Question

A board is 94 cm in length and must be cut so that one piece is 12 cm longer than the other piece. Find the length of each piece. Smaller piece, and longer piece.

Answer 1

The smaller piece will be half the difference shorter than the average length, so will be (94/2) -(12/2) cm = 41 cm.

Smaller piece: 41 cm
Longer piece: 53 cm

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There are several ways you can write equations for this.
Let s represent the length of the shorter piece. Then the longer one is (94-s), and the difference is
  (94-s) - s = 12
  94 - 2s = 12
  94 - 12 = 2s
  (94/2) - (12/2) = s = 41

Another way you can do it is to let u represent the length of the longer piece. Then you have
  u + s = 94
  u - s = 12
Adding the two equations gives you
  2u = 94 + 12
  u = (94 + 12)/2 = 53
Subtracting the second equation from the first gives you
  (u+s) - (u-s) = 94 - 12
  2s = 94 - 12
  s = (94 - 12)/2 = 41

Please note that this solution generalizes to other "sum and difference" problems. That's what I call a problem where you are given the sum of two numbers, along with their difference. One answer (usually, the largest) is half the sum of the given numbers. The other answer is half the difference of the given numbers, where the difference value is subtracted from the sum value.