nderstanding Multiplying Fractions, Mixed Numbers, and Simplifying Fractions

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.

Rational Numbers
Fraction
Mixed Numbers

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.

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.

$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$

Here,

b

and

d

are not

0 .

When multiplying fractions, it makes no difference whether they are like or unlike fractions. Consider multiplying

\frac{5}{6}

\frac{3}{4} .

\frac{5}{6} \cdot \frac{3}{4}

The result of this multiplication can be found in three steps.

Multiply the Numerators

The numerator of the first fraction is

5

and the numerator of the second is

3 .

The product of the numerators is then

15 .

\frac{5}{6} \cdot \frac{3}{4}

MultFrac

Multiply fractions

\frac{5 \cdot 3}{6 \cdot 4}

Multiply

\frac{1 5}{6 \cdot 4}

Multiply the Denominators

The product of the denominators is

6 \cdot 4 = 24 .

\frac{1 5}{6 \cdot 4}

Multiply

\frac{1 5}{2 4}

Simplify if Possible

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.

\frac{1 5}{2 4}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

\frac{1 5 / 3}{2 4 / 3}

SimpQuot

Simplify quotient

\frac{5}{8}

Therefore, the product of

\frac{5}{6}

and

\frac{3}{4}

simplified to its lowest terms is

\frac{5}{8} .

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

\frac{1}{2},

round the fraction to

1 .

If a fraction is less than

\frac{1}{2},

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 .

1 - \frac{1}{3}

NumberToFrac

$a = \frac{3 \cdot a}{3}$

\frac{3}{3} - \frac{1}{3}

SubFrac

Subtract fractions

\frac{3 - 1}{3}

SubTerm

Subtract term

\frac{2}{3}

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 \frac{1 0}{1 2} \times \frac{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

\frac{2}{3},

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.

\frac{1 0}{1 2} \times \frac{2}{3}

Now compare each fraction with

\frac{1}{2} .

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 $\frac{1}{2},$ or $\frac{6}{1 2}$
$\frac{1 0}{1 2}$	$\frac{1 0}{1 2}$	$\frac{1 0}{1 2} > \frac{6}{1 2}$
$\frac{2}{3}$	$\frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{1 2}$	$\frac{8}{1 2} > \frac{6}{1 2}$

Both fractions are greater than

\frac{1}{2} .

An estimate for the product can then be

1

because the fractions can be rounded to

1 .

Product : Estimate : \frac{1 0}{1 2} ↓ 1 \times \times \frac{2}{3} ↓ 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 $\frac{1 0}{1 2} \times \frac{2}{3}$
Round $\frac{1 0}{1 2}$ to $1$	$1 \times \frac{2}{3} = \frac{2}{3}$
Round $\frac{2}{3}$ to $1$	$\frac{1 0}{1 2} \times 1 = \frac{1 0}{1 2}$

The fraction $\frac{1 0}{1 2}$ can be simplified to $\frac{5}{6} .$ The answer is either $\frac{2}{3}$ or $\frac{5}{6} .$ The fractions in the options are in simplest form and $\frac{2}{3}$ is among the options. Therefore, the answer is $\frac{2}{3} .$

c To multiply the fractions, start with the multiplication of the numerators, followed by the multiplication of the denominators. Then, simplify the resulting fraction.

\frac{1 0}{1 2} \times \frac{2}{3}

MultFrac

Multiply fractions

\frac{1 0 \times 2}{1 2 \times 3}

Multiply

\frac{2 0}{1 2 \times 3}

Multiply

\frac{2 0}{3 6}

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.

\frac{2 0}{3 6}

ReduceFrac

$\frac{a}{b} = \frac{a / 4}{b / 4}$

\frac{2 0 / 4}{3 6 / 4}

CalcQuot

Calculate quotient

\frac{5}{9}

The product of the fractions is

\frac{5}{9} .

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

\frac{2}{3}

is equivalent to

\frac{6}{9} .

Result \frac{5}{9} \approx Estimate \frac{6}{9}

Since the actual result and the estimate are close to each other, the answer

\frac{5}{9}

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.

\frac{1 0}{1 2} \times \frac{2}{3}

The denominators are

12

and

3 .

The rectangle is then divided into

12

columns and

3

rows.

$area model of a product of fractions$

The numerators determine which parts will be shaded. Since $10$ is the numerator of $\frac{1 0}{1 2},$ $10$ of the $12$ columns will be shaded. Similarly, $2$ of the $3$ rows will be shaded.

$area model of a product of fractions$

In this model, the overlapping region represents the product. For this example, the product is $\frac{2 0}{3 6} = \frac{5}{9} .$

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.

$product of random fractions$

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

\frac{2}{2 7} .

