Probs of the Week v. 3.0

Yep. It's back.

1. Based on the following facts:
ab=1
b+c=0
b+c+d=0
bc=-9
c+a<d
What is a+b?

2. The first term of a geometric sequence is 12. If the geometric mean of the first 3 terms is 18, what's the third?

3. If three competing products have loyalties of 20%, 60%, and 80%, and half of the customers that switch go to one of the other products, then how many customers will go to each after they begin marketing regularly?

4. How many distinct integers are in the set of differences between numbers of the set, {1,3,5,7,9,10, 11,13}?

5. How much wood could a woodchuck chuck if a woodchuck could chuck wood?

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1. Let's gather our info:
a) a is the reciprocal of b, and the negative reciprocal of c
b) b is equal to negative c, and their product is -9
c) d is equal to 0, and is greater than the sum of a and c.
We can draw the conclusion that:
d) b and c are equal to 3 and -3, in some order
e) a is equal to either -1/3 or 1/3
f) a+c=-3+(1/3)<0
g) a=1/3, b=3, c=-3
h) a+b=3+(1/3)=10/3

2. By definition, the geometric mean is the second term. The ratio of 18 to 12 is 3/2; 3/2 times 18 is 27.

3. The matrix is...
[ 0.2  0.4  0.4]
[ 0.2  0.6  0.2]
[ 0.1  0.1  0.8]
To the power of 30 it reaches
[ 0.1429  0.2857  0.5714]
[ 0.1429  0.2857  0.5714]
[ 0.1429  0.2857  0.5714]

4. Let's remove the 10 for now. The differences between the numbers are the even integers from 2 to 12-- that makes 6. Now, the 10 is different- the difference between it and the other numbers is always odd, from one to nine. Six plus five, if you couldn't figure out, is 11.

5. Enough to spray splinters into your tongue... who really cares? According to a website I found, the answer is 22 cubic inches. That's more of a biology problem anyway.

So YEAH!

Bye sweet children!
John

PS: We're learning about conditionals and logic problems in class today, and it's taking absurdly long for some of my friends to figure it out. Oh, sweet rationale... where have you run off to now?