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Multiplying numbers from One set with another set

  1. Dec 24, 2014 #1
    1. The problem statement, all variables and given/known data

    Bob makes two sets: one with all the even integers between 1 and 30 inclusive, and another with all the odd integers inclusive. He called the sets Q and R. He multiplied each number from Q with each number in R. Then he added the 225 products together and called the result "S". What number does S represent?

    (A) 8,000 (B) 18,000 (C) 36,000 (D) 54,000 (E) 86,000

    2. Relevant equations


    3. The attempt at a solution

    Set Q = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}
    Set R = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}

    I noticed if I took 1 from Q and multiplied with every number in R, I get multiples of 2 from 2 to 30
    Likewise, if I took 3 from Q and multiplied with every number in R, I get multiples of 6 from 6 to 90 and so on.

    # from Q Range Sum (Added first term and last term and multiplied by 7 and then added the middle #)
    1 2 to 30 32 * 7 + 16 = 240
    3 6 to 90 96 * 7 + 48 = 720
    5 10 to 150 160 * 7 + 80 = 1200
    7 14 to 210 224 * 7 + 112 = 1680
    9 18 to 270 288 * 7 + 144 = 2160

    I stopped here because I noticed the sums were odd multiples of 240.

    So then I found the first term which is 240 and the last term which would be 240 * 29 = 6960. The middle # here would be 3600.

    6960 + 240 = 7200. There are 7 pairs of 7200 and the left out middle number 3600, thus the sum would be 54,000.

    I was wondering if there is a shorter or easier or alternative method of solving this problem.
     
  2. jcsd
  3. Dec 24, 2014 #2

    LCKurtz

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    Science Advisor
    Homework Helper
    Gold Member

    Yes. Multiply the sum of the odds by the sum of the evens. 225x240 = 54000.
     
  4. Dec 24, 2014 #3

    mfb

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    2016 Award

    Staff: Mentor

    And there is a formula you can use for those sums to make it even faster.
     
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