2. Multiply Fractions
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Chapter 3
2.
Multiply Fractions
| Student Learning Objectives: |
|---|
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Why
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.
Lesson Settings & Tools
| | 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. Along with these methods, estimation strategies will be used to assess whether the products are reasonable.
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 are not 0. When multiplying fractions, it makes no difference whether they are like or unlike fractions. Consider multiplying 56 by 34. 5/6 * 3/4 The result of this multiplication can be found in three steps.
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 simplified to its lowest terms is 58.
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Example
Finding What Fraction of the Water in a Bottle Is Drunk
Paulina drinks one-third of the water in her bottle before PE class. During class, she drinks ten-twelfths of the remaining water.
a Write a numeric expression to represent the amount of water in the bottle that 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 ten twelfths of 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, start with the multiplication of the numerators, followed by the multiplication of the denominators. Then, simplify the resulting fractions.
Solution
a Start by finding what fraction of the bottle is full before the physical education class. To do so, subtract one-third from 1.
This represents the fraction of the bottle that is full before class. This means that Paulina drinks ten-twelfths of two-thirds of the bottle of water during class. To find this amount, these fractions will be multiplied. Expression [0.5em] 10/12 * 2/3 A number line can be used to help with this concept. First, divide the number line between 0 and 1 into thirds. Fill in 23, representing the water that remains in the bottle before class. Then, divide this section into 12 smaller equal parts and color in 10 of them.
Notice that each third is divided into six smaller equal parts. The other third can also be divided into six parts. This will make it easier to find what fraction of the whole the red part represents.
The red part shows what fraction of the water bottle Paulina drinks during class.
b Estimation is a great tool to get a rough idea of the result of an operation — in this case, the result of a multiplication. Consider the value of each fraction in the expression separately.
10/12 * 2/3 Now compare each fraction with 12. These fractions do not have the same denominator. The denominators of the fractions are 12, 3, and 2. The least common denominator of these numbers is 12. Each fraction can be rewritten as an equivalent fraction with the common denominator to make it easier to compare them.
| 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. An estimate for the product can then be 1 because the fractions can be rounded to 1. rccc Product: & 10/12 & * & 2/3 [0.5em] & ↓ & & ↓ Estimate: & 1 & * & 1 Since 1 is the identity element of multiplication, this product is also equal to 1. However, that does not say much about the original product, and 1 is not one of the possible answer options. Now think of rounding only one of the fractions to 1 one at a time.
| Estimate for 1012* 23 | |
|---|---|
| Round 1012 to 1 | 1 * 2/3 = 2/3 |
| Round 23 to 1 | 10/12 * 1 = 10/12 |
The fraction 1012 can be simplified to 56. The answer is either 23 or 56. The fractions in the options are in simplest form and 23 is among the options. Therefore, the answer is 23.
c To multiply the fractions, start with the multiplication of the numerators, followed by the multiplication of the denominators. Then, simplify the resulting fraction.
Notice that the numerator and denominator both have 4 as a factor. This suggests that the resulting fraction can be reduced to its lowest form by dividing the numerator and denominator by this factor.
The product of the fractions is 59. This is the fraction of the bottle of water that Paulina drinks throughout the lesson. Notice also that this value is close the estimate found in 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 an Area Model to Represent the Product of FractionsIn addition to a number line, the product can also be represented by an area model. In this model, the denominators are used to divide a rectangular diagram into smaller parts. Consider modeling the following product.
10/12 * 2/3
The denominators are 12 and 3. The rectangle is then divided 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. For this example, the product is 2036= 59.
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Pop Quiz
Finding the Product of Fractions
The product of two fractions is equal to the product of the numerators divided by the product of the denominators. Practice finding the product of fractions. Simplify the answer to its lowest terms.
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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 in the form of a proper fraction or an improper fraction.
Multiplying Fractions By Whole Numbers
To multiply fractions with whole numbers, the whole number is written as a fraction with a denominator is 1. The following steps are identical to those for 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
Recall that mixed numbers are fractions that consist of a whole number and a proper fraction. 3 45 whole number: 3 proper fraction: 45 The mixed fraction must be converted into an improper fraction before it can be multiplied by a fraction. The multiplication process can be better understood with the help of an example.
3 45 * 1/3
▼
Write mixed number as a fraction
19/5 * 1/3
19/15
19/15
▼
Write mixed number as a fraction
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 Find how many minutes are devoted to the 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. To do so, add the fractions corresponding to the activities. Multiply the portion of the lesson spent on those activities by the total class time. Finally, use the fact that 1 hour is 60 minutes.
b Determine what fraction of the class time is allocated for the activities other than instruction.
Solution
a The fractions for both the warm-up and the cool-down to find the amount of time spent on them. Take a look 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 | |
The sum of the fractions is 25. 1/5 + 1/5 = 2/5 Next, the time spent on warming up and cooling down will be found by multiplying this fraction number by the lesson time 1 512. This will give the amount of time in hours. 2/5* 1 512 This is a multiplication of a fraction by a mixed number. Before performing the multiplication, the mixed number will be rewritten 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 multiply the numerators and denominators.
Since a common factor exists between numerator and denominator, the fraction can be simplified. 34 /2/60 / 2 = 17/30 The fraction simplifies to 1730. This means that 1730 of an hour is spent on warming up and cooling down. Now the number of minutes spent on these activities will be found by multiplying the fraction by 60 because 1 hour is 60 minutes. 60* 17/30 This is a multiplication of a whole number by an improper fraction. Start by rewriting 60 as an improper fraction, then multiply the fractions.
