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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

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?

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b What fraction of the cake is left for others to eat?

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Discussion

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

Two or more fractions that have the same denominator are called like fractions. For example, $83 ,$ $85 ,$ and $86 $ are all like fractions whose denominators are $8.$

Integers such as $6,$ $11,$ and $43$ are like fractions because they are considered to have a denominator of $1.$ Their fraction forms are $16 ,$ $111 ,$ and $143 ,$ respectively.

Fractions with different denominators are called unlike fractions. For example, $82 ,$ $32 ,$ and $53 $ are unlike fractions. The denominators of these fractions are $8,$ $3,$ and $5,$ which are different from each other.

Fractions like $31 ,$ $62 ,$ and $93 $ are also unlike fractions, even though they all are equivalent to $31 .$ 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

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.

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.

$ca +cb =ca+b ca −cb =ca−b $

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.

$ca +db =cdad+bc ca −db =cdad−bc $

$154 −127 $

The result can be found in four steps.
1

Find the Least Common Denominator

The least common denominator is the least common multiple of the numbers in the denominators.

The least common denominator is the product of the highest power of each prime factor.

$154 −127 $

The numbers need to be expressed as a product of their prime factors to find their LCM. Denominator | Prime Factorization |
---|---|

$15$ | $3⋅5$ |

$12$ | $2_{2}⋅3$ |

$LCD:2_{2}⋅3⋅5=60 $

2

Rewrite Each Fraction Using the LCD

The LCD was found to be $60.$ Now the fractions will be multiplied by the appropriate factors to make the denominators equal to $60.$ The first fraction must be multiplied by $4$ and the second fraction by $5$ to get the LCD.

$154 −127 $

ExpandFrac

$ba =b⋅4a⋅4 $

$15⋅44⋅4 −127 $

Multiply

Multiply

$6016 −127 $

ExpandFrac

$ba =b⋅5a⋅5 $

$6016 −12⋅57⋅5 $

Multiply

Multiply

$6016 −6035 $

3

Subtract the Numerators

4

Simplify if Possible

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.

$60-19 $

Example

Dylan's mother serves a berry pie, a peanut butter pie, and cake at the birthday party. She cuts the berry pie into $10$ 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?

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b Dylan's mother cut the peanut butter pie into $12$ 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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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.

a Dylan eats $3$ pieces of the berry pie that was cut into $10$ pieces. In other words, $103 $ of the pie is gone. Then, his sister eats another $2$ pieces, or $102 ,$ of the same pie. The sum of these fractions gives the amount of pie eaten.

$Amount of Pie Eaten 103 +102 $

Notice that both of these fractions have a denominator of $10.$ 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 $10.$
The denominator is a multiple of the numerator. That is, $10$ is a multiple of $5.$ This means the numerator and denominator have a common factor. Now, simplify the fraction. Start by rewriting $10$ as a product of its factors. Then, cancel out the common factors.
$105 $

SplitIntoFactors

Split into factors

$2⋅55 $

CancelCommonFac

Cancel out common factors

$2⋅5 5 $

SimpQuot

Simplify quotient

$21 $

b The party guests ate $5$ pieces of the $12-$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.

$Total Amount of Pie Eaten 21 +125 $

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 |
---|---|

$2$ | $2$ |

$12$ | $2_{2}⋅3$ |

$21 +125 $

ExpandFrac

$ba =b⋅6a⋅6 $

$2⋅61⋅6 +125 $

Multiply

Multiply

$126 +125 $

AddFrac

Add fractions

$126+5 $

AddTerms

Add terms

$1211 $

Every $12$ slices make a whole pie and there are $24$ slices or $2$ pies. When $11$ of those $24$ slices are eaten, they make $1211 $ of a pie as shown in the diagram.

Pop Quiz

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

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

The birthday party is going great. Right now $283 $ of the guests are dancing, while $71 $ of the guests are just listening to the music, and the others are playing games.

External credits: gpointstudio

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Subtract the total fraction of the guests who are listening to music and dancing from the whole, or $1.$

The given fractions $283 $ and $71 $ represent the fraction of the guests who are dancing and listening to music, respectively. The whole or $1$ represents all guests at the party.

All Guests | Fraction of the Guests Who Are Dancing | Fraction of the Guests Who Are Listening to Music |
---|---|---|

$1$ | $283 $ | $71 $ |

$1−(283 +71 ) $

Recall that every integer has $1$ as its denominator. $11 −(71 +283 ) $

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 |
---|---|---|

$1$ | $1$ | $2_{2}⋅7=28$ |

$7$ | $7$ | |

$28$ | $2_{2}⋅7$ |

$11 −(71 +283 )$

▼

Rewrite

ExpandFrac

$ba =b⋅28a⋅28 $

$1⋅281⋅28 −(71 +283 )$

MultByOne

$a⋅1=a$

$2828 −(71 +283 )$

ExpandFrac

$ba =b⋅4a⋅4 $

$2828 −(7⋅41⋅4 +283 )$

Multiply

Multiply

$2828 −(284 +283 )$

$2828 −(284 +283 )$

AddFrac

Add fractions

$2828 −(284+3 )$

AddTerms

Add terms

$2828 −287 $

SubFrac

Subtract fractions

$2828−7 $

SubTerm

Subtract term

$2821 $

$2821 $

SplitIntoFactors

Split into factors

$4⋅73⋅7 $

CancelCommonFac

Cancel out common factors

$4⋅7 3⋅7 $

SimpQuot

Simplify quotient

$43 $

Example

A guest at the party asks Dylan's mother for the basic recipe of the cake. The recipe she is using calls for $241 $ cups of flour, $132 $ cups of sugar, and $21 $ 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 $231 $ cups of sugar. How much extra sugar did she use compared to the original recipe? Simplify the answer if possible.

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

b Subtract $132 $ from $231 .$

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.

$241 +132 +21 $

Start by converting the mixed numbers into improper fractions. $acb $ | $ca⋅c+b $ | Simplify |
---|---|---|

$241 $ | $42⋅4+1 $ | $49 $ |

$132 $ | $31⋅3+2 $ | $35 $ |

$49 +35 +21 $

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 |
---|---|---|

$4$ | $2_{2}$ | $2_{2}⋅3=12$ |

$3$ | $3$ | |

$2$ | $2$ |

$49 +35 +21 $

▼

Rewrite

ExpandFrac

$ba =b⋅3a⋅3 $

$4⋅39⋅3 +35 +21 $

ExpandFrac

$ba =b⋅4a⋅4 $

$4⋅39⋅3 +3⋅45⋅4 +21 $

ExpandFrac

$ba =b⋅6a⋅6 $

$4⋅39⋅3 +3⋅45⋅4 +2⋅61⋅6 $

Multiply

Multiply

$1227 +1220 +126 $

AddFrac

Add fractions

$1227+20+6 $

AddTerms

Add terms

$1253 $

$1253 $

▼

Write fraction as a mixed number

WriteSum

Write as a sum

$1248+5 $

WriteSumFrac

Write as a sum of fractions

$1248 +125 $

CalcQuot

Calculate quotient

$4+125 $

AddTerms

Add terms

$4125 $

b The guest used $231 $ cups of sugar. That amount is greater than the amount Dylan's mom used. Rewrite $231 $ as an improper fraction so that it can be compared with $132 ,$ or equivalently $3<$