Fraction Division

The interpretation of multiplication and division operations, using fractions are different from those applicable in addition and subtraction. This is another reason why students find it difficult to deal with fractions.

Hence the popular jingle in schools - Ours is not to reason why, just invert and multiply'.

Division

How do we interpret ÷  ?

To understand, let us start with a whole number example of interpreting 8 ÷ 2. It can be interpreted as an “equal sharing” or “equal grouping” situation. In a situation involving fractions, the “equal sharing” metaphor does not make sense. So we use the “equal grouping” metaphor.

We can interpret 8 ÷ 2 as how many (groups of) 2’s there are in 8, the answer being 4. Similarly ÷ can be interpreted as how many (groups of) 1/2s of a whole are there in 1/3 of the same whole? This question makes sense only if the wholes of both the fractions are same!

We can take any quantity as the whole as long as it makes sense. Let us take the whole as 6, since we can take both ½ and 1/3 from it. ½ of 6 is 3 and 1/3 of 6 is 2. Now the problem is equivalent 3/2.

Hence we can say ÷ = .

If we examine the problem (and several other similar ones) and examine the results, it appears that dividing by is equivalent to multiplying by its inverse as shown below.

÷ = X = as per the rule for multiplication.

That this works in all cases can be verified with many other examples.

What Does it Mean?

We have got a result which satisfies rules of arithmetic. But what does it mean?

The result is saying that “asking how many 1/2s of a whole are there in 1/3 of the same whole” is equivalent to asking “how many 2s are there in 3?”

This can be interpreted in the following manner. 3 is 2 + 1, which is one 2 and a half 2 (1 being half of 2), which is 1 and half 2 which is 1. This is the same as saying “there are 1 ½ “two” s in 3.

This is also the meaning of writing 3/2 as 1 ½.

Another Mathematical Interpretation

There is also a mathematical interpretation for this process. The problem can itself be thought of finding the value of a fraction whose numerator is and the denominator is . Since we have a standard way of dealing with such fractions, let us try to make the denominator into 1 in which case we need to consider only the fraction in the numerator.

The easiest way of converting the denominator into 1 is to multiply it by its inverse. Since it is a fraction, in order to keep its value same, the numerator also has to be multiplied by the same fraction. Let us see the various steps below.

÷ = = X = = X =