Find the number of $X$ from start to end
Solution 1:
If you want to use pure subtraction, here's how you'd do it:
At the beginning, you were at the start of problem 45.
At the end, you were at the end of problem 55, which is the start of problem 56.
And $56-45=11$. You finished $11$ problems.
The reason for the offset is because you're measuring from different places from each problem - the start of 45, but the end of 46.
Solution 2:
Suggestion: if you'd done questions $1$ to $N$, you'd have done $N$ questions. So if you start at question $44$ and finish at question $55$, subtract $43$ from both $44$ and $55$ to reduce to the easy case where the question numbers begin with $1$.
Solution 3:
You'll always need to add one in such cases. Consider - if you do problems 45 through 45, you'll have done 45-45 + 1 = 1 questions.
Solution 4:
It depends on whether the range includes or not its endpoints. For example, let $n \in \mathbb{N}$. Then:
- $45 \lt n \lt 55\quad$ has $\;55-45-1=9$ solutions;
- $45 \le n \lt 55\;$ and $\;45 \lt n \le 55\quad$ both have has $\;55-45=10$ solutions;
- $45 \le n \le 55\quad$ has $\;55-45+1=11$ solutions.
[ EDIT ] In the original post,
the first question I do is question 45 and the last question I do is question 55 imply that the range is inclusive of both endpoints $45,55$ which falls in the latter case among the above, so the correct answer is $55-45+1=11$.
Solution 5:
If you started at the question 15 and finished at the question 15; how many question have you answered?
Imagine a list of exercises to be done in order. Some of them are marked as done already. You start from the first unmarked question.
Every time you complete an excercise you mark it, thus increasing the number of the first excercise waiting to be done.
This way the number of excercises done in some time is an incremet of a number of the first excercise not done yet.
Since you started with question 45 (so 45 was the first question NOT done then) and you stopped after question 55 (so the first question NOT done yet is $56 = 55+1$ now), you have answered $(55+1)-45=11$ questions.
Put it another way:
when you started your work, the last question answered was number $44 = 45-1$, and now it is $55$; the number of questions you answered today is an increment of the number of questions answered: $55 - (45-1) = 11$.