Counting Consecutive Multiples within a Range
Incorrect.
Make sure you are using the correct extreme values (the highest and lowest multiples of 7 within the range).
Correct!
How many integers are there between 115 and 120, inclusive?
Not quite.
Exactly!
The intuitive way to calculate this would be to subtract 115 from 120 and come up with 5, and that would, of course, be wrong. There are actually six integers between 115 and 120, including 115, 116, 117, 118, 119, and 120.
It is very easy to forget to include the extremes when confronted with this kind of seemingly easy question. In this case, simply subtracting 115 from 120 takes away the first extreme (115), which is subtracted along with all integers up to and including 114.
Remember that if both extremes should be counted, add one.
Now, if the question requires counting multiples of an integer within a range rather than just counting consecutive integers, a similar process applies. First, you need to subtract the extremes (the highest and lowest multiples of the integer that are within the range). Then, divide by the integer and add one at the end.
For example, how many multiples of 7 are there between 15 and 77, inclusive?
Incorrect.
Don't forget to add one!
To sum up:
[[summary]]
The process for answering this is to first find the extremes (i.e., the nearest multiples of 7 within the specified range). Since 15 is not a multiple of 7, the nearest multiple of 7 within the range is 21. On the other hand, since 77 is a multiple of 7, you leave it as the higher extreme of our range. Next, subtract the extremes and divide by 7. So you get the following.
$$\displaystyle (77 - 21) / 7 = 56 / 7 = 8$$
Then add one.
$$\displaystyle 8 + 1 = 9$$
5
6
8
9.86
9
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