## Finding the LCD, 1881 Style

Posted by mark schwartz on March 28, 2016

__Introduction__

Adding and subtracting fractions is sometimes a messy thing for a lot of students. At the core of adding or subtracting fractions is finding the lowest common denominator (LCD). There are a variety of approaches to finding the LCD (guessing, estimating, using prime factoring in some form, find the common denominator and reduce, etc.). I can’t possibly name them all since over time, math instructors have created many approaches to make the procedure of finding the LCD more available to students. There is, however, one procedure that doesn’t seem popular these days, but was popular in the 19^{th} century and you can see it in Mr. Ficklin’s *Elements of Algebra, 1881 (pg. 178)*.

__The Story__

In the lead up to operations with adding and subtracting fractions students usually are practiced in prime factoring, equivalent fractions, reducing fractions, greatest common factor, and lowest common denominator. These are all elements used in one way or another to find the LCD. Of the elements in this set, greatest common factor (GCF) gets the least attention, if any at all. In discussions with colleagues, I have discovered that some of them skip the GCF altogether, but it’s the core of Mr Ficklin’s method, so take a look at it.

*Mr. Ficklin notes that, given two numbers, the GCF multiplied by the LCD equals the product of those two numbers. From this it can be seen that the LCD can be found by dividing the product by the GCF.*

The GCF is that set of factors that is common to all the terms being considered. Again, there are a variety of ways to find it, typically using prime numbers. So, starting simply, the GCF of 2, 4, and 6 is 2 because if these numbers are factored, the outcome is 2•1, 2•2, and 2•3 and the GCF for these numbers is clearly 2, since each set of factors contains a 2. The numbers 5, 10, and 15 give a GCF of 5, and this can be seen almost without any factoring. What about the numbers 8, 12, and 32? At first glance, 2 is a common factor but taking another moment to look at these, 4 is really the GCF (by the way, if the GCF isn’t seen nor found immediately, I’ll show you how it can be recaptured later in the procedure). If these 3 numbers are prime factored, it will be seen that 4 is the GCF.

Now, start with this example: find the LCD for 12 and 18. The GCF is 6. There are common factors of 2 and 3, but these are just that; common factors, not the GCF. Using 12 and 18 and the GCF 6, and following Mr. Ficklin’s idea, the 12 and 18 are factors in the numerator of a fraction, with the GCF of 6 as the denominator …

(12*18)**/**6

Simplifying this fraction gives the LCD OF 36. You’re done.

Another example: find the LCD for 16 and 20. Again, it may take a moment to see the GCF of 4, but if 4 isn’t seen right away, I’ll talk about what happens, but to start with, use 4 as the GCF. The fraction is …

(16*20)**/**4

Simplifying this fraction gives the LCD of 80.

But what happens if 2, rather than 4, is seen as the GCF? The LCD would be found to be 160, not 80. Several things might happen. First, a student might say yuck, the number is too big and redo the work. Second, a student might decide to use 160 and although the numbers are big and messy, the ultimate answer to the problem would be the same, reduced to lowest terms of course.

This highlights the importance of spending time exploring how to find the GCF. If the text doesn’t dwell on it, it’s the instructor’s responsibility to help students get comfortable with the idea.

Does this method work when finding the LCD for 3 or more numbers? Of course it does. Mr. Ficklin demonstrates this by using any 2 of the numbers to generate an intermediate value and then uses this value with the third number. He suggests that we take a moment to decide which two numbers are to be used start the procedure, but in his examples, he seemed to use the two smaller numbers first.

For example, find the LCD for 8, 12, and 32. Using 8 and 12 first, the GCF is 4. The fraction is …

(8*12)/4

Simplifying this fraction gives 24. Now set up another fraction using this result of 24 and the remaining number, 32. First, what is the GCF for these two numbers? Again, after a few minutes of reflection (or prime factoring), the GCF is found to be 8, so the fraction is …

(24*32)/8

Simplifying this fraction, the result is 96, which is the LCD for the numbers 8, 12, and 32. I’d like to point out that any 2 of the 3 numbers in the problem could have been used to start the process.

Notice several things. First, it’s not necessary to prime factor all the numbers and use a procedure, to “build” the LCD. Second, it is a shorter procedure and in my view, a simpler procedure. Third, there aren’t as many numbers to handle and manipulate, even if prime factoring is used for finding the GCF.

By the way if 2, not 4, was used as the GCF to begin with the fraction would be …

(8*12)/2

which would give 48, not 24. But look what happens in the next fraction …

(48*32)/?

Again it may not be obvious at first, but given the size of both of those numbers, students will have to hesitate a bit and find the GCF a little more carefully. Its 16, so the fraction is now …

(48*32)/16

The LCD again is still 96. Again, if 16 isn’t seen to be the GCF, the result would be to a larger common denominator (not the GCF) but ultimately, the answer would be the same, having reduced the answer.

Try another one. Consider the problem where the 3 numbers are 15, 20, and 35. Starting with 15 and 20, the fraction would be …

(15*20)/5

which gives 60, and the next fraction is then …

(60*35)/5

which gives 420 as the LCD. These examples may seem awkward because of the numbers used, but the point is that the method works and I would propose that with an LCD this large, the effort to obtain it by other methods would likely be more awkward and time consuming.

One of the values of this method, in my view, is that there is less opportunity to err. Adding and subtracting fractions is complicated by having students find the LCD first, which actually seems to be a big pitfall for students. I believe Mr. Ficklin has the right approach.

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