9 \times \frac{2}{2 7}

Rewrite

Rewrite $9$ as $\frac{9}{1}$

\frac{9}{1} \times \frac{2}{2 7}

▼

Evaluate

MultFrac

Multiply fractions

\frac{9 \times 2}{1 \times 2 7}

Multiply

\frac{1 8}{1 \times 2 7}

MultByOne

$a \cdot 1 = a$

\frac{1 8}{2 7}

ReduceFrac

$\frac{a}{b} = \frac{a / 9}{b / 9}$

\frac{1 8 / 9}{2 7 / 9}

SimpQuot

Simplify quotient

\frac{2}{3}

Multiplying Fractions By Mixed Numbers

Recall that mixed numbers are fractions that consist of a whole number and a proper fraction.

3 \frac{4}{5} {whole number : 3 proper fraction : \frac{4}{5}

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 \frac{4}{5} \times \frac{1}{3}

▼

Write mixed number as a fraction

MixedToFrac

$a \frac{b}{c} = \frac{a \cdot c + b}{c}$

\frac{3 \cdot 5 + 4}{5} \times \frac{1}{3}

Multiply

\frac{1 5 + 4}{5} \times \frac{1}{3}

AddTerms

Add terms

\frac{1 9}{5} \times \frac{1}{3}

▼

Evaluate

MultFrac

Multiply fractions

\frac{1 9 \times 1}{5 \times 3}

MultByOne

$a \cdot 1 = a$

\frac{1 9}{5 \times 3}

Multiply

\frac{1 9}{1 5}

The result can also be written as a mixed number.

\frac{1 9}{1 5}

▼

Write mixed number as a fraction

Rewrite

Rewrite $19$ as $15 + 4$

\frac{1 5 + 4}{1 5}

WriteSumFrac

Write as a sum of fractions

\frac{1 5}{1 5} + \frac{4}{1 5}

CalcQuot

Calculate quotient

1 + \frac{4}{1 5}

Rewrite

Rewrite $1 + \frac{4}{1 5}$ as $1 \frac{4}{1 5}$

1 \frac{4}{1 5}

The product of

3 \frac{4}{5}

and

\frac{1}{3}

1 \frac{4}{1 5} .

Example

Finding Times Allocated For Activities

Paulina's PE class lasts for $1 \frac{5}{1 2}$ hours. The table shows what fraction of the class time is allocated for various activities.

Activity	Fraction
Warm-up	$\frac{1}{5}$
Instruction	$\frac{1}{2}$
Playing a game	$\frac{1}{1 0}$
Cool-down	$\frac{1}{5}$

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

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 $1 \frac{5}{1 2}$ -hour Class
Activity	Fraction
$Warm - up$	$\frac{1}{5}$
Instruction	$\frac{1}{2}$
Playing game	$\frac{1}{1 0}$
$Cool - down$	$\frac{1}{5}$

The sum of the fractions is

\frac{2}{5} .

\frac{1}{5} + \frac{1}{5} = \frac{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 \frac{5}{1 2} .

This will give the amount of time in hours.

\frac{2}{5} \times 1 \frac{5}{1 2}

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.

\frac{2}{5} \times 1 \frac{5}{1 2}

MixedToFrac

$a \frac{b}{c} = \frac{a \cdot c + b}{c}$

\frac{2}{5} \times \frac{1 \cdot 1 2 + 5}{1 2}

MultByOne

$a \cdot 1 = a$

\frac{2}{5} \times \frac{1 2 + 5}{1 2}

AddTerms

Add terms

\frac{2}{5} \times \frac{1 7}{1 2}

Now multiply the numerators and denominators.

\frac{2}{5} \times \frac{1 7}{1 2}

MultFrac

Multiply fractions

\frac{2 \times 1 7}{5 \times 1 2}

Multiply

\frac{3 4}{6 0}

Since a common factor exists between numerator and denominator, the fraction can be simplified.

\frac{3 4 / 2}{6 0 / 2} = \frac{1 7}{3 0}

The fraction simplifies to

\frac{1 7}{3 0} .

This means that

\frac{1 7}{3 0}

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 \times \frac{1 7}{3 0}

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 \times \frac{1 7}{3 0}

Rewrite

Rewrite $60$ as $\frac{6 0}{1}$

\frac{6 0}{1} \times \frac{1 7}{3 0}

MultFrac

Multiply fractions

\frac{6 0 \times 1 7}{1 \times 3 0}

Multiply

\frac{1 0 2 0}{3 0}

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 $1 \frac{5}{1 2}$ -hour Class
Activity	Fraction
$Warm - up$	$\frac{1}{5}$
$Instruction$	$\frac{1}{2}$
Playing game	$\frac{1}{1 0}$
$Cool - down$	$\frac{1}{5}$

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

\frac{1}{2} .

