1. Add and Subtract Fractions
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Chapter 3
1.
Add and Subtract Fractions
This lesson delves into the intricacies of adding and subtracting fractions, focusing on both like and unlike fractions. It starts with real-world scenarios, such as Dylan eating a fraction of a berry pie, to make the concepts relatable. The lesson explains that fractions with the same denominator are called 'like fractions,' while those with different denominators are termed 'unlike fractions.' It then guides you through the process of converting unlike fractions into like fractions so that they can be added or subtracted easily. The lesson also touches on the importance of finding the least common denominator (LCD) when dealing with unlike fractions. It provides step-by-step methods for performing these operations, even when mixed numbers are involved. The aim is to make you comfortable with fraction operations through practical examples and straightforward explanations.
Show less Show more expand_more| Student Learning Objectives: |
|---|
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| | 10 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Add and Subtract Fractions
Slide of 10
This lesson covers fractions with the same and different denominators and how to add and subtract them.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Challenge
What Fraction of the Cake Did Dylan Eat?
Dylan's mom baked a vanilla-strawberry cake for his birthday party. Dylan 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
Like and Unlike Fractions
Fractions can be classified based on whether or not they have the same denominator.
Like Fractions
Two or more fractions that have the same denominator are called like fractions. For example, 38, 58, and 68 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 61, 111, and 431, respectively.
Unlike Fractions
Fractions with different denominators are called unlike fractions. For example, 28, 23, and 35 are unlike fractions because they have different denominators.
Fractions like 13, 26, and 39 are also unlike fractions, even though they all are equivalent to 13. This is because their denominators are different.
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Pop Quiz
Identifying Like and Unlike Fractions
Determine whether the given fractions are like fractions or unlike fractions.
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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
To add or subtract like fractions, add or subtract the numerators and keep the denominator the same.
a/c + b/c=a + b/c [1.3em] a/c - b/c=a - b/c
Adding and Subtracting Unlike Fractions
Unlike fractions must first be converted to like fractions when adding or subtracting. 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.
a/c + b/d=ad + bc/cd [1.3em] a/c - b/d=ad - bc/cd
Another way is to find the least common denominator (LCD) of the fractions. Consider the example of subtracting 712 from 415. 4/15 - 7/12 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.
4/15 - 7/12
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 |
The least common denominator is the product of the highest power of each prime factor. LCD: 2^2 * 3 * 5 = 60
2
Rewrite Each Fraction Using the LCD
expand_more The LCD is 60. Now the fractions will be multiplied by the appropriate factor to make their denominators equal to 60. The first fraction must be multiplied by 4 and the second fraction by 5 to get the LCD.
4/15 - 7/12
ExpandFrac
a/b=a * 4/b * 4
4 * 4/15 * 4- 7/12
Multiply
Multiply
16/60 - 7/12
ExpandFrac
a/b=a * 5/b * 5
16/60 - 7 * 5/12 * 5
Multiply
Multiply
16/60 - 35/60
3
Combine the Numerators
expand_more The numbers in the numerators can now be subtracted because both fractions have the same denominator.
16/60 - 35/60
SubFrac
Subtract fractions
16-35/60
SubTerm
Subtract term
- 19/60
MoveNegNumToFrac
Put minus sign in front of fraction
- 19/60
4
Simplify if Possible
expand_more Check if the resulting fraction can be simplified or not. Our fraction cannot be simplified because the numerator and denominator do not have any common factors.
- 19/60
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Example
A Piece of Pie
A berry pie and a peanut butter pie are served at a dinner party. The berry pie is cut into 10 equal pieces and the peanut butter pie is cut into 12 equal pieces.
External credits: Toa Heftiba
a
Dylan takes three berry pie pieces. Emily takes two pieces for herself. What fraction of the berry pie did they eat altogether?
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b
The party guests ate five of the peanut butter pie pieces. 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 Emily eat?
b
Use the answer found in Part A to find the total amount of pie eaten.
Solution
a
Dylan ate 3 pieces of the berry pie that was cut into 10 pieces, so he ate 310 of the pie. Emily eats another 2 pieces, or 210, of the same pie. The sum of these fractions gives the amount of pie eaten.
Amount of Pie Eaten 3/10 + 2/10 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, so to add them, we simply add their numerators and keep the denominator the same.
