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    • Most math puzzles are a bit of magician’s distraction, “Look over there,” so you avert your eyes from what would lead you to the solution.

      In this puzzle, the distraction is an equation you often don’t see: multiple equalities. A more straightforward math problem would have been

      Calculate a-b if

      a+b = ab

      Alternatively, they could have provided one of the two other equalities that are true:

      a+b = a/b
      ab = a/b

      Instead, they provide you with

      a+b = ab = a/b

      Which means you have to get over this strange monstrosity that you’ve probably never had to deal with in high school Algebra.

      Then, if you do realize that you can set any two terms equal to each other to solve for a-b, you have to decide which of the three combinations to use.

      So let’s decide which combination is most useful.

      a+b = ab = a/b

      I need to find values for a and b such that the above equation is true.

      For example, b cannot equal 0 because a/b would be division by zero, which is undefined.

      In deciding on the combination, I go with whatever is easiest to work with. If addition is on one side of the equal sign, I want addition or subtraction on the other side. If there’s multiplication on one side of the equal sign than I want multiplication or division on the other.

      So the easiest combination to work with is

      ab = a/b

      ab = a/b
      b * ab = b * a/b

      b*b * a = a * b/b

      b^2 * a = a * 1

      b^2 * a = a

      b^2 * a/a =a/a

      b^2 * 1 = 1

      b^2 = 1

      sqrt( b^2 ) = sqrt( 1 )

      b = +/- 1

      Check
      b = +1

      a+b = ab
      a + 1 = a * 1
      a + 1 = a
      a-a + 1 = a-a
      0 + 1 = 0
      1 = 0

      Check
      b = -1

      a+b = ab
      a + -1 = a * -1
      a + -1 = -a
      a-a + -1 = -a-a
      0 + -1 = -2a
      -1 = -2a
      -1/-2 = -2a/-2
      1/2 = a

      a= 1/2
      b = -1

      a+b = ab = a/b

      1/2 + -1 = 1/2 * -1 = (1/2) / -1

      -1/2 = -1/2 = -1/2

      a - b
      = 1/2 - (-1)
      = 1/2 + 1
      = 3/2


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      Tagging @amacbean16