organizer divides 100 reference books equally among participating teams so as few books as possible are left over. If there are 4 books left over, which of the following could be the number of teams at the competition? So pause this video and try to work through it yourself, before we do it together. You will only have to pick one of these choices. Okay. Now, let's do it together. So what's happening is there's 100 books. They divide 'em equally amongst the various groups. So for example: if there was only 2 groups right over here, you could give 50 to one, 50 to the other. And, then you would have no left over. But if, for example, you had three groups. If you had three groups, and this is just an example. If you wanted to divide it evenly, you could give 33 there, you could give 33 there, you could give 33 there. But then you would have one left over. Cause this would be 99 and you would have one that you wouldn't be able to give any of the teams. Cause you have to equally divide them. Now they're telling us a situation where there are four books left over. So if there are four books left over, how many were given to the teams? Well, 100 minus four is equal to 96. So that means that 96 were equally distributed to the teams. That all of those could get distributed. And so, one way to think about it is, if that is true, then the number, the 96 has to be perfectly divisible by the number of teams. So look at the choices. So choice A here, says three teams. So first of all, is 96 divisible by three? Well, you could use a calculator if you like on the practice, or some of you might recognize that 96 divided by three, is exactly 32. And so you might say, "Hey! A looks like a good choice." Except for, an interesting situation. If you did indeed have three teams. So actually this is, this scenario over right here. How much would you have left over? Well, you would have one left over. Not four left over. Why is that? Because, if you had four left over, if you did this, if you had a hundred and you split it into three teams. And you were to just distribute these 96. So you would have 32, 32, and then 32, and then you have four left over. Remember, you're trying to give all the teams as many books as possible. Where it's equally divided, and so you minimize the amount that's left over. And so if you have four left over, with three teams, well, you can give everyone another book. So you could take three here. So you only have one left over. And go to 33 per team. So if you only had three teams because three is less than or equal to four, well, then you wouldn't have four left over. You could still give each team an extra book. So we rule that one out. And you can rule B out for the exact same reason. 96 is divisible by four. It would be, and you can do this on paper or use a calculator. But let's see, four goes into 80, 20 times and then four times four is 16. So, you could have four teams of 24. That would use, that have 24 books each. That would use up 96 books. But if you have four left over, well, then you would each give them another one. So you would actually wanna have four teams of 25. With no books left over. So we'll rule that one out. Now what about choice C? Well, 96 is not divisible by seven. You could use a calculator to verify that if you'd like. If you just say 96 divided by seven. Or it's not cleanly divisible by seven. Notice we get 13 point something here. So, I'll rule out our seven. Now is 96 divisible by eight? Well, sure. That's eight times 12. So you could have, eight teams that each get 12 books. And then you would have four left over. And so, this one looks great. Because 96 is perfectly divisible by eight. And eight is larger than the amount that you have left over. And then last, but not least is 96 divisible by 9? No! Nine times 10 is 90. Nine times 11 is 99. 96 divided by 9 would be, would be 10 point something or another. So we would rule that one out as well.