Why is the number of integers between $2$ integers not their difference? [duplicate]
Helping my kids in 3rd grade Math class today. We were rounding to the nearest $10$. So for $100$, the numbers that round to it are $95$ to $104$. $95, 96, 97, 98, 99, 100, 101, 102, 103, 104$ which are 10 numbers I asked how many numbers round to $100$ one kid says 9, I say 10.
Why is $104-95=9$ not the correct way to figure out the number of numbers?
What is the correct equation, if there is one?
Solution 1:
This is an example of the so-called fence post problem. It goes like this:
Suppose you are building a fence with rails suspended between fence posts. The fence might look like this: $$|=|=|=|=|=|$$ where $|$ denotes a post, and $=$ denotes a rail. If each rail is 10 feet long, how many posts do you need to make a fence 50 feet long?
The answer is that you need 6 posts. Why not 5? Because you need a post on both ends of the fence, so you always have one more post than you have rails.
The same idea applies to your situation. You want to count the number of integers that round to 100. This is all the integers between 95 and 104 (inclusive). The distance from 95 to 104 is 104-95=9. This is like building a fence 9 feet long with 1-foot long rails. The integers are the posts. How many posts do you need? You need one more post than you have rails, so 10 posts. There are 10 integers between 95 and 104 (inclusive). The equation to use is $$\text{# of integers between $a$ and $b$ (inclusive)}=b-a+1.$$
Solution 2:
A difference $a-b$ measures the amount of gaps between the numbers $a$ and $b$, and there is always one gap less than numbers surrounding it.
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This is because you can associate every gap to the number from which it starts, but there must always be another number at the end. No gap starts there.
This means a number sequence starting from $a$ and ending in $b$ contains $a-b\color{red}{+1}$ numbers.
Making steps
Think about making steps. When you made five steps, you left six footprints because there is also one in the place where you started.
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Solution 3:
If you want to count all of the integers from $95$ to $104$, then you want to take the integers from $1$ to $104$, and then exclude those from $1$ to $94$. Thus $104-94=10$. If you subtract $95$, then you're excluding everything up to $95$.
This is called the fencepost problem. See this reference for a detailed treatment.