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Question
(1)The product of two consecutive positive odd number is 195. By constructing a quadratic equation and solving it, find the two numbers.
(2)A Student bought some packets of pens for #2160. If she had paid #24 less for each packet,she could have bought three more packets. How many packets did she buy?
(3)find the positive number n such that twice its square is equal to six times the number?
(4)find the number n such that when 1/3 of it is added to 8, the result is the same as when 1/2 of it is subtracted from 18.

Answer
Let n be a positive integer, then 2n+1 is an odd number, and 2n+3 is the next odd number.

"The product of two consecutive positive odd number is 195."
(2n+1)(2n+3) = 195
4n + 8n + 3 = 195
4n + 8n - 192 = 0
n + 2n - 48 = 0

Solve by factoring:
n + 2n - 48 = (n+8)(n-6)
n = -8, 6  However, n must be positive, so n = 6.

2n+1 = 13
2n+3 = 15
The two consecutive odd numbers are 13 and 15.
:::::
"A student bought some packets of pens for #2160."
If x is the cost of one packet, then 2160/x is the number of packets bought.

"If she had paid #24 less for each packet, she could have bought three more packets."
2160/(x-24) = 3 + 2160/x = (3x+2160)/x
720/(x-24) = (x+720)/x
720x = (x+720)(x-24) = x + 720x - 24x - 17280
x - 24x - 17280 = 0

Solve using the Quadratic Formula:
x = [24 √(24 + 4117280)]/(21)
 = [24 √69696]/2
 = [24 264]/2
 = -120, 144
Since the price cannot be negative,
x = 144
2160/x = 15
She bought 15 packets of pens.
:::::
"find the positive number n such that twice its square is equal to six times the number"
2n = 6n
n = 3n
n = 3

Check:
23 = 18
63 = 18
:::::
"find the number n such that when 1/3 of it is added to 8, the result is the same as when 1/2 of it is subtracted from 18."
n/3 + 8 = 18 - n/2

n/3 + n/2 = 10
2n/6 + 3n/6 = 10
5n/6 = 10
n = 10(6/5) = 12

Check:
12/3 + 8 = 12
18 - 12/2 = 12  

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Janet Yang

Expertise

Word problems are my favorite type of math questions! I would not feel comfortable answering questions that require specialized knowledge (Physics, Statistics, etc.) because I have not studied these in depth.

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I tutor students (fifth through twelfth grades) and am a Top Contributor on Yahoo!Answers with over 24,000 math solutions.

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Co-author of An Outline of Scientific Writing: For Researchers With English as a Foreign Language.

Education/Credentials
I have a Bachelor's degree in Applied Mathematics from the University of California at Berkeley, and a Master of Business Administration degree from The Wharton School.

Past/Present Clients
George White Elementary School. Homework Help program at the Ridgewood Public Library, Ridgewood, NJ. Individual students.

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