Why 1/3 + 1/4 = 7/12?
Divide evenly seven (equal) apples between twelve boys subject to two restrictions:
- The only allowed operation is to cut an apple into a number of equal pieces.
- The number of pieces an apple may be cut into cannot exceed 6.
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 all have equal amount of the fruit, each received exactly 7/12 of an apple. We may conclude that
1/3 + 1/4 = 7/12.
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:
4/7 + 2/5 = 34/35,
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 fifths giving each kid 2/5ths.)
Regretfully, such an approach is not universal. Not every positive fraction can be represented as a sum of fractions with smaller denominators. For example,
1/3 = 1/12 + 1/4
which would correspond to the request to divide 1 apple between 3 boys which can be easily accomplished without dividing the apples into 12 or 4 pieces. So that examples must be thought up front before being handed to the students.
Let's call this approach to the fraction addition "organic". What does organic approach have to do with the common fashion of adding fractions? Say, to calculate
1/3 + 1/4 = 4/12 + 3/12 = 7/12.
Fractions 4/12 and 3/12 correspond to the two steps in the organic procedure. 4/12 signifies the division of 4 apples into 3 pieces each and getting 12 one thirds of an apple. 3/12 signifies the division of 3 apples into quarters and getting 12 quarter apples. Thus the process of organic addition reveals the hidden meaning of the operation of finding the common denominator of two fractions.
(There is a later follow-up, with nice illustrations.)
Here are additional pages related to the definitions, properties of and operations on, fractions:[an error occurred while processing this directive]