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I am trying to solve the problem :

A single fence is to be constructed from posts 6 inches wide and separated by lengths of chain 5 feet . If a certain fence begins and ends with a post.Which of following could be length of fence in feet ? a)17 b)28 c)35 d)39 e)50. (Ans:a,b,d and e)

Since the total space from one fence to another would be 6ft. I believe the fence size should be a multiple of 6 .None of the nos here are multiple of 6. How did they get the answer above ?

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    I don't see what the title has to do with the question.2012-07-25
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    I am open to suggestions for the title. In case it might help others.2012-07-25

2 Answers 2

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Let $n$ be the number of posts. Then there are $n-1$ lengths of chain, since there has to be a post at both ends. Because the post is $\frac{1}{2}$ of a foot, the total length of the fence, in feet, is $$\frac{1}{2}n +5(n-1).$$ This is $5.5n -5$. So we want to check which of our numbers is of the shape $5.5n-5$, for some integer $n$.

For example, can we have $5.5n-5=17$? Sure, $22$ is an integer multiple of $5.5$.

In general, we are interested in whether, for given $k$, there is an integer $n$ such that $5.5n-5=k$, or equivalently $11n=2k+10$. So what we care about is whether $11$ divides $2k+10$.

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    Interesting. How did you get that equationv - I mean is that from probability ?2012-07-25
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    I (mentally) drew a fence, with post. Each of the $n$ posts takes up half a foot. Each of the $n-1$ lengths of chain takes up $5$ feet. So total length is $(0.5)(n)+(5)(n-1)$. Just geometry, or carpentry, not probability.2012-07-25
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    Yes that makes sense - I was just going over the edit. And the second equation $11n=2k + 5$ . I assume means 2 posts and one chain ? right ? so how did u get 11 ?2012-07-25
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    Our equation simplifies for $5.5n-5=k$. Multiply by $2$ to get integers. We get $11n-10=2k$. Earlier I had (mistake) forgotten to multiply the $5$ by $2$.2012-07-25
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    Great thank you that makes sense. Plus @Branden post made me realize why I was wrong2012-07-25
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The inital length is 6ft, but every time you add a post after that it increases by 5.5ft |=post --=chain |--| is 6ft but |--|--| is 11.5ft, which isn't divisible by 6 obviously. Adding another post, makes it 17ft. |--|--|--|

Knowing this, you can make a model. Keep in mind the number of posts will always be 1 more than the number of chains. x=#posts

x(.5) + (x-1)(5) = length

.5 is length of post, 5 is length of chain. Subbing in for the available lengths, and solving for x, will give you the number of posts each length requires, if it is a whole number it works, if x comes out as a fraction, then it is not a valid fence.

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    Thanks I was curious why my answer was wrong.2012-07-25