How many integers are there from 6 to 10? Four, right? Wrong! There are actually five integers from 6 to 10. Count them and you will see: 6, 7, 8, 9, 10. It is easy to forget that you have to include extremes. In this case, both extremes (the numbers 6 and 10) must be counted.
You don't have to methodically count each term in a long consecutive pattern. Just remember that if both extremes should be counted, you need to take the difference of the last and first numbers and add 1 before you are done.
How many integers are there from 14 to 765, inclusive?
Just remember: for consecutive integers, the formula is (Last − First + 1). Thus: 765 − 14, plus 1, yields 752.
This works easily enough if you are dealing with consecutive integers. Sometimes, however, the question will ask about consecutive multiples. For example, “How many multiples of 4 …” or “How many even numbers …” are examples of sequences of consecutive multiples.
In this case, if you just subtract the largest number from the smallest and add 1, you will be overcounting. For example, “All of the even integers between 12 and 24” yields 12, 14, 16, 18, 20, 22, and 24. That is seven even integers. However, (Last − First + 1) would yield (24 − 12 + 1) = 13, which is too large. How do you amend this? Because the items in the list are going up by increments of 2 (counting only the even numbers), you need to divide (Last − First) by 2. Then, add the 1 before you are done:
(Last − First) ÷ Increment + 1 = (24 − 12) ÷ 2 + 1 = 6 + 1 = 7
Just remember: for consecutive multiples, the formula is (Last − First) ÷ Increment + 1. The bigger the increment, the smaller the result, because there is a larger gap between the numbers you are counting.
Sometimes, however, it is easier to list the terms of a consecutive pattern and count them, especially if the list is short or if one or both of the extremes are omitted.
How many multiples of 7 are there between 100 and 150?
Note that the first and last items in the sequences are omitted—they must be determined by you. Here it may be easiest to list the multiples: 105, 112, 119, 126, 133, 140, 147. Count the number of terms to get the answer: 7. Alternatively, you could note that 105 is the first number, 147 is the last number, and 7 is the increment, thus:
Number of terms = (Last − First) ÷ Increment + 1 = (147 − 105) ÷ 7 + 1 = 6 + 1 = 7
How many integers are there from 1,002 to 10,001?
How many multiples of 11 are there between 55 and 144, exclusive?