Addition and Subtraction of Fractions

1. Adding and Subtracting Fractions with Common Denominators

If you remember from an earlier lesson, you learned that fractions were a part of a whole and are called “rational” numbers.

We had explained that fractions can be thought of as pieces of a pizza and showed the following example:

The total amount of pieces the pizza was cut into is called the denominator. The amount of pieces you ate is called the numerator. So for this example, the pizza was cut into 8 pieces and you ate 3. The fraction for pieces eaten would be .

Now, what if you ate 3 pieces of pizza on the first night and 4 pieces of pizza the next night? How much of the pizza have you eaten now? To solve this question, we can now discuss how to add or subtract fractions.

When you add or subtract fractions with the same, or common, denominators—like our pizza example—you simply add or subtract the numerators (or pieces of pizza eaten).

EXAMPLE

Consider the equation

You’ll notice that the denominators stayed the same. Why didn’t we add them too to make 16? This is because the denominator represents how many pieces the pizza was cut into. Just because you ate 7 pieces of pizza doesn’t mean you started cutting the pizza into smaller pieces before you ate it, right? The pizza slice sizes and amount cut into the pizza remained the same. The only thing that changed was how many pieces of the pizza you ate.

EXAMPLE

Of course, we don’t always need to use pizzas to think about fractions. In the example below, you are adding two fractions with common, or the same, denominators (the 5s). Therefore, you simply add the numerators of each fraction (2 and 1), which equals 3. The denominators stay the same.



To think about this problem in a different way, look at the picture representation below. The bar is split into five pieces, which matches the 5 in the denominator. The pieces are all the same size because the denominators are all the same size. You start with 2/5, or 2 out of 5 pieces, shaded in, and then you add 1/5, or 1 piece, which equals a total of 3/5, or 3 out of 5 pieces, shaded in.



You can subtract the two fractions in the same way. When you subtract fractions, you subtract the numerators.

EXAMPLE

Referring to our original equation, 2 minus 1 is 1. Again, the denominators stay the same.



Looking at the bar representation, you start with 2 out of 5 pieces, or 2/5, shaded in, then subtract 1 piece (1/5), leaving 1 piece, or 1/5 of the bar, shaded in.

2. Adding and Subtracting Fractions with Uncommon Denominators

But what if you want to add or subtract two fractions that do not have the same denominator.

EXAMPLE

Suppose you want to add 1/2 and 1/4. Looking at the picture representation of these two fractions, you can see that there is a problem.



You can’t simply combine the pieces together as you did in the last example because the pieces are different sizes. However, if the denominators were the same, the pieces would be the same size, and you could add or subtract your numerators.

big idea

When adding or subtracting fractions with uncommon denominators, you need to convert them into equivalent fractions with common denominators.

Consider the last example again to see how you can find the least common denominator.

EXAMPLE

Let's keep our original addition problem.

	The expression
	The multiples of our first denominator (2) are 2, 4, 6, etc. The multiples of our second denominator (4) are 4, 8, 12, etc. The smallest common multiple is 4, so we can rewrite our fractions using 4 as our common denominator. To convert the denominator of 1/2 to 4, you multiply by 2 in the denominator and numerator. The second fraction can stay the same, as it already has a denominator of 4.
	The first fraction is equivalent to .
	Finally, add together your numerators; 2 and 1 equals 3.

try it

Consider the expression .

Evaluate the expression to find the solution.

	The expression
	Find the least common denominator. The multiples of our first denominator (3) are 3, 6, 9, 12, 15, etc. The multiples of our second denominator (5) are 5, 10, 15, 20, 25, etc. The smallest common multiple is 15, so we can rewrite our fractions using 15 as our common denominator. To convert the denominator of 1/3 to 15, you multiply by 5 in the denominator and numerator. To convert the denominator of 1/5 to 15, you multiply by 3 in the denominator and numerator.
	The first fraction can be rewritten as and the second fraction can be rewritten as .
	Finally, subtract your numerators. 5 minus 3 equals 2.

hint

Do you have a panic attack whenever you hear “least common denominator?” When in doubt, you can always just multiply your denominators together and then simplify your answer. To do this, we can use the Butterfly Method to add or subtract fractions with unlike denominators.

Suppose you were evaluating .



With the Butterfly Method, you would first multiply the denominators of the two fractions together (2 x 4). This will be your common denominator (8). Then, you multiply the numerator of the first fraction with the denominator of the second fraction (1 x 4). This will be the numerator of your first common denominator fraction. To find the numerator of the second common denominator fraction, multiply the denominator of the first fraction with the numerator of the second fraction (2 x 3). You will now have two new fractions with a common denominator (4/8) and (6/8).

Remember, this method of finding a common denominator will always work, but it may not give you the least or smallest common denominator, and some simplification might be necessary at the end, which you will learn in this next section.

