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Fractions with the same or different denominators will be named in this lesson. Additionally, procedures for adding and subtracting fractions will be explored.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

## What Fraction of the Cake Did Dylan Eat?

Dylan is ecstatic to celebrate his birthday party. His mom baked a vanilla-strawberry cake for the party. Dylan cannot control himself and eats one-tenth of the cake before the party even starts. He also eats two-fifteenths of the cake during the party.

a What fraction of the cake did he eat?
b What fraction of the cake is left for others to eat?
Discussion

## Like and Unlike Fractions

Fractions can be classified based on whether they have the same denominator or not.

### Like Fractions

Two or more fractions that have the same denominator are called like fractions. For example, and are all like fractions whose denominators are

Integers such as and are like fractions because they are considered to have a denominator of Their fraction forms are and respectively.

### Unlike Fractions

Fractions with different denominators are called unlike fractions. For example, and are unlike fractions. The denominators of these fractions are and which are different from each other.

Fractions like and are also unlike fractions, even though they all are equivalent to This is because their denominators are different.

The diagram shows pies representing equivalent fractions. The fractions have the same value. However, they are unlike fractions because the pies are divided into different numbers of pieces.
Pop Quiz

## Identifying Like and Unlike Fractions

Determine whether the given fractions are like fractions or unlike fractions.

Discussion

The first step in adding and subtracting fractions is to check if they share the same denominator. Here, the methods of performing these operations on fractions and how to convert unlike fractions to like fractions will be discussed with examples.

### Adding and Subtracting Like Fractions

The numerators of like fractions are added or subtracted when finding the sum or difference of the like fractions, respectively. The denominator remains the same in these situations.

### Adding and Subtracting Unlike Fractions

Unlike fractions must first be converted to like fractions when the operation deals with the sum or difference. One way to convert them is to multiply the numerator and denominator of each fraction by the denominator of the other. Then, the given operation can be performed.

Another way is to find the least common denominator (LCD) of the fractions. Consider the example of subtracting from
The result can be found in four steps.
1
Find the Least Common Denominator
expand_more
The least common denominator is the least common multiple of the numbers in the denominators.
The numbers need to be expressed as a product of their prime factors to find their LCM.
Denominator Prime Factorization
The least common denominator is the product of the highest power of each prime factor.
2
Rewrite Each Fraction Using the LCD
expand_more
The LCD was found to be Now the fractions will be multiplied by the appropriate factors to make the denominators equal to The first fraction must be multiplied by and the second fraction by to get the LCD.
3
Subtract the Numerators
expand_more
The numbers in the numerators can now be subtracted because both fractions have the same denominator.
4
Simplify if Possible
expand_more
Check if the resulting fraction can be simplified or not. It cannot be simplified since the numerator and denominator do not have any common factors.
The process for adding unlike fractions is similar to the just performed example of subtracting unlike fractions.
Example

## How Much Pie Is Left?

Dylan's mother serves a berry pie, a peanut butter pie, and cake at the birthday party. She cuts the berry pie into equal pieces. Dylan takes three berry pie pieces. Dylan's sister smells the sweet aroma of the berry pie and takes two pieces for herself.

External credits: Toa Heftiba
a What fraction of the berry pie did Dylan and his sister eat all together?
b Dylan's mother cut the peanut butter pie into equal pieces. Party guests only ate five of the peanut butter pie pieces. Dylan and his sister did not eat any of the peanut butter pie. What fraction of the total amount of pie, both berry and peanut butter, was eaten at the party?

### Hint

a What fraction of the berry pie did Dylan eat? What fraction of the pie did his sister eat?
b Use the answer found in Part A to find the total amount of pie eaten.

