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    Fgrsdg Fgrsdg Presentation Transcript

    • Solution to Problem of the Week!
      • Let X be the original amount of coconuts
      • The amount of coconuts after on monkey has finished will equal b = (6/7)*(x-1) where b is the number of coconuts.
      • Use the same equation for the remaining monkeys
      • So the equation set would be
      • b = (6/7)*(x-1) = amount after 1 monkey,
      • c = (6/7)*(b-1) = amount after 2 monkeys,
      • d = (6/7)*(c-1) = amount after 3 monkeys,
      • e = (6/7)*(d-1) = amount after 4 monkeys,
      • f = (6/7)*(e-1) = amount after 5 monkeys,
      • g = (6/7)*(f-1) = amount after 6 monkeys,
      • h = (6/7)*(g-1) = amount after 7 monkeys,
      • i = (1/7)*(h-1) = amount each got at the end.
      • This works as the original number of coconuts must be able to be subtracted from 6/7 when one is subtracted from the original number and form an integer
      • Now the equations from the last slide need to be simplified
      • b = (6/7)*(x-1) Multiply each side by 7
      • 7B=6*(x-1) Distribute out the 6
      • 7B=6x-6 Put the six on one side of the equation and the variables on the other
      • 6=7B+6x
      • Repeat this process with the remaining equations; you get
      • 6a - 7b = 6,
      • 6b - 7c = 6,
      • 6c - 7d = 6,
      • 6d - 7e = 6,
      • 6e - 7f = 6,
      • 6f - 7g = 6,
      • 6g - 7h = 6,
      • h - 7i = 1
      • As you eliminate all the variables except X and I you will get
      • 6^1x - 7^1b = 6*(7^1 - 6^1)
      • 6^2x - 7^2c = 6*(7^2 - 6^2)
      • 6^3x - 7^3d = 6*(7^3 - 6^3)
      • 6^4x - 7^4e = 6*(7^4 - 6^4)
      • 6^5x - 7^5f = 6*(7^5 - 6^5)
      • 6^6x - 7^6g = 6*(7^6 - 6^6)
      • 6^7x - 7^7h = 6*(7^7 - 6^7)
      • 6^7x - 7^8i = 7^8 – 6^8
      • In the last problem X: Is going to be -6(match the 6^7X on the left side to the -6^8 on the right. You need a negative and one more six to make the two terms equal)
      • In the last problem;I is going to equal -1(again match the -7^8i on the left to the 7^8 on the right)
    • Solution
      • So to find the solution
      • X=-6 + 7^8*T
      • And
      • I=-1 + 6^7*T
      • T is any integer
      • Set T equal to one, as that is the lowest positive integer
      • X=-6+7^8(1)
      • X=5,764,795
      • I=-1+6^7(1)
      • I=279,935
    • Solution
      • So there was at least 5,764,795 coconuts to start with and in the end, and each monkeys received at least 279,935 coconuts in the morning.