The denominator is a multiple of the numerator. Let's simplify the fraction. We will rewrite 10 as a product of its factors, then cancel out the common factors.
5/10
SplitIntoFactors
Split into factors
5/2 * 5
CancelCommonFac
Cancel out common factors
5/2 * 5
SimpQuot
Simplify quotient
1/2
Dylan and Emily ate 12 of the berry pie.
b
The party guests ate 5 pieces of the 12-piece peanut butter pie, so 512 of this pie was eaten. In Part A, we found that half of the berry pie was also eaten. The sum of these two fractions is the total amount of pie that was eaten at the dinner party.
Total Amount of Pie Eaten 1/2 + 5/12 To add these factions, we need to find equivalent fractions with a common denominator. Let's find the prime factors of these numbers to help us find the least common denominator.
| Denominators | Prime Factorization |
|---|---|
| 2 | 2 |
| 12 | 2^2 * 3 |
The least common denominator is 2^2 * 3, or 12. The second fraction already has a denominator of 12, so we will multiply the first fraction's numerator and denominator by 6. Then, let's add the fractions.
1/2+5/12
ExpandFrac
a/b=a * 6/b * 6
1 * 6/2 * 6+5/12
Multiply
Multiply
6/12+5/12
AddFrac
Add fractions
6+5/12
AddTerms
Add terms
11/12
The dinner party guests ate 1112 of a pie! We can also explain the process of addition visually. The brown part is equal to 11 of the 112 sized pieces.
Each slice represents 112 of a pie, so the 11 brown slices make 1112 of a pie eaten.
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Pop Quiz
Finding the Sum and Difference of Fractions
Perform the indicated operation. Simplify the result if possible.
Extra
Need an Extra Hand?We can only add and subtract fractions when they have the same denominators. Once we have a common denominator, combine the numerators and keep the denominator the same.
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Example
Students Playing Games
At a school festival, 328 of the students are dancing, while 17 of the students are just listening to the music. The remaining students are playing games. What fraction of students are playing games?
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Hint
Subtract the total fraction of the students who are listening to music and dancing from the whole, or 1.
Solution
The given fractions 328 and 17 represent the portion of the students who are dancing and listening to music, respectively. The whole, or 1, represents all students at the festival.
| All Students | Fraction of the Students Who Are Dancing | Fraction of the Students Who Are Listening to Music |
|---|---|---|
| 1 | 3/28 | 1/7 |
Let's subtract the sum of the two fractions from 1 to find the fraction of students who are playing games. 1 - ( 3/28 + 1/7 ) Every integer has 1 as its denominator. 1/1 - ( 3/28 + 1/7 ) The fractions are unlike fractions because they have different denominators. The fractions must have a common denominator to add or subtract them. Let's find the prime factors of the denominators so we can determine the least common denominator.
| Denominator | Prime Factorization | LCD |
|---|---|---|
| 1 | 1 | 2^2 * 7 =28 |
| 28 | 2^2 * 7 | |
| 7 | 7 |
Next, let's convert the fractions into equivalent fractions with a denominator of 28.
1/1 - ( 3/28 + 1/7 )
▼
Rewrite
ExpandFrac
a/b=a * 28/b * 28
1* 28/1 * 28 - ( 3/28 + 1/7 )
MultByOne
a * 1=a
28/28 - ( 3/28 + 1/7 )
ExpandFrac
a/b=a * 4/b * 4
28/28 - ( 3/28 + 1 * 4/7 * 4 )
Multiply
Multiply
28/28 - ( 3/28 + 4/28 )
Now we can evaluate the expression.
28/28 - ( 4/28 + 3/28 )
AddFrac
Add fractions
28/28 - ( 4+3/28)
AddTerms
Add terms
28/28 - 7/28
SubFrac
Subtract fractions
28-7/28
SubTerm
Subtract term
21/28
The numerator and denominator of this fraction have a common factor of 7. Let's simplify it.
21/28
SplitIntoFactors
Split into factors
3 * 7/4 * 7
CancelCommonFac
Cancel out common factors
3 * 7/4 * 7
SimpQuot
Simplify quotient
3/4
The fraction of the students playing games is 34.
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Example
Finding the Cups of Ingredients
A cake recipe calls for 2 14 cups of flour, 1 23 cups of sugar, and 12 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
Suppose someone uses 2 13 cups of sugar instead. How much extra sugar did they 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 1 23 from 2 13.
