For example, if we want to know a simpler form for the fraction 12/16, we can draw a diagram representing 12/16; we can then see if every two of the shaded parts and the unshaded parts could be merged. See the diagram below:
After the first merger, there should be no single shaded or unshaded part remaining. If a single part remains, it is clear that the merger generating equal bigger parts is not possible and we have to retain the earlier diagram.
Our aim here is to figure out if there were unnecessary smaller parts. After a successful merger, we identify the new simpler fraction obtained from the complicated fraction, 12/16. We keep repeating this process of merging 2 or 3 of equal smaller parts in the diagram we have, until there is no further possibility for merging smaller parts left. Every time, we figure out the fraction associated with the diagram that emerges. The fractions we obtain with every merging are a simpler version of the preceding equivalent fraction.
The last simplest equivalent fraction that we obtain is called a reduced fraction, or a fraction in lowest terms. For this fraction, we will not find a whole number greater than one, which can divide both its numerator and denominator.
In the example we have, there are three equivalent fractions, viz., 12/16, 6/8, and ¾. Only one fraction of these is in the simplest form having lowest terms, i.e. ¾. The equivalent fractions of 12/16 with higher terms will be 24/32, 48/64, 96/128, etc. However, all these equivalent fractions will have only one equivalent fraction with lowest terms.
When we carry out addition, subtraction, multiplication or division on fractions, we all might land up with different looking answers, but we all will get only one fraction with lowest terms. This could be one of the reasons for insistence on reporting answers always in terms of lowest reduced form of fraction.
The problem of finding reduced form of a fraction like 12/16 can also be viewed in terms of discrete objects. Suppose we have a big piece of a cardboard, and we cut it into 16 equal pieces.
We take 12 of those equal pieces and shade them, to identify the pieces that we have chosen. We can now see that we have some unnecessary cutting, which could have been avoided to get the required portion of the cardboard. We then try to reunite the pieces to get bigger equal number of pieces of the cardboard. The situation is shown in the diagram below:
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