What does LCD stand for?
In math, LCD stands for either the Lowest Common Denominator, or Least Common Denominator. They both mean the same thing. Before getting too deep into Lowest Common Denominators, let’s review fractions, and some of the words and ideas that we’ll use.
What is a Fraction?
A fraction is a tool (OK — a number) that helps you solve problems involving pieces of units. In a given fraction, those pieces are all the same size. The two numbers that compose a fraction show two things: the number of pieces that each unit is broken into, and the number of those pieces present. It’s OK for there to be more pieces present than the number of pieces each unit is broken into; all that means is that the fraction represents a portion bigger than a single unit’s worth.
What is a Denominator?
The denominator is the number of pieces that a unit is broken into in a fraction. We write it on the bottom part of the fraction.
What is a Numerator?
The numerator is the number of pieces present in a fraction. We write it on the top part of the fraction.
What are Equivalent Fractions?
Equivalent fractions are fractions that are written with different numbers, but that represent the same actual value. How can that be? For example, if you divide a unit into two pieces, and keep both pieces, you’ve still got a unit worth. 1 = 2/2. You could also divide it into 3 pieces and keep all three: 1 = 2/2 = 3/3. They are equivalent fractions. Another example: one half is the same as two quarters. You can do this with any fraction. If you multiply (or divide) both the numerator and the denominator by the same number you get an equivalent fraction. (No — you can’t use zero.) It’s an incredibly useful trick.
Using equivalent forms makes it possible to add, subtract, and compare fractions that would otherwise be incompatible. But, they can also cause us to do arithmetic with numbers that are bigger than necessary. To avoid problems caused by equivalent fractions, we simplify them.
Why would you want the LCD of a Fraction?
That’s a trick question! Actually, we don’t look for the lowest common denominator of a fraction, we look for it between two or more fractions. Most of the time, we find the LCD of just two fractions. The word “common”, in this case, means that we are trying to find a single number that can serve as the denominator for two different fractions. (The numerators are usually going to be different.)
When fractions have different denominators, we can’t always work with them. Things that we can’t do include adding them, subtracting them, and even comparing them. Finding a Common Denominator makes it possible to do those things. Finding the Lowest Common Denominator is even better, because it lets us do those things with the least work we can get away with.
What is the Lowest Common Denominator?
Fractions have two parts — the numerator and the denominator. In fractions, units are always broken into pieces the same size. The denominator tells what that size is. If you try to add, subtract, or compare fractions with the same size pieces, all you do is add, subtract, or compare the pieces. Easy! Just add, subtract, or compare the numerators, and don’t change the denominator.
But, if you’ve got fractions with different size pieces what can you do? Break the pieces into even smaller pieces! Using a bit of cleverness, you can choose the right size for the little pieces so they’re the same for both fractions.
The denominator is the size of the pieces; since we want to change the size of the pieces, that’s our main focus. Common means it’s the same between two fractions. Our goal is to convert both fractions into equivalent fractions that have the same denominators. When we find a denominator that is common between the two fractions we will be able to do different things with them (generally we want to add, subtract, or compare them).
Lowest or Least means that we don’t overdo it by breaking our units into pieces which are smaller than necessary.
How do you find a Common Denominator?
There is a really obvious and easy way to find a common denominator for two fractions. Unfortunately, it will generally not be the least common denominator.
- Multiply the two denominators. The result will be a common denominator (but usually not the lowest one).
- Convert the first fraction into an equivalent fraction with the common denominator, by multiplying both the top and bottom by the denominator of the second fraction.
- Convert the second fraction into an equivalent fraction with the common denominator, by multiplying both the top and bottom by the denominator of the first fraction.
This technique is simple and always works, but it has its weaknesses. For many fractions, it uses larger numbers than necessary, which means spending too much time and work doing arithmetic with big numbers. The chance of making “careless” mistakes goes up when working with big numbers too. To make life easier, it is worth using the Lowest Common Denominator.
How do you find the Lowest Common Denominator?