60 * 17/30
Rewrite
Rewrite 60 as 60/1
60/1 * 17/30
MultFrac
Multiply fractions
60 * 17/1* 30
Multiply
Multiply
1020/30
CalcQuot
Calculate quotient
34
The warm-up and cool-down activities last for 34 minutes.
b Consider the given table again, this time focusing on the instruction time.
| Part of 1 512-hour Class | ||
|---|---|---|
| Activity | Fraction | |
| Warm-up | 1/5 | |
| Instruction | 1/2 | |
| Playing game | 1/10 | |
| Cool-down | 1/5 | |
As shown, half of the class time is spent on instruction. In other words, one-half of the lesson is not spent on instruction. With this in mind, the total lesson time will be multiplied by 12. 1/2 * 1 5/12 Rewrite the mixed number as an improper fraction to find the product.
1/2* 1 512
▼
Write mixed number as a fraction
1/2* 17/12
Next, multiply the numerators and denominators.
This fraction cannot be simplified further. It means that 1724 of an hour is spent on activities other than instruction. Finally, multiply 1724 by 60 to write the amount of time in terms of minutes. 60* 17/24 This is a multiplication of a whole number by a fraction. An easy way to find this product is to move the whole number to the numerator of the fraction as a factor.
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
Paulina loves a photo of her playing volleyball and prints it. The diagram shows the dimensions of the photograph.
External credits: pch.vector
a Estimate the area of the photograph.
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b What is the 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-centimeter 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
External credits: pch.vector
The length of the photo is 8 34 inches and the width of the photo is 6 23 inches. Area = 8 34 * 6 23 Since the fractional parts of the mixed numbers are greater than 12, the mixed numbers can be rounded up. Area = & 8 34 * 6 23 & ↓ ↓ & 9 * 7 Therefore, the area of the photo is about 9* 7, or 63 square inches.
b Recall the expression for the area of the photo from Part A. It is the product of the two mixed numbers.
Area = 8 34 * 6 23 To multiply these mixed numbers, they first need to be rewritten as improper fractions.
8 34 * 6 23
MixedToFrac
a bc=a* c+b/c
8* 4 +3/4 * 6 * 3 +2/3
Multiply
Multiply
32+3/4 * 18 +2/3
AddTerms
Add terms
35/4 * 20/3
Recall that the product of two fractions is equal to the product of the numerators divided by the product of the denominators.
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. Note that this is about the same as the estimate found in Part A, so this answer is reasonable.
c In this part, start by adding a 15-inch border on each side of the photo. This will extend each side by 2* 15 inches.
External credits: pch.vector
The expression for the area of the photo with the frame is then the product of the side lengths shown in the diagram. (8 34+2* 1/5) * (6 23+2* 1/5) ⇕ (8 34+2/5) * (6 23+ 2/5) Once again, rewrite the fractions so that all of them are either proper or improper fractions.
(8 34+ 2/5) * (6 23+2/5)
▼
Write mixed number as a fraction
MixedToFrac
a bc=a* c+b/c
(8* 4+3/4+ 2/5) * (6 * 3+2/3+2/5)
Multiply
Multiply
(32+3/4+ 2/5) * (18+2/3+2/5)
AddTerms
Add terms
(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
When multiplying fractions, the product is equal to the product of the numerators divided by the product of the denominators.
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
CalcQuot
Calculate 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 Mixed Numbers
To multiply a fraction by a whole number, the whole number is multiplied by the numerator of the fraction. To multiply mixed numbers, the mixed numbers can be converted into improper fractions before multiplying. Practice finding the product of fractions. Simplify the answer to its lowest terms.
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Closure
Finding Time Paulina Runs
The important point in multiplying fractions is to ensure that the fractions are either proper fractions or improper fractions. 1 26 * 2 = 8/6 * 2/1 The final step usually involves simplifying the resulting fraction. However, to make calculations easier, first check if the two fractions are already in their lowest forms. If not, the fractions can be simplified first 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 devotes two-sixths of her free time to exercise.
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
It is given that Paulina has 5 hours of free time per day and that two-sixths of this time is devoted to physical activities. The time spent on running can be found by finding two-sixths of 5.
2/6 * 5
Notice that the fraction can be simplified to its lowest terms.
Paulina runs for 53 hours every day. Since the number of hours she runs in 4 days is required, the daily amount of activity should be multiplied by 4 to find the total amount of time.
This final fraction can also be written as 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 6 23 hours.
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Multiply Fractions
Exercise 3.1
Diego is discussing the possible outcomes of multiplying fractions and mixed numbers with his friends. He claims that the following sentence is correct.
|
The product of two positive mixed numbers is less than 1. |
Is Diego's claim correct?
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Let's start by recalling what a mixed number is. A mixed number is a number that consists of an integer number and a proper fraction — a fraction whose numerator is less than the denominator. a bc, where b < c and c ≠ 0 We can think of a mixed number as the sum of its two components. a bc = a+b/c We are told that the mixed numbers are both positive. This means that a is greater than or equal to 1 for both mixed numbers. A positive mixed number is always greater than 1. Therefore, multiplying two positive mixed numbers can be considered as the same as multiplying two positive numbers that are greater than 1. For example, 2* 2 is equal to 4, which is greater than 1. 2 * 2 = 4 and 4 > 1 With this in mind, we can conclude that Diego's claim is not correct because the product of two positive mixed numbers is always greater than 1. a bc * d ef > 1
Let's arbitrarily choose two positive mixed numbers and check if their product is greater than 1.
Since the numerator is greater than the denominator, this fraction is greater than 1.
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Multiply Fractions
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