2. Multiply Fractions
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Chapter 3
2.
Multiply Fractions
This lesson delves into the methods for multiplying fractions and mixed numbers, as well as simplifying fractions. It covers various techniques, including the use of area models and number lines, to make the concepts more understandable. The lesson also emphasizes the importance of estimation strategies to assess the reasonableness of answers. Practical examples are provided to demonstrate how these mathematical concepts can be applied in everyday situations like calculating time spent on activities or determining the area of objects.
| Student Learning Objectives: |
|---|
|
| | 10 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Multiply Fractions
Slide of 10
This lesson will cover methods for multiplying fractions by whole numbers, fractions, and mixed numbers.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Challenge
Running Time
Paulina runs for two-sixths of her free time each day.
External credits: pikisuperstar
If Paulina has 5 hours of free time per day, how many hours does she run in 4 days? Give the exact answer.
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Discussion
Multiplying Fractions
The product of two fractions is equal to the product of the numerators divided by the product of the denominators. The resulting fraction is then simplified to its lowest terms, if possible.
a/b * c/d = a * c/b * d
Here, b and d cannot equal 0. When multiplying fractions, it makes no difference whether they are like or unlike fractions. Consider multiplying 56 by 34. 5/6 * 3/4 Follow three steps to find the product.
1
Multiply the Numerators
expand_more 2
Multiply the Denominators
expand_more The product of the denominators is 6 * 4 = 24.
3
Simplify if Possible
expand_more Note that 3 is the greatest common factor of 15 and 24. Divide both the numerator and the denominator by 3 to simplify the fraction.
Therefore, the product of 56 and 34 is 58 when simplified to its lowest terms.
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Example
Finding What Fraction of the Water in a Bottle Is Drunk
Paulina drinks 13 of the water in her bottle before PE class. During class, she drinks 1012 of the remaining water.
a
Write a numeric expression that represents the part of the bottle of water Paulina drinks during class.
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b
Which of the following is the best estimate for the value of the expression written in Part A?
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c
Find the value of the expression found in Part A. Simplify the answer if possible.
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Hint
a
What fraction of the bottle is full before the class? What mathematical operation must be performed to find 1012of a number?
b
If a fraction is greater than or equal to 12, round the fraction to 1. If a fraction is less than 12, round it to 0.
c
To multiply fractions, multiply the numerators and the denominators separately. Then, simplify the resulting fractions.
Solution
a
First, we find what portion of the bottle is full before the PE class by subtracting 13 from 1.
Next, we know that Paulina drinks 1012 of the 23 of the water in the bottle. To find this amount, we multiply these fractions. 10/12 * 2/3
b
Now, we want to estimate the value of the expression 1012 * 23 that we found in Part A. To do this, we compare each fraction with 12. To make it easier, let's write 23 with a denominator of 12 and write 12 as 612.
| Rewrite | Compare with 12, or 612 | |
|---|---|---|
| 10/12 | 10/12 | 10/12 > 6/12 |
| 2/3 | 2*4/3*4 = 8/12 | 8/12 > 6/12 |
Both fractions are greater than 12, so we can round both fractions to 1. rccc Product: & 10/12 & * & 2/3 [0.5em] & ↓ & & ↓ Estimate: & 1 & * & 1 The estimate is 1 * 1 = 1. However, this does not tell us much about the original product, and 1 is not one of the possible answer choices. Let's try rounding only one of the fractions to 1.
| Estimate for 1012* 23 | |
|---|---|
| Round 1012 to 1 | 1 * 2/3 = 2/3 |
| Round 23 to 1 | 10/12 * 1 = 10/12 |
We can simplify 1012 to 56. The answer is either 23 or 56, but only 23 is among the answer choices. This means our answer is 23.
c
To multiply the fractions, we multiply the numerators and denominators separately. Then, we simplify the resulting fraction.
The numerator and denominator both have 4 as a factor, so we can simplify the fraction by dividing both by this factor.
Since 5 and 9 have no common factors other than 1, the product of the fractions, 59, is in simplest form. Paulina drinks 59 of the bottle of water throughout the lesson. This value is close to our estimate from Part B because 23 is equivalent to 69. ccc Result & & Estimate 5/9 & ≈ & 6/9 Since the actual result and the estimate are close to each other, the answer 59 is reasonable.
Extra
Using a Number Line to Represent the Product of FractionsTo represent the product of the fractions from Part A, we can also use a number line.
10/12 * 2/3
First, we divide the number line between 0 and 1 into 3 equal parts and mark 23. Then, we divide the section from 0 to 23 into 12 equal parts and color 10 parts to represent 1012 of 23.
Notice that each third is divided into 6 equal parts. Finally, we divide the last third into 6 equal parts and count how many parts of 1, or the whole water bottle, are red.
