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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.

Challenge

## Running Time

Paulina runs for two-sixths of her free time each day.

External credits: pikisuperstar

If Paulina has hours of free time per day, how many hours does she run in days? Give the exact answer.

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.

Here, and are not When multiplying fractions, it makes no difference whether they are like or unlike fractions. Consider multiplying by
The result of this multiplication can be found in three steps.
1
Multiply the Numerators
expand_more
The numerator of the first fraction is and the numerator of the second is The product of the numerators is then
2
Multiply the Denominators
expand_more
The product of the denominators is
3
Simplify if Possible
expand_more
Note that is the greatest common factor of and Divide both the numerator and the denominator by to simplify the fraction.
Therefore, the product of and simplified to its lowest terms is
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.
b Which of the following is the best estimate for the value of the expression written in Part A?
c Find the value of the expression found in Part A. Simplify the answer if possible.

### 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 round the fraction to If a fraction is less than round it to
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
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.
A number line can be used to help with this concept. First, divide the number line between and into thirds. Fill in representing the water that remains in the bottle before class. Then, divide this section into smaller equal parts and color in 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.
Now compare each fraction with These fractions do not have the same denominator. The denominators of the fractions are and The least common denominator of these numbers is Each fraction can be rewritten as an equivalent fraction with the common denominator to make it easier to compare them.
Rewrite Compare with or
Both fractions are greater than An estimate for the product can then be because the fractions can be rounded to
Since is the identity element of multiplication, this product is also equal to However, that does not say much about the original product, and is not one of the possible answer options. Now think of rounding only one of the fractions to one at a time.
Estimate for
Round to
Round to

The fraction can be simplified to The answer is either or The fractions in the options are in simplest form and is among the options. Therefore, the answer is

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 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 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 is equivalent to
Since the actual result and the estimate are close to each other, the answer is reasonable.

### Extra

Using an Area Model to Represent the Product of Fractions
In 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.
The denominators are and The rectangle is then divided into columns and rows.

The numerators determine which parts will be shaded. Since is the numerator of of the columns will be shaded. Similarly, of the rows will be shaded.

In this model, the overlapping region represents the product. For this example, the product is

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.

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 The following steps are identical to those for multiplying fractions. Consider multiplying and
Evaluate

### Multiplying Fractions By Mixed Numbers

Recall that mixed numbers are fractions that consist of a whole number and a proper fraction.
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.
Write mixed number as a fraction
Evaluate
The result can also be written as a mixed number.
Write mixed number as a fraction
The product of and is
Example

## Finding Times Allocated For Activities

Paulina's PE class lasts for hours. The table shows what fraction of the class time is allocated for various activities.

Activity Fraction
Warm-up
Instruction
Playing a game
Cool-down
a Find how many minutes are devoted to warming up and cooling down.
b Find how many minutes are devoted to the activities other than instruction.

### 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 hour is 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 -hour Class
Activity Fraction
Instruction
Playing game
The sum of the fractions is
Next, the time spent on warming up and cooling down will be found by multiplying this fraction number by the lesson time This will give the amount of time in hours.
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.
Now multiply the numerators and denominators.
Since a common factor exists between numerator and denominator, the fraction can be simplified.
The fraction simplifies to This means that 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 because hour is minutes.
This is a multiplication of a whole number by an improper fraction. Start by rewriting as an improper fraction, then multiply the fractions.
The warm-up and cool-down activities last for minutes.
b Consider the given table again, this time focusing on the instruction time.
Part of -hour Class
Activity Fraction
Playing game
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
Rewrite the mixed number as an improper fraction to find the product.
Write mixed number as a fraction
Next, multiply the numerators and denominators.
This fraction cannot be simplified further. It means that of an hour is spent on activities other than instruction. Finally, multiply by to write the amount of time in terms of minutes.
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.