Fraction Multiplication

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In whole numbers, operation procedures of multiplication &amp; division are closely related to that of addition &amp; subtraction. But in fractions the logic of multiplication and division operations are different from those applicable to addition and subtraction. This is another reason why students find it difficult to deal with fractions.

Multiplication

How do we interpret X? We can think of it as taking of a Whole and then taking  of the new quantity (of a Whole) and not the original Whole.

Hence in the second operation of taking 1/3rd, the whole is the new quantity and not the original whole from which we took &frac12;.

Let us take an actual example. Once again for convenience we take a discrete representation example.

A father buys a chocolate, which is divided into 6 parts as in a cadbury&rsquo;s chocolate. He gives &frac12; the chocolate to his son. So the son gets 3 parts. The son takes this chocolate (which has 3 parts) to school. He gives 1/3rdof what he has to his friend. Hence his friend gets 1 part.

The question now is what part of the original whole (chocolate which the father bought) has the friend got? The friend has got 1 part out of the original 6 parts and hence 1/6th.The whole process can be visualized as given below.

The Whole                                            of the Whole             of the new Whole       of the original whole

Change in Part to Whole

Please note that in the above transaction, what was a part in the first transaction, becomes the Whole for the next transaction.

Points to Note

Here we have been able to solve the problem from first principles. Several points need to be noted

Multiplication Procedure
 * 1) Why did we choose a strip of 6 as the Whole? 6 is a number from which as we have seen above, both and subsequently   of the new whole 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 new whole which would be 4.
 * 2) Another way of looking at it would be to see that 6 is a product of 2 &amp; 3. We could have also used any multiple of 6. For example if we had taken a chocolate of 18 parts as the whole the answer would be which can be expressed as  after a few arithmetic manipulations. We choose 6 since it is the smallest whole which will satisfy the condition.
 * 3) We can also see that we could have got the result by multiplying both the numerators to get the numerator of the result and the denominator in a similar way.

The arithmetic procedure for doing the above problem can be written as given below.

X  =   =


 * 1) Multiply the numerators and write as the numerator of the product.
 * 2) Multiply the denominators and write as the denominator of the product.
 * 3) Simplify to smallest numbers where necessary.

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