Addition of Fractions 2

Using Discrete Representations

Let us again take the problem of interpreting ½ + 1/3.

Let us look for a "whole" from which both these fractions can be easily taken. We can see that any "common multiple" of 2 & 3 will serve our purpose. That is we can take a collection of 6, 12, 18 .... etc as a convenient whole. For convenience sake let us take 6 (say 6 toffees) as the whole.

Then ½ of this whole would be 3. 1/3rdof this whole would be 2. Their sum would be 3 + 2 = 5. Hence, we can say that the total would be 5 toffees. In terms of fractions it would be 5/6thof the whole. Hence, we can say that ½ + 1/3 would be 5/6.

We can say 1/2 of a Whole + 1/3 of the same Whole = 5/6 of the Whole

Lowest Common Multiple

We have been able to solve the problem from first principles, without following any “rules”. We will summarize the process and also answer several questions that may arise in the mind of the learner.

Why did we choose a strip of 6 as the Whole?

6 is a number from which as we have seen above, both and can be taken. IF we had, for example taken a strip of 8 tablets as the whole, we would not be able to take of that whole. In other words, 6 is a multiple of both 2 & 3. 6 is a common multiple of 2 & 3.

Any common multiple would do

It is also clear that we could have solved the problem by taking strips of 12, 18, 24 or 600 tablets also. In fact any multiple of 6 will serve the purpose.

Lowest Common Multiple

We choose 6 since it is the smallest whole which will satisfy the condition. Using other bigger numbers will serve the purpose but increase the number computations. For example, if we had taken a strip of 18 tablets as the whole the answer would be which can be expressed as after a few arithmetic manipulations.

We also find that 6 is the LCM of 2 and 3.

A new interpretation of LCM

From this we get a new understanding of LCM as the smallest number from which both the given fractions can be taken without having to break the tablets.

Addition Procedure

With this the arithmetic procedure for doing the above problem also can be written as given below.

½ + 1/3 = ?

1. Find the LCM of the denominators of both the fractions

2. Find the equivalent fractions for both the fractions such that their denominators are equal to the LCM

3. Add the numerators

4. Write the sum of the fractions as another faction whose numerator would be equal to the sum of the numerators of the equivalent fraction and the denominator would be the LCM

5. Rewrite, if necessary, as an equivalent factions using the smallest possible numbers.

Summary

1. Fractions in rational number form can only be added if both are seen as parts of the same whole

2. The addition procedure is quite different from that applicable to whole numbers.

3. Discrete representation is easier while performing addition or subtraction.

If this is not understood and repeatedly practiced, students may use the addition procedure that they have learnt for whole numbers and get a wrong answer as shown below.

½ + 1/3 = 2/5 (by adding both the numerators & denominators separately)

Fraction Subtraction

The same procedure, suitable modified, is applicable also for Subtraction.