Solution
a
We want to find the sum of the given mixed numbers and fraction.
2 14 + 1 23 + 1/2 We will start by converting the mixed numbers into improper fractions.
| a bc | a* c+b/c | Simplify |
|---|---|---|
| 2 14 | 2* 4+1/4 | 9/4 |
| 1 23 | 1* 3+2/3 | 5/3 |
So, we want to find the value of the following expression. 9/4 + 5/3 + 12 Since the denominators are different, our fractions are unlike fractions. We will convert them to like fractions before adding them. Let's find the least common denominator of the three fractions.
| Denominator | Prime Factorization | LCD |
|---|---|---|
| 4 | 2^2 | 2^2 * 3 = 12 |
| 3 | 3 | |
| 2 | 2 |
We can write the given fractions as equivalent fractions with a denominator of 12. Now let's evaluate the expression.
9/4 + 5/3 + 1/2
▼
Rewrite
ExpandFrac
a/b=a * 3/b * 3
9 * 3/4 * 3 + 5/3 + 1/2
ExpandFrac
a/b=a * 4/b * 4
9 * 3/4 * 3 + 5* 4/3 * 4 + 1/2
ExpandFrac
a/b=a * 6/b * 6
9 * 3/4 * 3 + 5* 4/3 * 4 + 1 * 6/2 * 6
Multiply
Multiply
27/12 + 20/12 + 6/12
AddFrac
Add fractions
27+20+6/12
AddTerms
Add terms
53/12
Finally, let's write our fraction as a mixed number.
53/12
▼
Write fraction as a mixed number
WriteSum
Write as a sum
48 + 5/12
WriteSumFrac
Write as a sum of fractions
48/12+5/12
CalcQuot
Calculate quotient
4+5/12
AddTerms
Add terms
4 512
A total of 4 512 cups of ingredients are needed to bake the cake.
b
This time someone used 2 13 cups of sugar to bake the cake. That amount is greater than the recipe calls for. Let's rewrite 2 13 as an improper fraction so that we can compare it to 1 23, or 53.
2 13
▼
Write mixed number as a fraction
7/3
The numerator of 73 is greater than the numerator of 53, so 73 is greater than 53. 7/3 > 5/3 We can use these fractions to find how much extra sugar was used. Let's subtract 53 from 73.
An extra 23 cup of sugar was used.
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Closure
Finding the Amount of Cake in Fractions
Consider the challenge presented at the beginning of the lesson. [-1.1em] Dylan eats $ 110$ of his birthday cake before the party starts. He also eats $ 215$ 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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Hint
a
Add the given fractions.
b
Subtract the Part A's answer from 1.
Solution
a
Let's add the given fractions to find the fraction of the cake that Dylan ate. The fractions do not have a common denominator, so we cannot add them immediately. We will find their least common denominator by splitting the denominators into their prime factors.
| Denominator | Prime Factorization | LCD |
|---|---|---|
| 10 | 2 * 5 | 2 * 3 * 5 = 30 |
| 15 | 3 * 5 |
Let's find equivalent fractions with denominator 30 and add them.
1/10 + 2/15
▼
Rewrite
ExpandFrac
a/b=a * 3/b * 3
1 * 3/10 * 3 + 2/15
ExpandFrac
a/b=a * 2/b * 2
1 * 3/10 * 3 + 2* 2/15 * 2
Multiply
Multiply
3/30 + 4/30
AddFrac
Add fractions
3+4/30
AddTerms
Add terms
7/30
Dylan ate 730 of the cake.
b
Now we want to find the fraction of the cake that is remaining. From Part A we know that Dylan ate 730 of the cake, so let's subtract 730 from 1. Here, 1 or 3030 represents one whole cake.
1- 7/30
Rewrite
Rewrite 1 as 30/30
30/30 - 7/30
SubFrac
Subtract fractions
30-7/30
SubTerm
Subtract term
23/30
Dylan ate 730 of the cake, so 2330 of the cake remains for his guests.
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Add and Subtract Fractions
Exercise 1.1
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{"type":"choice","form":{"alts":["Like fractions","Unlike fractions"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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Let's compare the denominators of the fractions. 4/14 and 4/24 We see that one fraction has a denominator of 14 and the other has a denominator of 24. The fractions do not have the same denominator. Therefore, we can say that they are unlike fractions.
