Question 567915
This problem is mainly an exercise in interpreting the relations among three consecutive odd numbers as they are written in the problem.
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Begin with an understanding of what the relationships among three consecutive odd numbers are. An example is 5, 7, and 9. They are consecutive (no odd number is skipped in the series) and note that they each are separated from the preceding odd number by 2 to get to the next consecutive odd number. Therefore, if we let 5 be represented as n, the next consecutive odd number is represented by n+2 ... that will give you the 7. And the next consecutive odd number is (n+2)+2 or n+4. (That gives you 5+4 = 9, the third consecutive odd number.
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You can use this same pattern of n, n+2, and n+4 to represent the three consecutive odd numbers you are to find in this problem.
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The problem starts out by saying, "If 8 is subtracted from the third of 3 consecutive odd integers ..." So take the third of the three consecutive odd integers, which is n+4, and subtract 8 from that, meaning n+4-8, and the result is n-4. 
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The problem goes on to say, "...and the result is multiplied by 2..." So take the result of n-4 and multiply it by 2. That is, 2*(n-4), and this answer is 2n-8.
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Regarding this answer the problem says that, "...the answer is 33 less than the sum of the first and twice the second of the integers."
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So find the sum of the first, which is n, and twice the second, which is 2*(n+2). This sum is given by:
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n+2*(n+2) 
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First do the distributed multiplication by multiplying 2 times each of the terms in the parentheses. When you do that the entire expression becomes:
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n+2n+4
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and combining the two terms that contain n simplifies the expression to
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3n+4
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This equals the sum of the first term and twice the second of the integers. So return to the problem statement and replace the words "the sum of the first term and twice the second of the integers" with 3n+4 to get:
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"...the answer is 33 less than 3n+4."
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Recall that "the answer" was what was obtained earlier ... namely 2n-8. So substitute 2n-8 for "the answer" as follows:
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2n-8 is 33 less than 3n+4
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This means that by subtracting 33 from 3n+4 the result should be equal to 2n-8. In equation form this is:
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2n-8=3n+4-33
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On the right side combine the +4 and -33 to get:
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2n-8 = 3n-29
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Get rid of the -8 on the left side by adding +8 to both sides and you have:
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2n = 3n-21
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Get rid of the 3n on the right side by subtracting 3n from both sides:
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-n = -21
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Solve for +n by multiplying both sides by -1 and the answer becomes:
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n = 21
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Since the answer is to be three consecutive odd numbers and since 21 is the first of these, the three consecutive odd numbers must be 21,23, and 25. And this is the answer to the problem. 
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Check it as follows:
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1. Subtract 8 from the third of these numbers (25 -8 = 17)
2. Multiply the 17 by 2 to get 34.
3. Now add the first (21) to twice the second (2 times 23) and you get 21 + 46 which equals 67.
4. Then determine if the 34 from step 2 is 33 less than the 67 from step 3. It is, so the answers of 21,23, and 25 work and the solution is correct.
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Hope this helps you understand the problem and demonstrates to you how you have to pick these problems apart to solve them.
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