شمارش فراگیرفصل: بخش ریاضی / درس: مسائل کلمه / درس 19
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So now we’ll talk about inclusive counting. First of all, consider this very easy question. On a warehouse shelf, there are 27 boxes, numbered from 1 to 27. John carried the first eight, so he carried boxes 1 through 8 and Mary carried the rest. How many boxes did Mary carry?
Well, obviously, what we would do is simply subtract, 27 minus 8 is 19. That would be the number of boxes that Mary carried. So that’s a really easy question. Now consider this superficially similar question. A workshop’s first day was April 8th, and it’s last day was April 27th. How many days did the workshop run?
We might be tempted to do the same subtraction, but that will not produce the correct answer. There’s a crucial difference between this problem and the previous box carrying problem. Now, we think about it. In the box-carrying problem, Mary carried box number 27, but box number 8 was not included among the boxes she carried.
Box number 8 was the last box the John carried. Ordinary subtraction works absolutely fine there. So in other words, one of the numbers was not included and one of the numbers was included. In the workshop problem, both April 8th and April 27th both endpoints are included as part of the workshop.
Ordinary subtraction is not equipped for this because the difference that results from ordinary subtraction automatically excludes the lower endpoint. Let’s think about this, if we subtract 27 minus 8 equals 19 that gives the whole group of days and in this group it excludes April 8th but includes April 9th and all the days up to and including April 27th. So we have the whole workshop except for the first day.
So all we have to do really is just add one more for this excluded day at the beginning. So the workshop was 27 minus 8 which is 19 plus 1 which is 20, it was 20 days long. This way of counting is known as inclusive counting. We always use inclusive counting when both endpoints, the starting value and the ending value, are included in what we are counting.
We perform the ordinary subtraction for this situation, then we add one to the difference to account for the included starting value. So think about this question for a minute. You can pause the video if you need to and then we’ll talk about it. Okay.
Contract negotiations opened on the morning of March 20th cons, continued every day without break and ended late in the evening of May 10th, or how many calendar days were the contract negotiations in session? Well, let’s see. First of all, in March, we’re talking about the 27th to the 31st that’s the end of March.
So how many days there, that would be 31 minus 20 plus 1 for the inclusive counting that’s 12 days. So 12 days in March, the whole month of April, so that’s 30 days, and then the first ten days in May. So all together, that’s 12 plus 30 plus 10, 52 days. So that’s how many days the contract negotiations were in session.
Here’s another one, pause the video and then we’ll talk about this. Okay, how many multiples of 8 are there from 200 to 640, inclusive? Well 8 goes into 200, 25 times, 8 goes into 640, 80 times. So it’s really like counting the integers from 25 to 80 and both of those are included. So we’re just gonna do 80 minus 25 plus 1 is 56, and that’s how many multiples of 8 there are,.
Starting at 200 and going all the way up to 640. We use inclusive counting whenever the situation demands that both endpoints, the lowest value and the highest value, are part of what we are counting. We perform the ordinary subtraction of high minus low, then add one for the included lower endpoint.
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