Geometry of Division of Fractions – Analysis and Examples

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. . . . A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

George POLYA

Welcome to the blog Math1089 – Mathematics for All.

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The division of fractions is one of the important concepts in school mathematics.  We all are familiar with the rule: to divide by a fraction, multiply by its reciprocal (so-called multiply by the reciprocal). Let us call this the known method.

But division actually means sharing equally into groups. For example, 18 ÷ 3 essentially means that how many groups of 3 can we find in 18? Since there are 6 groups, we write 18 ÷ 3 = 6. A geometric interpretation of this division is given below.

In the same way, we now turn our attention to the division of fractions and in this division, what the answer means. As before, when we divide a fraction we’re asking how many groups of the divisor (second member) can be found in the dividend (first member). To illustrate the concept geometrically, consider our first question.

Of course, the answer to this question 1 ÷ ½ is 1 × 2 = 2. Why the solution is a bigger number than the fractions involved? This is because we’re asking how many ½ appear in 1.

To answer this visually, consider a birthday cake. Now, how many half cakes are there in a full cake? Definitely 2, and that’s the answer. It’s the same as asking, how many parts will be there if each serving ½ of the whole?

Similarly, asking the question to find the value of 1 ÷ ¼ is same as asking how many parts will be there, if each serving ¼ of the whole? Certainly, the answer is 4, as there are 4 pieces of the cake of the said type.

Let’s have a look at the example ½ ÷ ⅙ = 3. Why the answer is a bigger number than the fractions involved?

In the above equation, we’re asking how many appear in ½. To answer this visually, consider a birthday cake. Suppose, half of the cake is remaining. If each serving of the cake is of the whole, how many servings do we have left? Clearly, it is 3.

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