3. Simplifying Fractions

So far, we’ve been adding or subtracting smaller fractions, but there are times once we’re done with our math that we could come up with a fraction like 50/100. That’s really large and could be simplified into a smaller number. Think about it this way. This means you have sliced a pizza into 100 slices and then ate 50 of them...and nobody would believe you ate 50 slices. Simply put, a simplified fraction means you’ve cut the pizza into as few pieces as possible. Mathematically, this means that a simplified fraction is a fraction in which the numerator and the denominator have no common factors other than 1.

You always want to write fractions in their simplest form so they are easier to compare and calculate with. And you may remember from your school days that a solution that has not been simplified is considered wrong, so let’s cover how we can simplify.

EXAMPLE

50/100 can be simplified to 1/2, because both the numerator and the denominator are divisible by 50. It’s much easier to say that you ate half of your pizza rather than fifty-hundredths of your pizza!

EXAMPLE

Simplify the fraction .

	The fraction
	See if the numerator and denominator have any common factors. Expand both numbers into their prime factors. In this case, the prime factors of 6 are 3 and 2. The prime factors of 8 are 2, 2, and 2.
	Since there is at least one 2 in both the numerator and the denominator, you can cancel them out.
	This leaves 3 in the numerator, and 2 times 2 in the denominator.
	Evaluate 2 times 2, which is 4. is the simplest form of .

hint

If the number is really large and you’re struggling to reduce your fraction to its simplest form, use the following tricks:

• Are both the numerator and denominator even numbers? You can always keep dividing each by 2 until one of the values is no longer even.
• Can you divide the numerator and denominator by 3? What about 5?
• Or, list out the factors of your denominator and numerator (i.e., 10 is 1, 2, 5, and 10) and see if your numerator and denominator have any of the same factors. If they do, divide by that value.

try it

Consider the following table of fractions:

Fraction	Simplified Fraction
	?
	?
	?

Find the simplified form of each of the above fractions.

Fraction	Simplified Fraction	Explanation
		The numbers 12 and 36 share many factors: 2, 3, 4, 6, 12. You can start by dividing each by the smaller factors, like 2 or 3, but since we know they share 12, we can divide them both by this number and end up with the most simplified form.
		Both 40 and 92 share a factor of 4, so you can divide by this number. We know this is the most simplified form because 10 and 23 do not share any factors.
		This is the most simplified form because 17 and 31 do not share any factors.

think about it

Did you notice anything weird about our answer in the Butterfly Method example? The solution was 5/4. Thinking back to fractions as pizzas, what does 5/4 mean?

• Numerator: number of pieces eaten
• Denominator: number of pieces a pizza is cut into

So 5/4 means that you’ve eaten 5 pieces, but you only cut the pizza into 4 pieces. How did you get that other piece?! Well, this means you ate all of your pizza, and some of someone else’s.



We can see that you’ve eaten 1 whole pizza and ¼ of another, or in other words: 1 ¼. But we’re not always going to draw out a pizza to see how to convert what we call an improper fraction (a numerator that is larger than the denominator) into a mixed fraction (a whole number and a fraction). To simplify an improper fraction into a mixed fraction, all you have to do is divide.



You have a remainder of 1 after you’ve divided. That 1 is our numerator and what we divided by (4) is our denominator. So, we have 1 ¼.

Let’s try some other problem where we can put all of our tips and tricks together by adding or subtracting fractions and making sure we have a simplified fraction as the answer.

EXAMPLE

You have a 1 inch ribbon. You cut off a ¼-inch piece of ribbon and then cut off a ⅜-inch piece of ribbon. How much ribbon did you cut? How much of your 1 inch ribbon do you have left?

	We need to add ¼ and ⅜ together to determine how much ribbon has been cut off.
	The least common denominator for both fractions is 8. We multiply the numerator and denominator of the first fraction (¼) by 2 to equal 2/8. Now both fractions have the same denominator.
	We add the numerators of the fractions to get ⅝. So, ⅝ inches was cut from the ribbon.
	To find out how much ribbon is left, we would subtract ⅝ inch from 1 inch.
	We can rewrite the whole number 1 as a fraction, by placing the whole number over 1. So, 1 becomes 1/1.
	We will find a common denominator between 1 and 8. This will be 8. We multiply the numerator and denominator of the first fraction by 8.
	Now that both fractions have the same denominator, we can subtract the numerators. There is ⅜ inch of our 1 inch of ribbon left.

try it

You have pound of sugar and your friend gives you pounds of sugar.

How much sugar do you have (in pounds)?

	You will need to add and together.
	The least common factor of the two fractions is 6. We can use the butterfly method or multiply the numerator of the first fraction by 3 and the numerator and the denominator of the second fraction by 2.
	Adding the numerators together gives us the fraction . This solution is an improper fraction.
	We can simplify this by converting our solution into a mixed fraction. We do this by dividing 6 into 7. We get 1 with a remainder of 1.
	We now have 1 ⅙ pounds of sugar.