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1) Mr. Smith and Mr. Jones have decided to separate their property by placing a fence along the property line. They ordered enough fence posts so that the fence would have
posts placed 8 feet apart. However, five of the posts were not usable. They were still able to put the fence up by placing the remaining posts 10 feet apart. How many fence posts
did they originally order?
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Let n = the number of original fence posts and
f = the total length of the fence.
This is a 2 equations with 2 unknowns problem.
- The equation for the original fence is:
f = (n - 1) x 8
You subtract 1 because there is one more fence post on the end.
- The equation for the second fence is:
f = (n - 6) x 10
- Substituting the first equation into the second:
(n - 1) x 8 = (n - 6) x 10
8n - 8 = 10n - 60
-2n = -60 + 8 = -52
n = 26
They originally ordered 26 posts.
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2) Kate creates a new sequence of numbers: 2, 3, 5, 25, 200 .... After the first three terms, 2, 3, and 5, each new term is found by adding the first two of three previous terms and then multiplying this sum by the final previous term. For example, the fourth terms is (2+3)(5) = 25. Now, Carrie uses Kate's technique but starts with new terms (a, 4, 6) to create a new sequence: a, 4, 6, (a+4)(6), ..... If the fifth term of Carrie's sequence is 300, what is the fourth terms of Carrie's sequence?
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Calling the first 3 terms a,b, and c, the first 5 terms of Kate's sequence are:
a b c (a+b) c (b + c) (a + b)c
So, with the new terms a, 4, 6, the 5th term is
(4+6) ((a+4)6) = 300
(10)(6a+24) = 300
60a + 240 = 300
60a = 60
a = 1
Therefore, the 4th term of Kate's sequence is
(a+4)6 = 5x6 = 30
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