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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.
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a What fraction of the cake did he eat?
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b What fraction of the cake is left for others to eat?
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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.
Pop Quiz
Identifying Like and Unlike Fractions
Determine whether the given fractions are like fractions or unlike fractions.
Discussion
Adding and Subtracting Fractions
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.
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.
The least common denominator is the product of the highest power of each prime factor.
| Denominator | Prime Factorization |
|---|---|
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 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.
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.
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a What fraction of the berry pie did Dylan and his sister eat all together?
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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?
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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.
SplitIntoFactors
Split into factors
CancelCommonFac
Cancel out common factors
SimpQuot
Simplify quotient
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.
| Denominators | Prime Factorization |
|---|---|
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.
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What is the fraction of guests who are playing games? {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{21}{28}","\\dfrac{3}{4}"]}}
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 |
|---|---|---|
| Denominator | Prime Factorization | LCD |
|---|---|---|
SplitIntoFactors
Split into factors
CancelCommonFac
Cancel out common factors
SimpQuot
Simplify quotient
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.
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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.
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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.
| Simplify | ||
|---|---|---|
| Denominator | Prime Factorization | LCD |
|---|---|---|
▼
Write fraction as a mixed number
WriteSum
Write as a sum
WriteSumFrac
Write as a sum of fractions
CalcQuot
Calculate quotient
AddTerms
Add terms
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
Closure
Finding the Amount of Cake in Fractions
Their is a key point in adding and subtracting fractions. Make sure that the denominators of the fractions are the same. Additionally, the following three steps are applied to perform arithmetic operations on mixed numbers. Remember, such operations include addition and subtraction.
- Convert mixed numbers into improper fractions.
- Check if they are unlike fractions. Write them as equivalent fractions with the common denominator if they are.
- Perform the operation.
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a What fraction of the cake did he eat?
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b What fraction of the cake is left for others to eat?
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Hint
a Add the given fractions.
b Subtract the Part A's answer from
Solution
a The given fractions must be added to find the fraction of the cake that Dylan ate. Write the verbally expressed fractions in mathematical terms to get started.
| Denominator | Prime Factorization | LCD |
|---|---|---|
b Now, it is time to find the fraction of the cake that is remaining. Begin by subtracting from The value of represents the whole cake.
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