Fractions are difficult—and this type of unit, similarly to most in math, usually just builds on itself. This means that if you don’t address your confusions right away, you might find that even after you move on to new material, you’ll continue to use the concept that you didn’t understand.
If you’re struggling with fractions in class, you’re also likely having difficulty tackling your homework in math, which may certainly add to your stress levels.
This may particularly apply to you if you’re a student experiencing math anxiety, which means that dealing with any math concept leaves you with anxiety and doubt.
Luckily, there are some handy methods you can implement to make nailing your fractions a bit easier.
Adding fractions by calculating least common denominator
First things first, let’s recall what the denominator is.
The denominator is the bottom number in a fraction and indicates how many equal parts the item is divided into.
When two—or more—fractions have the same denominators, they have common denominators. Why are common denominators important?
Because in order to add fractions, you need common denominators. But specifically, you’ll want to be using the least common denominators.
Why do you need the least common denominator (LCD), specifically? You don’t actually need to use the LCD, but it will certainly make simplifying and working with any given fraction easier—as opposed to using any common denominator.
In order to determine the LCD…
List all multiples
Let’s say you have two fractions: ⅘ and ⅔. You’ll want to list all the multiples of both denominators, which are 5 and 3. Then, you can determine the LCD. This should look something like this:
While these aren’t all the multiples, you don’t need to list them all since you can already see that there is a common denominator: 15. And since you’re looking for the least common denominator, there’s no need to keep listing them. This means that you want both fractions to have a denominator of 15.
Multiply both parts of fractions
The next step is to multiply both the top and bottom of each fraction by the number that will make each denominator equal 15.
So for ⅘, you’ll need to multiply the whole fraction by 3 (because the denominator is 5, and 5 x 3 = 15). For ⅔, you’ll multiple the whole fraction by 5 (the denominator is 3, and 3 x 5 = 15). Now both fractions have the same denominator, and you’re ready to add the two together.
In this example, the final answer should come out to be 22/15.
Adding fractions without calculating least common multiple
If you’re not fan of the LCD method for adding fractions, you’ll be happy to know that it’s not the only option. With this trick, you’ll initially be using multiplication, but the result will be the same as if you were to add the fractions using the LCD method.
For a detailed explanation, check out the video below:
Step 1: Multiply the denominators
First, you’ll multiply the denominators together. We’ll use the same fractions as in the example above: ⅘ and ⅔.
So, 5 x 13 = 15.
Step 2: Cross multiply fractions
This is where it gets a tad bit more complicated. Multiply the numerator of the first fraction (4) with the denominator of the second fraction (3), and then multiply the denominator of the first fraction (5) with the numerator of the second fraction (2). This is cross multiplication. To visualize this better, picture two diagonal lines going from each numerator to the opposite denominator.
So, your first results are 12 and 10.
Next, add those two products together. The result here is 22.
Now go back to step 1, where you have your denominator of 15. Place the result from the cross multiplication (22) over the denominator from step 1 (15). Now, you have 22/15, which is the same result we got using the LCD method.
Dividing fractions by flipping
Thinking about dividing fractions can be confusing, since you may primarily associate division with one number over another—which is also how you probably picture a fraction.
This trick may make the concept of dividing fractions much easier to comprehend. The method is simple, and really only involves a few quick steps.
Step 1: Change the sign
Again, we’ll use the same fractions: ⅘ ÷ ⅔.
First, you’ll want to swap the division sign with a multiplication sign. So your new equation is: ⅘ x ⅔.
Step 2: Find the reciprocal
The next step is to flip the second fraction (⅔), creating the reciprocal—a reciprocal is simply the number you’d have to multiply another number by to come up with 1. In simple terms, the reciprocal is what you create when you divide something by 1. So for instance, if you have 3, then divide it by 1, the reciprocal is ⅓.
In this case, you need to multiply ⅔ by 3/2 to get 1. So your reciprocal is 3/2.
Step 3: Multiply the fractions
After you’ve flipped the sign and changed the second fraction to its reciprocal, all you’ve got to do is multiply the numerators and denominators, and you’ll have your result:
⅘ x 3/2 = 12/10
Follow the circle method to create improper fractions
The circle method is a great tool to utilize—particularly for all you visual learners out there.
You can use the circle method when you have a number and a fraction, which is a mixed number, (such as 1½) and need to transform it into an improper fraction. An improper fraction is simply a fraction that contains a numerator that is larger than the denominator.
This is called the “circle” method, because you’re essentially traveling in a circle around the fraction.
Let’s look at the case of 1½:
You’ll start at the bottom, 2. From the bottom number, go around, clockwise, to the next number, which is 1.
The first step is to multiply those two numbers: 1 x 2 = 2. Next, you keep going in the circular clockwise motion, and arrive at the next number, which is 1. The second step is to add the product from the previous step (2) to the numerator of the fraction (1) : 2 + 1 = 3.
And voilà. You don’t change the denominator at all. Simply follow the circular steps, and you’ll get your improper fraction result, 3/1.
Simplify by finding the greatest common factor
Another challenge when learning fractions is nailing the simplification process. Here we’ll go over the greatest common factor (GCF) method. The GCF is the largest number that divides evenly into two or more numbers.
With this strategy, you’ll be looking for the GCF between the numerator and denominator—once you find it, you’ll be able to efficiently simplify the fraction.
Why do we want to simplify a fraction? The reason is simple (pun intended). Simplifying fractions just makes the subsequent work easier for you compared to if you leave a fraction in a larger-than-simplified version. You’ll be dealing with smaller numbers, which means less work for you.
Step 1: Find the GCF
First, you need to determine the GCF between the numerator and denominator. Let’s look at the fraction 9/12.
For some fractions, you’ll know right away what the GCF is, and for others you’ll have to do a little more brain work. This “brain work”, just refers to listing the factors of the numerator and denominator until you find the GCF
If you can’t tell right away, you can start by seeing if 9 (the greatest factor of 9) is a common factor of 9 and 12, and then work down from there. You should eventually determine that the GCF between 9 and 12 is 3.
For a visual explanation, check out the video below:
Step 2: Divide by the GCF
Once you’ve found the GCF, divide both the numerator and denominator by this number. And then you’re done! In the case of 9/12, after dividing by the GCF (3), you should arrive at a final answer of ¾ (9 ÷ 3 = 3; 12 ÷ 3 = 4).
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Author: Lydia Schapiro