GED Mathematical Reasoning: Comparing Fractions: LCM and LCD

 

Adding and subtracting fractions can be a challenging topic, but we can make things a bit easier if we talk FIRST about comparing fractions, the Least Common Multiple (also known as the “LCM”) and the Least Common Denominator (or “LCD” for short).

Fractions are easy to compare when they are “like” fractions; “like” fractions are fractions that have the same denominator. Fractions with the same denominator are often referred to as having a common denominator. For example: \frac{4}{7} and \frac{1}{7} are examples of “like” fractions since they have a common denominator of 7.

When you have fractions with common denominators, the fraction with the greater numerator is the greater fraction.

But what if we want to compare fractions that are not “like” fractions – without common denominators? For instance, suppose we want to compare: \frac{2}{3} and \frac{3}{4}

To compare “unlike” fractions, we transform the fractions INTO “like” fractions so that they HAVE a common denominator. We’ll do this using a concept called the Least Common Multiple as well our knowledge of equivalent fractions.

The first step in creating fractions with common denominators is to find the Least Common Multiple of the original denominators. To do this, list the first several multiples of each denominator and then identify the smallest one in common. That value is the Least Common Multiple, which we’ll use as the Least Common Denominator.

And just as a side note: if the Least Common Multiple doesn’t appear when you list the first few multiples of each number, you need to continue listing multiples until you see the one they have in common.

You may be thinking: How is all this stuff going to be important for adding and subtracting fractions?

Well, to be able to add or subtract fractions, we may only add or subtract apples to apples and oranges to oranges. After all, if I asked you to add 4 apples and 6 oranges, you would probably give me a quizzical look in return :).

It is often necessary to add fractions that don’t have common denominators. Therefore, we need to understand this process of transforming “unlike” fractions into fractions that have common denominators. Only then, can we add or subtract.

 

Example 1

Compare \frac{4}{7} and \frac{1}{7}

 

So if we were to compare: \frac{4}{7} and \frac{1}{7}, \frac{4}{7} has the greater value because it has the larger numerator. Picture two pizzas, each containing 7 slices. If Marty eats 4 slices of one pizza and Carl eats 1 slice of the other, Marty has obviously eaten the greater amount.

 

Example 2

Compare \frac{2}{3} and \frac{3}{4}

 

Since the two fractions have different denominators, we will transform the fractions into “like” fractions so that they have a common denominator. Let’s find the Least Common Multiple for our denominators 3 and 4.

The first several multiples of 3 are:  3, 6, 9 and 12

The first several multiples of 4 are: 4, 8, 12 and 16

The smallest number common to both lists is: 12

So the Least Common Multiple of the denominators is: 12

Next, rewrite both fractions using 12 as the common denominator. Let’s create an equivalent fraction for \frac{2}{3} with a denominator of 12.

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First, we’ll write a 12 in the denominator of a new fraction. Then, we’ll ask ourselves: What do we multiply our old denominator of 3 by to arrive at our new denominator of 12? The answer is 4. So we’ll multiply the numerator by that same value.

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2 times 4 equals 8. Therefore our new, equivalent fraction that’s equal to \frac{2}{3} but with a denominator of 12 is \frac{8}{12}.

Now we’ll create an equivalent fraction for \frac{3}{4} with a denominator of 12.

math 138

First, let’s write a 12 in the denominator of a new fraction. Then, we’ll ask ourselves: What do we multiply our old denominator of 4 by to arrive at our new denominator of 12? The answer is 3. So we’ll multiply the numerator by that same value.

math 139

3 times 3 equals 9. Therefore our new, equivalent fraction that’s equal to \frac{3}{4} but with a denominator of 12 is \frac{9}{12}.

Since \frac{9}{12} has the larger numerator, it is the larger fraction. Going back to our original fractions, \frac{9}{12} is equivalent to \frac{3}{4}, so \frac{3}{4} is larger than \frac{2}{3}

 

 

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