Let's first determine which fractions the diagrams represent. Each diagram consists of 10 parts. The number of shaded parts is 3 for the diagram on the left and 7 for the other diagram.
The diagram on the left with three shaded parts represents 310. The diagram on the right with seven shaded regions parts 710. Both fractions have the same denominator. Therefore, they are like fractions.
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We see that both fractions have a denominator of 12. 7/12 + 5/12 The fractions are like fractions. The sum of the fractions is the sum of the numerators divided by 12.
We can simplify this fraction to 1 because the numerator and denominator are the same. 12/12 = 1
We want to find the difference of two fractions that have the same denominator.
15/16-3/16
We subtract the numerators to find this difference.
The numerator and denominator have a common factor, which is 4. We can simplify the fraction. Let's rewrite the numerator and denominator as a product of its factors. Then we will cancel out the common factors.
The difference of the fractions is 34.
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Solution
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We see that the fractions have different denominators. 11/12 - 5/8 To subtract fractions with different denominators, we need to rewrite the fractions as equivalent fractions with the least common denominator. We need to determine the prime factors of the denominators to find the least common denominator.
| Denominators | Prime Factorization |
|---|---|
| 12 | 2^2 * 3 |
| 8 | 2^3 |
The least common denominator is the product of the highest power of each factor. Least Common Denominator 2^3 * 3 = 24 We can now rewrite the denominators. We will multiply the numerator and the denominator of the first fraction by 2. Then, we will multiply the numerator and the denominator of the second fraction by 3. Once the fractions have the same denominator, we can subtract them.
The difference of the fractions is 724.
We want to find the sum of two fractions that are unlike fractions.
7/8+1/2
To add fractions with different denominators, we need to rewrite the fractions as equivalent fractions with the least common denominator. To do so, let's first determine the prime factors of the denominators.
| Denominators | Prime Factorization |
|---|---|
| 8 | 2^3 |
| 2 | 2 |
The least common denominator is the product of the highest power of each factor. Least Common Denominator 2^3 = 8 We see that the first fraction already has this as its denominator. We need to rewrite the second fraction. To do so, we will multiply the numerator and the denominator of it by 4. Once the fractions have the same denominator, we can add them.
The sum of the fractions is an improper fraction, 118. We can write it as a mixed number.
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Solution
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We want to evaluate the sum of the mixed numbers. - 4 34 + ( - 3 14 ) We will start by rewriting the mixed numbers as improper fractions.
| - a bc | - (a* c+b/c ) | Simplify |
|---|---|---|
| - 4 34 | - (4* 4+3/4 ) | - 19/4 |
| - 3 14 | - (3* 4+1/4 ) | - 13/4 |
The mixed numbers are converted into improper fractions. - 4 34 + ( - 3 14 ) ⇕ - 19/4 + (- 13/4 ) Fractions should have the same denominator when we add or subtract them. In this case, the fractions have the same denominators. That means the sum of the fractions is the sum of the numerators divided by 4.
The sum of the mixed number is - 8.
We want to find the given difference.
3 15 - 5 13
Again, we will start by rewriting the mixed numbers as improper fractions.
| a bc | a* c+b/c | Simplify |
|---|---|---|
| 3 15 | 3* 5+1/5 | 16/5 |
| 5 13 | 5* 3+1/3 | 16/3 |
The mixed numbers are converted into improper fractions. 3 15 - 5 13 ⇕ 16/5 - 16/3 Fractions should have the same denominator when they go through addition or subtraction. In this expression, the fractions have different denominators. We multiply the numerator and denominator of each fraction by the denominator of the other to make them like fractions. Let's do it.
We will rewrite the numerator as 30 plus 2 with the aim of writing the fraction as a mixed number.
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Solution
Evaluate the expression. (- 3/8 + 2/11 ) + 11/8
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We see that the expression contains two like fractions, - 38 and 118. Let's write these fractions next to each other. We will use the Commutative Property of Addition to do this. Also, we can change the order of the fractions in the parentheses by the same property.
Next, we can group the last two fractions by the Associative Property of Addition.
We can now add the like fractions.
The value of the expression is 1 211.
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Solution
Add and Subtract Fractions
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