Expert: Abe Mantell - 1/13/2005

Question
I am not really (insistent) on using this method for more than two fractions.  I am simply curious to know whether or not this method would provide the least common multiple for the denominators of more than two fractions as I have indicated in my first follow-up question. I just wanted additional verification.
There is no need to be confused by my curiosity.
-------------------------
Followup To
Question -
Hello:

I want to thank you for the reply.

I assume that it would be wise not to use this calculation to determine the least common multiple for more than two fractions. However, if this method is used for more than two fractions, calculate the least common multiple for two denominators, 1/2 and 3/4, then use this least common multiple, the new denominator, with the third fraction, 5/6, to determine the final least common multiple.

Example: 1/2, 3/4, and 5/6

(2 X 4)/ gcd 2 = lcm 4
1/2 + 3/4 = 5/4

Now determine the least common multiple for the denominators from 5/4 and 5/6.
(4 X 6)/ gcd 2 = lcm 12
5/6 + 5/4 = 25/12

12 is the least common multiple for all three denominators from the fractions.

Is this idea or method correct?

I thank you for your follow-up reply.

-------------------------
Followup To
Question -
Hello:

The following calculation can be used to determine the least common multiple (lcm) of two fractions by dividing the denominators of the fractions by thier greatest common divisor (gcd):
See the following: http://en.wikipedia.org/wiki/Least_common_multiple

Example: 3/4 and 5/12

(4 X 12)/ 4 (gcd of 4, 12) = lcm 12

Using this same method to find the least common multiple for 3 denominators produces some mixed results.  

Example: 1/2, 2/3, and 4/5 Find the lcm:
(2 X 3 X 5)/gcd 1 = lcm 30, but with the fractions 1/2, 3/4, and 5/6, the calculation provides (2 X 4 X 6)/gcd 2 = 24, but 12 is the lcm for these three denominators not 24.

Can you explain why the method/calculation works with two denominators but not generally with three or more denominators?

I thank you for your reply.


Answer -
Hello,

The reason it does not work (in general) with more
than 2 denominators is because each denominator
*may* have the same common factor...and when we
multiply them (in the numerator) we *may* be having
multiple factors of the GCD which will not be
"divided out" by the denominator.

For example, with 1/2, 3/4, and 5/6:
there is a GCD of 2...so when we multiply
2, 4, and 6...we are multiply the 2 three times,
but only "dividing it out" once!

See?

Abe
Answer -
I am confused by your insistance on using this method for
more than 2 fractions after we see that it is, as you said,
"not wise" to use it for such problems...

Answer
I was only confused by your "curiosity" because I felt
I already answered your question...so I will clearly
state that:

The method you outlined DOES NOT WORK (in general) for
more than 2 fractions...however, it CAN WORK WITH
MODIFICATION -- but in my opinion not worth the trouble.