The Fence-Post Formula

Today I want to discuss a math concept that can be used to solve certain types of problems that appear every now and then on standardized tests. Tutors tend not to spend much time (if any) helping students prepare for this type of problem as it seldom appears. But if you want a perfect math score, then you want to be able to recognize all problem types and be able to answer them correctly (and quickly!).

This problem type involves what I like to call “Fence-Posting,” a technique that allows us to count the number of integers in a consecutive list.

Fence-post formula: The number of integers from a to b, inclusive, is

b – a + 1.

The word inclusive means that we include the extreme values.

For example, if we want to count the integers from 1 to 3, inclusive, we can easily see that there are 3 of them. Note that we do include the extreme values 1 and 3. Using the fence-post formula we get

3 – 1 + 1 = 3.

The most common error when attempting to count the integers in a list is to simply subtract the smallest value from the largest value (without adding 1). Note that in the previous example, this would give an incorrect answer of 2.

As another simple example, let’s count the number of integers from 5 to 12, inclusive, in two ways – directly and by fence-posting. First directly – the integers from 5 to 12 are 5, 6, 7, 8, 9, 10, 11, 12, and we see that there are 8 of them. Now using the fence-post formula we have

12 – 5 + 1 = 8.

Note that the computation 12 – 5 gives an incorrect answer of 7.

If you ever happen to forget this little formula test it out on a small list of numbers as I just did in the two examples above. But it’s nice to have this one committed to memory so that it is there for you when you need it.

Let’s take a look at a couple of examples where we can use fence-posting to obtain an answer efficiently.

Example 1

Set X contains only the integers 0 through 180 inclusive. If a number is selected at random from X, what is the probability that the number selected will be greater than the median of the numbers in X?

So first of all we need to find out how many integers there are in the set X. By the fence-post formula there are

180 – 0 + 1 = 181

integers in this set.

The median of the numbers in set X is 90 (note that in a set of consecutive integers the median is equal to the average of the first and last integer). Again, by the fence-post formula, we see that there are

180 – 91 + 1 = 90

integers greater than the median.

From this we see that the desired probability is

90/181 (approximately 0.4972375691).

So we grid in .497. (Note that this is a grid-in question because I have not provided answer choices).

Remember with grid in problems, we have only four slots available. We can simply truncate the final answer that we get in our calculator to fit in the four slots.

Example 2

How many numbers between 72 and 356 can be expressed as 5x + 3, where x is an integer?

This problem is more complicated than the first example, but we can use fence-posting to solve this one too: Let’s start by guessing x-values until we find the smallest and largest values of x satisfying

72 < 5x + 3 < 356.

Since we have 5(13) + 3 = 68 and also 5(14) + 3 = 73 we see that 14 is the smallest value of x satisfying the inequality.

Since 5(70) + 3 = 353 and 5(71) + 3 = 358, the largest value of x satisfying the inequality is 70.

It follows by fence-posting that the answer is 70 – 14 + 1 = 57.

Note that we also used the strategy of taking a guess to solve this problem. I discussed this strategy in a previous blog post. You can find that information by clicking on the link.

Algebraic solution:

72 < 5x + 3 < 356
69 < 5x < 353
69/5 < x < 353/5
13.8 < x < 70.6
14 ≤ x ≤ 70

We get the last inequality because x must be an integer. Notice that this last inequality is not strict. For those of you not familiar with this piece of math lingo, an inequality that is not strict means that it is ‘less than or equal to’ rather than just ‘less than’.

Finally, t follows by fence-posting that the answer is 70 – 14 + 1 = 57.

Tomorrow I will be showing a quick method for computing differences of large sums. You will also see how the fence-post formula can sometimes be useful in solving these types of problems. Click the following link to read this article: Differences of Large Sums

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