### Solution

a Dylan eats pieces of the berry pie that was cut into pieces. In other words, of the pie is gone. Then, his sister eats another pieces, or of the same pie. The sum of these fractions gives the amount of pie eaten.
Notice that both of these fractions have a denominator of This denominator represents the whole pie's number of slices. These fractions are like fractions. Therefore, the sum of these fractions will be the sum of the numerators divided by
The denominator is a multiple of the numerator. That is, is a multiple of This means the numerator and denominator have a common factor. Now, simplify the fraction. Start by rewriting as a product of its factors. Then, cancel out the common factors.
Dylan and his sister ate of the berry pie.
b The party guests ate pieces of the piece peanut butter pie. In other words, five-twelfths of this pie was eaten. In Part A, it was found that half of the berry pie was also eaten. The sum of these two amounts is the total amount of pie that was eaten at the party.
Recall what to do when adding fractions with different denominators. The fractions must be expressed as equivalent fractions with the least common denominator. The prime factors of these numbers must be found first.
Denominators Prime Factorization
The least common denominator is That can also be expressed as The second fraction already has this as its denominator. Thus, only the first fraction must be adjusted. Multiply its numerator and denominator by The fractions can be added once they have the same denominator.
Dylan, his sister, and their guests ate a total of what would amount to be of a pie! The process of addition can also be explained visually. The brown parts is equal to of the sized pieces.
Every slices make a whole pie and there are slices or pies. When of those slices are eaten, they make of a pie as shown in the diagram.
Pop Quiz

## Finding the Sum and Difference of Fractions

Perform the indicated operation. Simplify the result if it is possible.

### Extra

Need an Extra Hand?

Fractions must have the same denominators when adding and subtracting two or more fractions. What if two fractions with equal denominators are given? In that case, the denominator is kept the same and the sum or difference of the numerators is written into the numerator.

Example

## Finding the Fraction of Guests Playing Games

The birthday party is going great. Right now of the guests are dancing, while of the guests are just listening to the music, and the others are playing games.

External credits: gpointstudio
What is the fraction of guests who are playing games?

### Hint

Subtract the total fraction of the guests who are listening to music and dancing from the whole, or

### Solution

The given fractions and represent the fraction of the guests who are dancing and listening to music, respectively. The whole or represents all guests at the party.

All Guests Fraction of the Guests Who Are Dancing Fraction of the Guests Who Are Listening to Music
Now, subtract the sum of the two fractions from to find the fraction of guests who are playing games.
Recall that every integer has as its denominator.
The fractions are unlike fractions because they have different denominators. The fractions must have a common denominator to perform the operations. The prime factors of the denominators must be found first for this purpose. The least common denominator can then be determined.
Denominator Prime Factorization LCD
The next step is to convert the fractions into equivalent fractions with a denominator of
Rewrite
Now the value of the resulting expression can be evaluated.
Notice the similarities in and The numerator and denominator of the fraction have a common factor of This means that the result can be simplified.
The fraction of the guests playing games is
Example

## Finding the Cups of Ingredients

A guest at the party asks Dylan's mother for the basic recipe of the cake. The recipe she is using calls for cups of flour, cups of sugar, and cup of oil.

a What is the total amount of the listed ingredients needed to bake the cake? Write the answer as a mixed number.
b The guest wants to make the cake a few days later. However, she forgot how many cups of sugar to use. She decides to use cups of sugar. How much extra sugar did she use compared to the original recipe? Simplify the answer if possible.

### Hint

a Start by converting the given mixed numbers into improper fractions.
b Subtract from

### Solution

a The sum of the given mixed numbers and the fraction must be found to find the total number of cups of ingredients required for the cake.
Start by converting the mixed numbers into improper fractions.
Simplify
The mixed numbers are converted into improper fractions.
Fractions should have the same denominator when adding or subtracting them. The fractions have different denominators in this expression. That means they should be made into like fractions before performing the operation. Find the least common denominator to do that.
Denominator Prime Factorization LCD
The given fractions can be written as equivalent fractions with a denominator of Next, the value of the expression can be evaluated.
Rewrite
This fraction now needs to be written as a mixed number. Begin by expressing the numerator as a multiple of plus a number.
Write fraction as a mixed number
A total of cups of ingredients are needed to bake the cake.
b The guest used cups of sugar. That amount is greater than the amount Dylan's mom used. Rewrite as an improper fraction so that it can be compared with or equivalently