Q) I can see what happens when fractions are multiplied together, but for division I have always used the rule "turn upside down and multiply"; eg, 23 V 35 becomes 23 X 53 = 109 = 119.
Please can you tell me why this rule works.
A) There are many maths techniques that have been taught in the past as a "rule". My enlightenment of the basis for the division rule for fractions came when I was giving a parents' maths club session on fractions. I began by ripping up pieces of paper - one parent commented: "so that's what fractions are. Why didn't my teacher do that when I was at school?" We progressed from addition to equivalent fractions and multiplication, and then thought about what we are actually doing in dividing fractions.
First, we explored division by a whole number, eg 23 V 4. This is, in fact, asking us to find one-quarter of two-thirds. So we replaced 23 with 13 + 13, and discussed the fact that we could further reduce the fractions to sixths, giving: 23 = 16 + 16 + 16 + 16 and we can easily see that 23 V 4 = 16.
Then we looked at this in terms of creating the same denominator in both fractions, just as we do in addition: 1X21X3 V 4X31X3 = 23 V 123.
This is the equivalent of asking how many twelve-thirds there are in two-thirds, and we can see that we will have a fractional answer. The question thus becomes: 2 V 12 = 212 = 16.
In terms of "turn upside down and multiply", 23 V 41 becomes: 32 X 14 = 212 = 16.
This can be extended to look at a fraction divided by a fraction.
Let's consider your problem of 23 V 35. To deal with this, the fractions need, once again, to have the same denominator. This is achieved as before, and so23 V 35 becomes: 2X53X5 V 3X35X3 = 1015 V 915.
Now we can reduce the problem: 10 V 9 = 109 = 119.
We can also represent this problem diagrammatically. The two-thirds section is shown in orange. In the second diagram, the this section has been subdivided to become ten-fifteenths.
The next diagram shows nine-fifteenths in purple. This is the crucial part of the concept, because, in the division, we consider the purple section as a whole piece - 915 becomes a whole section made up of nine pieces.
One-fifteenth thus becomes one-ninth of nine-fifteenths. In a sense, we have recreated a new whole.
You could draw another diagram to demonstrate the answer to the question how many times 915 will fit on 1015. It would give 915 as an overlay. It fits over 1015 (23) exactly once, with a 19 section of the 915 left uncovered. To completely cover the 1015 (23), 119 of the 915 is required.
Now the "turn upside down and multiply" method should be clearer and so 23 V 35 becomes 23 X 53 = 109 = 119 as before.
Looking at division of fractions in this way makes you reflect on what we are actually doing when carrying out a division. It highlights the discussion in this column on September 5, in which the solution to 8 V 5 was discussed as being written, not as "1 remainder 3" but as "135". The use of the word "remainder" can lead to misconceptions about division when the question is out of context.
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