There are 10 red parts out of 18 parts representing the whole bottle, so 1018 of the bottle is filled. Since 10 and 18 are both divisible by 2, we can simplify the fraction. 10 ÷ 2/18 ÷ 2 = 5/9
Extra
Using an Area Model to Represent the Product of FractionsAnother way to represent the product is with an area model. In this model, we use the denominators to divide a rectangular diagram into smaller parts. The denominators are 12 and 3, so we divide the rectangle into 12 columns and 3 rows.
The numerators determine which parts will be shaded. Since 10 is the numerator of 1012, 10 of the 12 columns will be shaded. Similarly, 2 of the 3 rows will be shaded.
In this model, the overlapping region represents the product. Here, the product is 2036= 59.
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Pop Quiz
Finding the Product of Fractions
Multiply the fractions. Simplify the product if possible.
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Discussion
Multiplying Fractions by Whole Numbers and By Mixed Numbers
When multiplying fractions by whole numbers or mixed numbers, both factors should be written as proper fractions or as improper fractions.
Multiplying Fractions By Whole Numbers
To multiply fractions by whole numbers, the whole number is written as a fraction with a denominator is 1. The procedure after that is identical to multiplying fractions. Consider multiplying 9 and 227.
9 * 2/27
Rewrite
Rewrite 9 as 9/1
9/1 * 2/27
18/27
ReduceFrac
a/b=.a /9./.b /9.
18 /9/27 /9
SimpQuot
Simplify quotient
2/3
Multiplying Fractions By Mixed Numbers
Mixed numbers are fractions that consist of a whole number and a proper fraction. 3 45 whole number: 3 proper fraction: 45 The mixed number must be converted into an improper fraction before it can be multiplied by a fraction. Let's consider an example.
3 45 * 1/3
▼
Write mixed number as a fraction
19/5 * 1/3
19/15
19/15
▼
Write fraction as a mixed number
Rewrite
Rewrite 19 as 15+4
15+4/15
WriteSumFrac
Write as a sum of fractions
15/15+4/15
CalcQuot
Calculate quotient
1 + 4/15
Rewrite
Rewrite 1+4/15 as 1 415
1 415
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Example
Finding Times Allocated For Activities
Paulina's PE class lasts for 1 512 hours. The table shows what fraction of the class time is allocated for various activities.
| Activity | Fraction |
|---|---|
| Warm-up | 1/5 |
| Instruction | 1/2 |
| Playing a game | 1/10 |
| Cool-down | 1/5 |
a
Find how many minutes are devoted to warming up and cooling down.
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b
How much time is spent on activities other than instruction?
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Hint
a
Start by determining what fraction of the class time is allocated for warm-up and cool-down.
b
What fraction of the class time is spent on activities other than instruction?
Solution
a
First, we find the fractions of the class spent on the warm-up and the cool-down by looking at the given table.
| Part of 1 512-hour Class | ||
|---|---|---|
| Activity | Fraction | |
| Warm-up | 1/5 | |
| Instruction | 1/2 | |
| Playing game | 1/10 | |
| Cool-down | 1/5 | |
Now, we find the portion of the class spent on these activities by adding the fractions. 1/5 + 1/5 = 2/5 We can find the time spent warming up and cooling down by multiplying the portion of the class it took by the lesson time, 1 512. Let's first rewrite the mixed number as an improper fraction.
2/5* 1 512
MixedToFrac
a bc=a* c+b/c
2/5* 1 * 12 + 5/12
MultByOne
a * 1=a
2/5* 12 + 5/12
AddTerms
Add terms
2/5* 17/12
Now we multiply the numerators and denominators.
We found that warming up and cooling down takes 3460 of an hour. We can find the time in minutes by multiplying it by 60, since there are 60 minutes in an hour.
The warm-up and cool-down activities last for 34 minutes.
b
Now, we want to find the time spent on activities other than instruction.
| Part of 1 512-hour Class | ||
|---|---|---|
| Activity | Fraction | |
| Warm-up | 1/5 | |
| Instruction | 1/2 | |
| Playing game | 1/10 | |
| Cool-down | 1/5 | |
Since instruction takes 12, or one-half, of the time, other activities also take half of the lesson time. This means that we can find the time spent on other activities by multiplying the lesson time by 12. Let's use 1712 for the lesson time, which we found in Part A.
Finally, we multiply 1724 by 60 to find the amount of time in minutes.
60 * 17/24
MoveLeftFacToNum
a*b/c= a* b/c
60* 17/24
Multiply
Multiply
1020/24
CalcQuot
Calculate quotient
42.5
The activities other than instruction last for 42.5 minutes.
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Example
Finding the Area of a Printed Photograph
The diagram shows the dimensions of a photograph.
External credits: pch.vector
a
Estimate the area of the photograph.
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b
What is the actual area of the photograph? Write the answer as a mixed number.
