<p>I thought I would post the following question I received from a student since I think many students would like to see the solution:</p>
<p>I was just perusing through your advanced math strategies and was wondering if you would mind explaining the steps to this problem ?</p>
<p>How many numbers between 72 and 356 can be expressed as 5x+3, where x is an integer?</p>
<p>Lets guess x-values until we find the smallest and largest values of x satisfying the inequality 72 < 5x + 3 < 356. </p>
<p>Since we have 5(13) + 3 = 68 and 5(14) + 3 = 73 we see that 14 is the smallest value of x satisfying the inequality. </p>
<p>Since 5(70) + 3 = 353 and 5(71) + 3 = 358 we see that 70 is the largest value of x satisfying the inequality. </p>
<p>Therefore it follows that the answer is 70 14 + 1 = 57.</p>
<p>Remarks: (1) We used two strategies here: "taking a guess" and the "fence-post formula"</p>
<p>(2) We could have also found the two extreme x-values algebraically as follows.
72 < 5x + 3 < 356
69 < 5x < 353
69/5 < x < 353/5
13.8 < x < 70.6
14 ≤ x ≤ 70</p>
<p>We get the last inequality because x must be an integer. Notice that this last inequality is not strict.</p>
<p>If an integer assumes the form 72 < 5x + 3 < 356, then if follows that 69 < 5x < 351. So now we include every multiple of 5 in that domain. This is every 5 numbers starting at 70: 350-70 = 280. 280/5 = 56. Since the values are from 70 to 350 inclusive, add 1. The answer is 57.</p>
<p>I see why it’s called the “fence-post” formula, but I don’t like creating obscure names of theorems or counting techniques (plus it’s not really a “formula”). Just say the counting is inclusive.</p>
<p>I do not recommend memorizing the “fence-post” formula either. But I do recommend being AWARE of it. Without this awareness many students will get a “fence-post” type problem wrong. As long as you are aware that the number of integers from a to b, inclusive, is a bit tricky, you won’t be tricked. You can just check a small sample. For example, let’s count the number of integers from 3 to 7: </p>
<p>3, 4, 5, 6, 7</p>
<p>There are 5 of them. But 7 - 3 = 4, not 5. What’s going on? Well you have to add back 1. </p>
<p>7 - 3 is the number of spaces between fence posts, so the number of fence-posts is 7 - 3 + 1.</p>
<p>Having a conceptual understanding of the situation is always much better than memorizing the formula.</p>
<p>Yes, one should definitely should be aware of the inclusive-ness. I wouldn’t be at all surprised if 56 was an answer choice.</p>
<p>I might solve it by counting the number of elements in the set {73, 78, …, 353}. You can subtract 68 from every number and divide by 5, so the set becomes {1, 2, …, 57}, 57 elements. Either way works; most problems it’s just a matter of personal preference.</p>