1/3 Plus What Equals 1

Why 1/3 + 1/4 = seven/12?

Dissever evenly vii (equal) apples between twelve boys bailiwick to two restrictions:

The merely 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 half-dozen.

Equally the championship in a higher place suggests, fractions ane/3 and 1/4 may be expected to be pertinent to the solution. They are.

First split four apples into three equal parts each and manus each boy a 3rd of an apple.

Side by side dissever the remaining three apples into four equal parts each and hand each boy one quarter of an apple.

At this fourth dimension each of the boys has one/iii of an apple and i/iv of an apple. Since all have equal amount of the fruit, each received exactly seven/12 of an apple tree. We may conclude that

ane/three + 1/iv = 7/12.

Detect at present that that result has been obtained without using whatever property, non even a definition, of the add-on of the fractions. Information technology may thus serve as a motivation for such a definition.

There is great latitude in selecting examples. Here is another one:

four/7 + 2/5 = 34/35,

which corresponds to the problem of dividing evenly 34 apples among 35 kids, with apples existence divided into, say, at near 10 pieces. (20 applies are divided into vii parts, giving every kid four/seven^th. fourteen apples are divided into the fifths giving each kid two/5^ths.)

Regretfully, such an approach is not universal. Not every positive fraction tin can be represented as a sum of fractions with smaller denominators. For example,

1/three = 1/12 + one/iv

which would stand for to the request to divide ane apple between iii boys which tin be easily accomplished without dividing the apples into 12 or 4 pieces. So that examples must be thought up front earlier being handed to the students.

Let's call this arroyo to the fraction addition "organic". What does organic approach have to do with the common style of calculation fractions? Say, to calculate one/3 + ane/4, nosotros first find the common denominator which is 12, and next rewrite the fractions with their common denominator: one/three = four/12, 1/4 = iii/12. Afterwards which addition is easy:

1/3 + 1/4 = 4/12 + iii/12 = seven/12.

Fractions 4/12 and 3/12 correspond to the two steps in the organic process. 4/12 signifies the sectionalisation of iv apples into three pieces each and getting 12 ane thirds of an apple. three/12 signifies the segmentation of three apples into quarters and getting 12 quarter apples. Thus the procedure of organic addition reveals the hidden meaning of the operation of finding the common denominator of two fractions.

(There is a after follow-up, with nice illustrations.)

Hither are additional pages related to the definitions, properties of and operations on, fractions: