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Why 1/3 + 1/4 = 7/12?Divide evenly seven (equal) apples between twelve boys subject to two restrictions:
As the title above suggests, fractions 1/3 and 1/4 may be expected to be pertinent to the solution. They are. 
First divide four apples into three equal parts each and hand each boy a third of an apple. Next divide the remaining three apples into four equal parts each and hand each boy one quarter of an apple. At this time each of the boys has 1/3 of an apple and 1/4 of an apple. Since each have equal amount of the fruit, each received exactly 7/12 of an apple. We may conclude that
Observe now that that result has been obtained without using any property, not even a definition, of the addition of the fractions. It may thus serve as a motivation for such a definition. There is great latitude in selecting examples. Here is another one:
which corresponds to the problem of dividing evenly 34 apples among 35 kids, with apples being divided into, say, at most 10 pieces. (20 applies are divided into 7 parts, giving every kid 4/7th. 14 apples are divided into the fifth giving each kid 2/5ths.) However, not every fraction is representable as a sum of two positive fractions. For example,
which would correspond to the request to divide 1 apple between 12 boys, so that examples must be thought up front before being handed to the students.
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