6-3 proving that a quadrilateral is a parallelogram

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  24 or 30

Step-by-step explanation:

If the rectangle dimensions are x by y, then there are xy sandwiches in the rectangle and (x+2)(y+2)-xy sandwiches in the border. If those are the same numbers, we require ...

  xy = (x+2)(y+2)-xy

  2xy = xy +2x +2y +4

  xy = 2(x +y +2)

There are no integer solutions for x=1 or 2. For x=3, we have ...

  3y = 2(3 +y +2)

  3y = 2y +10

  y = 10 . . . . . . . . 3·10 = 30 sandwiches will make a 3×10 rectangle

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For x=4, we have ...

  4y = 2(4 +y +2)

  4y = 2y +12

  y = 6 . . . . . . . 4·6 = 24 sandwiches will make a 4×6 rectangle

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There are no other integer solutions.

Ken and Sarah could each have 24 or 30 sandwiches.