Lowest Common Denominator (LCD)

This post is about meaning and math

First we’ll learn a math concept - Lowest (or Least) Common Denominator (or LCD). Then we’ll talk about how it’s sometimes used in everyday life.

In layman’s (non-mathematician’s, in this case) terms, the LCD is the largest partition of something that will also go into another thing of the same kind.

What the heck does that mean? Well, if you have, say, 1/4 of a apple blueberry crumble pie (you can tell where my mind is this balmy upstate Memorial day), and you have 1/5 of another apple blueberry crumble pie, how much apple blueberry crumble pie do you have altogether?

Changing the Denominators

You probably know that you can’t add fourths and fifths without turning them into something else, because the denominators (lower numbers of the fractions) are different.

What do you change those denominators into? You change them into the lowest common multiple. That means the lowest number that both of those denominators go into.

The lowest number that both five and four go into (without remainders) is 20. So the lowest common multiple of 4 and 5 is 20, which means that the lowest common denominator of 1/4 and 1/5 is 1/20.

Now, how many groups of 1/20 do you need to make up 1/4? Well, when you turned that 4 (the denomenator) into a 20, you had to multiply it by 5, so you have to also multiply the 1 (the numerator) by 5, giving us five twentieths. That means that in 1/4 can be expressed as 5/20.

A good way to look at it, is to imagine that you want to keep to things in proportion (the numerator and the denominator). If you make one bigger, you have to make the other equally bigger. Like when you grow; If your head grows, the rest of your body should grow along with it. Usually.

So far we’ve used the lowest common multiple to turn the fourths in to twentieths. Now we need to turn the fifths in to twentieths, so both numbers will have a common (same) denomenator.

Since twenty is the lowest common multiple, we also want to turn the one fifth into twentieths. To turn 5 (the denominator) into 20, you have to multiply by 4. So you aslo have to multiply the 1 (the numerator) by 4, thus giving you four twentieths. That means that in 1/5 can be expressed as 4/20.

Adding with the new (common) Denominators

So if you want to add one-fourth (which, as we now know is 5/20), to one fifth (which, as we now know is 4/20), instead of saying 1/4 + 1/5, you can state it as 4/20 + 5/20. That is easy. Four of anything plus five of the same thing is always nine of them. So 4/20 + 5/20 = 9/20.

You couldn’t have added two uncommon things without changing them into groups of the smallest unit they had in common.

One more quick example:

2/3 + 6/7 =

• The lowest number that 3 and 7 have in common is 21. So that is our LCD.
• You had to multiply the 3 by 7 to get 21, so you must do that to the 2 in the numerator as well, which gives you 14. So the first fraction would be changed to 14/21.
• You had to multiply the 7 by 3 to get 21, so you must do that to the 6 in the numerator as well, which gives you 18. So the first fraction would be changed to 18/21.
• 14/21 + 18/21 = 32/21, which is the answer.

This answer can be turned into a mixed fraction (a whole number and a fraction) which makes it more convenient to work with. It’s also usually what schools want you to do (although they are seldom sure why they want you to do it). This isn’t a lesson about simplifying and mixed fractions, but here’s the skinny on it:

32/21 is the same thing as 21/21 + 11/21.  21/21 is the same thing as 1, so 21/21 + 11/2 = 1  11/21 (that’s “one and eleven twenty-firsts.”)

You basically divide the numerator by the denominator, and write the answer (the quotient) as a whole number. If there is a remainder, that becomes the new numerator of the fraction.

One more example of that? OK:

53/16 (it even looks like a division problem).

• 16 goes into 53 3 times, giving you 3, remainder 5 (because 16 x 3 = 48)
• The 3 is the whole number, the remainder goes over the 16, giving you 3  5/16 (“three and five sixteenths.”)

What about the meaning? I’ll get to that when I update this post.

I sure hope this helps you understand the basics of using the Lowest Common Denominator to add fractions.

All the best,

Brian (a.k.a. Professor Homunculus)