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.
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Lesson Settings & Tools
| | 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
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.
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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?
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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, 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. The denominators of these fractions are 8, 3, and 5, which are different from each other.
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
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.
a/c + b/c=a + b/c [0.8em] a/c - b/c=a - b/c
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.
a/c + b/d=ad + bc/cd [0.8em] 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 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.
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
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.
- 19/60
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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 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.
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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 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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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 3 pieces of the berry pie that was cut into 10 pieces. In other words, 310 of the pie is gone. Then, his sister 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. 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.
5/10
SplitIntoFactors
Split into factors
5/2 * 5
CancelCommonFac
Cancel out common factors
5/2 * 5
SimpQuot
Simplify quotient
1/2
Dylan and his sister ate 12 of the berry pie.
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 1/2 + 5/12 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 |
The least common denominator is 2^2 * 3. That can also be expressed as 12. The second fraction already has this as its denominator. Thus, only the first fraction must be adjusted. Multiply its numerator and denominator by 6. The fractions can be added once they have the same denominator.
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
Dylan, his sister, and their guests ate a total of what would amount to be 1112 of a pie! The process of addition can also be explained visually. The brown parts is equal to 11 of the 112 sized pieces.
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 1112 of a pie as shown in the diagram.
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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.
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Example
Finding the Fraction of Guests Playing Games
The birthday party is going great. Right now 328 of the guests are dancing, while 17 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?
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Hint
Subtract the total fraction of the guests who are listening to music and dancing from the whole, or 1.
Solution
The given fractions 328 and 17 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 | 3/28 | 1/7 |
Now, subtract the sum of the two fractions from 1 to find the fraction of guests who are playing games. 1 - ( 3/28 + 1/7 ) Recall that every integer has 1 as its denominator. 1/1 - ( 1/7 + 3/28 ) 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 |
The next step is to convert the fractions into equivalent fractions with a denominator of 28.
1/1-( 1/7 +3/28 )
▼
Rewrite
ExpandFrac
a/b=a * 28/b * 28
1* 28/1 * 28 - ( 1/7 + 3/28 )
MultByOne
a * 1=a
28/28 - ( 1/7 + 3/28 )
ExpandFrac
a/b=a * 4/b * 4
28/28 - ( 1 * 4/7 * 4 + 3/28 )
Multiply
Multiply
28/28 - ( 4/28 + 3/28 )
Now the value of the resulting expression can be evaluated.
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
Notice the similarities in 21 = 3 * 7 and 28 = 4 * 7. The numerator and denominator of the fraction have a common factor of 7. This means that the result can be simplified.
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 guests playing games is 34.
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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 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 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 2 13 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 1 23 from 2 13.
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.
2 14 + 1 23 + 1/2 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 |
The mixed numbers are converted into improper fractions. 9/4 + 5/3 + 12 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 |
The given fractions can be written as equivalent fractions with a denominator of 12. Next, the value of the expression can be evaluated.
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
This fraction now needs to be written as a mixed number. Begin by expressing the numerator as a multiple of 12 plus a 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 The guest used 2 13 cups of sugar. That amount is greater than the amount Dylan's mom used. Rewrite 2 13 as an improper fraction so that it can be compared with 1 23, or equivalently 53.
2 13
▼
Write mixed number as a fraction
7/3
The numerator of 73 is greater than the numerator of 53. Therefore, 73 is greater than 53. 7/3 > 5/3 These fractions can now be used to find how much extra sugar Paulina's mother used. Subtract 53 from 73.
The guest used 23 cups more sugar.
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Closure
Finding the Amount of Cake in Fractions
A key point in adding and subtracting fractions is to 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.
Now, consider the challenge presented at the beginning of the lesson. [-1.1em] Dylan eats one-tenth of his birthday cake before the party 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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Hint
a Add the given fractions.
b Subtract the Part A's answer from 1.
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.
One-tenth & → 1/10 [1em] Two-fifteenths & → 2/15 The fractions do not have a common denominator. The fractions should have the same denominator to add them. Finding the least common denominator of the fractions will be helpful at this point. The denominators can then be written as products of their prime factors.
| Denominator | Prime Factorization | LCD |
|---|---|---|
| 10 | 2 * 5 | 2 * 3 * 5 = 30 |
| 15 | 3 * 5 |
The given fractions can be written as equivalent fractions with denominator 30.
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, it is time to find the fraction of the cake that is remaining. Begin by subtracting 730 from 1. The value of 1 represents the whole cake.
1- 7/30 Start by rewriting the number 1 as 3030 to perform the subtraction. The difference can be found after that.
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, and 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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Solution
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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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