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c
Paulina frames the photo using a frame with a 15-inch border. What is the area of the photo including the frame? Write the answer as a mixed number.
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Hint
a
Use the formula for the area of a rectangle. Round the mixed numbers to the nearest whole numbers.
b
To multiply fractions, multiply the numerators and denominators with each other.
c
Add 2* 15 to each side of the photo. Then repeat the process from Part A.
Solution
Area = 8 34 * 6 23 Since the fractional parts of the mixed numbers are greater than 12, we can round the mixed numbers up. Area = & 8 34 * 6 23 & ↓ ↓ & 9 * 7 = 63 The area of the photo is about 9* 7, or 63 square inches.
b
Now, we want to find the exact area of the photograph. First, we rewrite the mixed number dimensions of the photograph as improper fractions.
Area = 8 34 * 6 23
MixedToFrac
a bc=a* c+b/c
Area = 8*4 +3/4 * 6 *3 +2/3
Multiply
Multiply
Area = 32+3/4 * 18 +2/3
AddTerms
Add terms
Area = 35/4 * 20/3
Next, we can multiply the fractions by multiplying their numerators and denominators. Then, we rewrite the result as a mixed number.
35/4 * 20/3
MultFrac
Multiply fractions
35* 20/4* 3
Multiply
Multiply
700/12
ReduceFrac
a/b=.a /4./.b /4.
700 ÷ 4/12 ÷ 4
CalcQuot
Calculate quotient
175/3
▼
Write fraction as a mixed number
Rewrite
Rewrite 175 as 174+1
174+1/3
WriteSumFrac
Write as a sum of fractions
174/3+1/3
CalcQuot
Calculate quotient
58+1/3
Rewrite
Rewrite 58+1/3 as 58 13
58 13
The area of the photo is 58 13 square inches. This is about the same as the estimate we found in Part A, so our answer is reasonable.
c
In this part, we will start by adding a 15-inch border to each side of the photo. This extends each side by 2* 15 = 25 inches.
External credits: pch.vector
Next, we can write the expression for the area of the photograph including the frame. This is the product of the extended side lengths. (8 34+2/5) * (6 23+ 2/5) Before finding the area, let's rewrite the mixed numbers as improper fractions.
| Number | Conversion | Improper Fraction |
|---|---|---|
| 8 34 | 8*4+3/4 | 35/4 |
| 6 23 | 6 *3+2/3 | 20/3 |
Now that we have all the numbers written as fractions, we can calculate the exact area of the extended photograph. First, we add the fractions in parentheses by finding common denominators.
(35/4+ 2/5) * (20/3+2/5)
▼
Add fractions
ExpandFrac
a/b=a * 5/b * 5
(35 * 5/4 * 5+ 2/5) * (20/3+2/5)
ExpandFrac
a/b=a * 4/b * 4
(35 * 5/4 * 5+ 2*4/5*4) * (20/3+2/5)
ExpandFrac
a/b=a * 5/b * 5
(35 * 5/4 * 5+ 2*4/5*4) * (20 * 5/3 * 5+2/5)
ExpandFrac
a/b=a * 3/b * 3
(35 * 5/4 * 5+ 2*4/5*4) *(20 * 5/3 * 5+2 * 3/5* 3)
Multiply
Multiply
(175/20+ 8/20) * (100/15+6/15)
AddFrac
Add fractions
175+8/20 * 100+6/15
AddTerms
Add terms
183/20 * 106/15
Finally, we multiply the fractions and write the result as a mixed number.
183/20 * 106/15
MultFrac
Multiply fractions
183* 106/20* 15
Multiply
Multiply
19 398/300
ReduceFrac
a/b=.a /6./.b /6.
19 398 ÷ 6/300 ÷ 6
SimpQuot
Simplify quotient
3233/50
▼
Write fraction as a mixed number
Rewrite
Rewrite 3233 as 3200+33
3200+33/50
WriteSumFrac
Write as a sum of fractions
3200/50+33/50
CalcQuot
Calculate quotient
64+33/50
Rewrite
Rewrite 64+33/50 as 64 3350
64 3350
The area of the photo with the frame is 64 3350 square inches.
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Pop Quiz
Finding the Product of Various Fractions
Practice finding the product of fractions. Give the answer as a proper or improper fraction in simplest form.
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Closure
Finding Time Paulina Runs
The important point in multiplying fractions and mixed numbers is to ensure that everything is written as either a proper fraction or an improper fraction. 1 26 * 2 = 8/6 * 2/1 The final step usually involves simplifying the resulting fraction. However, it can be easier to first check if the two fractions can be simplified before multiplying them. 4 & 8/6 &* 2/1 = 4/3 * 2/1 = 8/3 3 & Consider the challenge presented at the beginning of the lesson. Paulina spends 26 of her free time exercising.
