Why does the least common denominator work?Take for instance the following problem. You have two beakers of the same height. One has tick marks that break it into thirds. The other has tick marks that separate it into fourths. The water levels are 1/3 and 1/4 respectively. If I did not know about the concept of LCDs, how would I figure out how much water there is all together? Please walk me through your reasoning.Note: I understand the need to find a common scale between the two beakers. I don't know how I would find that 12 is the smallest possible common scale, if I had never been introduced to the concept of LCDs/LCMs.

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Why does the least common denominator work?Take for instance the following problem. You have two beakers of the same height. One has tick marks that break it into thirds. The other has tick marks that separate it into fourths. The water levels are 1/3 and 1/4 respectively. If I did not know about the concept of LCDs, how would I figure out how much water there is all together? Please walk me through your reasoning.Note: I understand the need to find a common scale between the two beakers. I don&#039;t know how I would find that 12 is the smallest possible common scale, if I had never been introduced to the concept of LCDs/LCMs.

aniizl · Accepted Answer

You are looking for numbers say x and y such that:  x  ×      (          1      3        +          1      4        )  i.e.      x    3    +      x    4    =  y  ,where y is an integer. Assuming you do not know about LCM, you will try numbers   x  =  1  ,  2  ,  3  ,  … and   x  =  12 will be the first number for which you will get an integer (7 in this case) as an answer.So you have  12  ×      (          1      3        +          1      4        )    =  7.Hence      1    3    +      1    4    =      7    12    .Continuing this way, we find for   x  =  24, we have    y  =  14, for   x  =  36, we have   y  =  21, etc. And clearly 12 is the least value of x for which y is an integer.

Rocatiwb · Accepted Answer

Because it readily bridges the gap between different measures. If you didn&#039;t know about least common denominators, you&#039;d probably wind up rediscovering the concept and wondering why no one else had thought of it before.Let&#039;s say you swap the contents of the two beakers. You find that      2    4    &amp;gt;      1    3    &amp;gt;      1    4    &amp;gt;  0.But this doesn&#039;t tell you precisely how much more       1    3   is than       1    4  . It&#039;s clear that you need finer tick marks. If you&#039;re smart, you won&#039;t bother trying fifths. But you might try sixths, in which case you find that      3    6    &amp;gt;      2    6    &amp;gt;      1    4    &amp;gt;      1    6    &amp;gt;  0.So the total content is more than       3    6    but less than       4    6  . So you need finer tick marks still:      6    12    &amp;gt;      4    12    &amp;gt;      3    12    &amp;gt;  0.The denominators are all the same now. Since       1    3    =      4    12   and       1    4    =      3    12   you can now add up these fractions with ease:      4    12    +      3    12    =            4      +      3        12    =      7    12    .You might figure out that 6 did not work because   6  =  2  ×  3 yet   4  =      2    2  , and that 12 did work because   12  =      2    2    ×  3. We could say that 12 brings together 3 and 4.You might name this concept &quot;Joe&#039;s finest tick mark denominator method,&quot; write up a pamphlet about it and give it to a lot of people. Then someone comes up to you and tells you that the ancient Egyptians or the ancient Chinese had already figured this out a long time ago.

Why does the least common denominator work? Take for instance the following problem. You have two b

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