The trick is that the LCD has to have all the prime factors of each denominator, with no other extraneous factors. You can find the LCD of two fractions by following these steps:
- Find the prime factorization of both denominators. Doing this requires being able to “see” what the factors are of the numbers. We noticed that students sometimes get lost in this step, so we developed an educational game to help. The game is a fun way to become proficient at factoring.
- The LCD will be found via its factors. Start by copying all the prime factors of one of the denominators.
- Now copy prime factors of the other denominator into the LCD, but (here’s the important part) you don’t need to copy any that are already there. This takes a little explaining for prime numbers with multiple copies in one or both of the denominators. If there are multiple copies of some prime number in one of the denominators, that many copies have to be put in the LCD. But if the other denominator only has that many or less, additional copies don’t need to be added on its behalf. So, for instance, if the first denominator has three 2s and the second denominator has four 2s, the LCD needs four 2s.
- Multiply together all the prime factors found in the previous step. The product in the LCD.
A Time Saving Shortcut for Using The LCD
Once you find the LCD, most likely, you’ll convert both of your starting fractions into equivalent fractions with the LCD in the denominators. There’s an excellent timesaving shortcut you can use if you like. You already know the prime factorizations of both denominators, and of the LCD. For the first fraction, collect the prime factors in the LCD that came purely from the second fraction, and multiply top and bottom of the first fraction by those. For the second fraction multiply top and bottom by prime factors of the LCD that came purely from the first fraction.
How do you find the LCD of three or more Fractions?
The process is similar to that for the LCD of two fractions. The LCD has to have as many copies of each prime factor as any of the denominators does.
- Find the prime factorization of all the denominators.
- Start finding the LCD’s prime factors by writing all the prime factors of one of the denominators.
- Add additional prime factors to the LCD from one of the remaining denominators. Do this by looking for prime factors that aren’t in the LCD yet, or prime factors that are there, but have less copies than the denominator in question.
- Repeat step 3 with each of the remaining denominators.
- Multiply together all the prime factors of the LCD to represent it as a number.
If you want to use the shortcut for converting your starting fractions into equivalent forms with the LCD, you can do it the same way as for two fractions. However, for each starting fraction, collect all the LCD prime factors that were not in that starting fraction’s original denominator.
Remembering how to find LCDs
It can feel like finding LCDs is a bunch of nit-picky steps that are hard to remember. One of the best tricks for remembering things is to find a way to visualize them. Here’s a suggestion for visualizing Lowest Common Denominators, and how to find them:
Start by reading about the Fundamental Theorem of Arithmetic, and focusing on the idea of prime numbers as building blocks. Once that’s clear in your mind, think of the LCD as a kit that has only enough parts (prime number building blocks) to build either one of the denominators at a time, but not necessarily enough to make them both at the same time. Visualize those numbers as actual building blocks, like the blocks young children play with.
The first common denominator (which wasn’t a lowest common denominator) that we made, by just multiplying both denominators, was like making a kit with enough pieces (building blocks) to build both denominators at the same time. That’s nice, and easy to do, because you just throw in all the pieces required to make both numbers. But, it’s also more expensive than necessary, because the person using the kit only needs to make one or the other denominator at a time. The trick is to make the kit smaller by removing duplicate parts but making sure that all the non-duplicate parts are there.
Is there anything you can do with Fractions, without finding the LCD?
It can start to feel like we need to find Lowest Common Denominators for everything we do with fractions. However, there’s actually plenty of things that we can do without it.
The big ones are multiplications and division.
Multiplying by a fraction means breaking a number into pieces the size of the fraction, even if your original number is a fraction itself. Dividing by a fraction means divvying up the quotient into a number of portions, or portions of a certain size. You don’t need the LCD for either of those things.
You can also factor out part of a fraction from a sum or difference of fractions using the “distributive property” without figuring out the LCD. However, when you do that, you are usually looking at what the factors of multiple denominators are, so even though this isn’t finding the LCD, it’s related because you’ll be interested in prime factorization.
Next Topic: Greatest Common Factors