External credits: pikisuperstar
If Paulina has 5 hours of free time per day, how many hours does she run in 4 days?
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Hint
Start by finding the number of hours Paulina spends running in a day.
Solution
We can find the amount of time Paulina spends running by finding 26 of the 5 hours of free time she has each day. We start by simplifying the fraction and then multiply.
Since we want to find the total time she spends running over 4 days, we multiply 53 by 4.
Lastly, we will convert this to a mixed number.
20/3
▼
Write fraction as a mixed number
Rewrite
Rewrite 20 as 18+2
18+2/3
WriteSumFrac
Write as a sum of fractions
18/3 + 2/3
CalcQuot
Calculate quotient
6+2/3
Rewrite
Rewrite 6+2/3 as 6 23
6 23
In 4 days, Paulina runs for 6 23 hours.
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Multiply Fractions
Exercises
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We want to find the product of 36 and 811. 3/6 * 8/11 When we multiply fractions, we need to remember that the product of two fractions is equal to the product of the numerators divided by the product of the denominators. Let's find the given product!
Now we will split the numerator and denominator into prime factors. This will enable us to find the greatest common factor (GCF) of the numbers.
| Number | Prime Factorization | GCF |
|---|---|---|
| 24 | 2^3 * 3 | 2* 3=6 |
| 66 | 2* 3 * 11 |
We can simplify the fraction by dividing the numerator and denominator by their GCF.
We will start again by multiplying the numerators, followed by multiplying the denominators. Then we will reduce the fraction if necessary. Let's do it!
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Solution
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We want to multiply a fraction and a whole number. To do so, we will first rewrite the whole number as a fraction with a denominator of 1. Then, we will multiply the numerators and denominators with each other. If necessary, we will also simplify the obtained product. Let's do it!
To find the given product, we will rewrite the whole number as a fraction. Then, we will multiply the numerators and denominators with each other like we did in Part A.
We can simplify this fraction because they have a common factor. 14 = 2 * 7 21 = 3 * 7 The greatest common factor (GCF) of the numbers is 7. Let's dividing the numerator and denominator by the GCF.
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We want to find the product of a mixed number and a whole number. Let's start by rewriting them as improper fractions. Then, their product will be equal to the product of the numerators divided by the product of the denominators.
Now, we need to write it as a mixed number.
This time we are asked to find the product of two mixed numbers. To do so, we will first rewrite them as improper fractions. Let's do it!
Notice that the numerator of 2110 and the denominator of 1514 have a common factor of 7. Additionally, the numerator of 1514 and the denominator of 2110 have a common factor of 5. We can use this information to simplify the fractions before we multiply them. Let's do it by factoring the numbers! 21/10 * 15/14 = 3 * 7/2 * 5 * 3* 5/2 * 7 The fractions in the product can be simplified to 32. Let's find this product!
We are asked to write the answer as a mixed number. Let's convert it!
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We want to estimate the product of a mixed number and a whole number. 5 45 * 4 To do so, we will round the mixed number. Recall that if the fractional part of a mixed number is greater than or equal to 12, we round the mixed number up. If not, we round it down. Now let's check the fractional part of our mixed number! 4/5 >1/2 because 8/10 > 5/10 Since the fractional part 45 is greater than 12, we can round the mixed number up. This means that 5 45 is about 6. 5 45 * 4 ↓ 6 * 4 Since 6*4 is equal to 24, the given product is about 24.
We want to estimate the product of two mixed numbers.
1 67 * 6 16
We can round the first mixed number up because 67 is greater than 12. We can round the other mixed number down because its fractional part is less than 12.
6/7 >1/2 & because & 12/14 > 7/14 [0.8em]
1/6 <1/2 &because & 1/6 < 3/6
Let's now multiply the rounded numbers.
1 67 * 6 16
↓ ↓
2 * 6
The product of the mixed numbers is about the product of 2 and 6, or 12.
1 67 * 6 16 ≈ 12
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In area model, the shaded columns represents one fraction and the shaded rows represents another. If we break down the model into two parts, we can understand what this means. Let's do it!
The figure on the left is divided into 5 parts and 3 of them are shaded. This means that the shaded columns represent 35. In the same way, the figure on the right is divided into 4 parts and 1 of them is shaded, so the shaded rows represent 14. The diagram models the multiplication of these fractions. 3/5 * 1/4
Let's multiply the fractions. Remember that we multiply the numerators with each other and the denominators with each other.
The product of the fractions is 320, which cannot be simplified further.
We can verify that the product is 320 by looking at the diagram. Let's focus on the region where the blue and yellow regions overlap.
This region consists of 3 of the 20 small squares. Notice also that 3 and 20 are the product of the numerators and denominators, respectively. 3/20 The green region represents the product of the multiplication written in Part A and matches the product of the two fractions. 3/5* 1/4 = 3/20 ✓
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Multiply Fractions
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