\frac{1}{2} \times 1 \frac{5}{1 2}

Rewrite the mixed number as an improper fraction to find the product.

\frac{1}{2} \times 1 \frac{5}{1 2}

▼

Write mixed number as a fraction

MixedToFrac

$a \frac{b}{c} = \frac{a \cdot c + b}{c}$

\frac{1}{2} \times \frac{1 \cdot 1 2 + 5}{1 2}

MultByOne

$a \cdot 1 = a$

\frac{1}{2} \times \frac{1 2 + 5}{1 2}

AddTerms

Add terms

\frac{1}{2} \times \frac{1 7}{1 2}

Next, multiply the numerators and denominators.

\frac{1}{2} \times \frac{1 7}{1 2}

MultFrac

Multiply fractions

\frac{1 \times 1 7}{2 \times 1 2}

Multiply

\frac{1 7}{2 4}

This fraction cannot be simplified further. It means that

\frac{1 7}{2 4}

of an hour is spent on activities other than instruction. Finally, multiply

\frac{1 7}{2 4}

60

to write the amount of time in terms of minutes.

60 \times \frac{1 7}{2 4}

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 \times \frac{1 7}{2 4}

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

\frac{6 0 \times 1 7}{2 4}

Multiply

\frac{1 0 2 0}{2 4}

CalcQuot

Calculate quotient

42.5

The activities other than instruction last for

42.5

minutes.

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.

A photo with a length of 8 and 3/4 inches and a width of 6 and 2/3 inches

External credits: pch.vector

a Estimate the area of the photograph.

b What is the area of the photograph? Write the answer as a mixed number.

c Paulina frames the photo using a frame with a

\frac{1}{5} -

centimeter border. What is the area of the photo including the frame? Write the answer as a mixed number.

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 \times \frac{1}{5}

to each side of the photo. Then repeat the process from Part A.

Solution

a The photo is a rectangle so its area is equal to the product of its length and its width.

External credits: pch.vector

The length of the photo is

8 \frac{3}{4}

inches and the width of the photo is

6 \frac{2}{3}

inches.

Area = 8 \frac{3}{4} \cdot 6 \frac{2}{3}

Since the fractional parts of the mixed numbers are greater than

\frac{1}{2},

the mixed numbers can be rounded up.

Area = 8 \frac{3}{4} \cdot 6 \frac{2}{3} ↓ ↓ 9 \cdot 7

Therefore, the area of the photo is about

9 \cdot 7,

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 \frac{3}{4} \cdot 6 \frac{2}{3}

To multiply these mixed numbers, they first need to be rewritten as improper fractions.

8 \frac{3}{4} \cdot 6 \frac{2}{3}

MixedToFrac

$a \frac{b}{c} = \frac{a \cdot c + b}{c}$

\frac{8 \cdot 4 + 3}{4} \cdot \frac{6 \cdot 3 + 2}{3}

Multiply

\frac{3 2 + 3}{4} \cdot \frac{1 8 + 2}{3}

AddTerms

Add terms

\frac{3 5}{4} \cdot \frac{2 0}{3}

Recall that the product of two fractions is equal to the product of the numerators divided by the product of the denominators.

\frac{3 5}{4} \cdot \frac{2 0}{3}

MultFrac

Multiply fractions

\frac{3 5 \cdot 2 0}{4 \cdot 3}

Multiply

\frac{7 0 0}{1 2}

ReduceFrac

$\frac{a}{b} = \frac{a / 4}{b / 4}$

\frac{7 0 0 / 4}{1 2 / 4}

CalcQuot

Calculate quotient

\frac{1 7 5}{3}

▼

Write fraction as a mixed number

Rewrite

Rewrite $175$ as $174 + 1$

\frac{1 7 4 + 1}{3}

WriteSumFrac

Write as a sum of fractions

\frac{1 7 4}{3} + \frac{1}{3}

CalcQuot

Calculate quotient

58 + \frac{1}{3}

Rewrite

Rewrite $58 + \frac{1}{3}$ as $58 \frac{1}{3}$

58 \frac{1}{3}

The area of the photo is

58 \frac{1}{3}

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

\frac{1}{5} -

inch border on each side of the photo. This will extend each side by

2 \cdot \frac{1}{5}

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 \frac{3}{4} + 2 \cdot \frac{1}{5}) \cdot (6 \frac{2}{3} + 2 \cdot \frac{1}{5}) ⇕ (8 \frac{3}{4} + \frac{2}{5}) \cdot (6 \frac{2}{3} + \frac{2}{5})

Once again, rewrite the fractions so that all of them are either proper or improper fractions.

(8 \frac{3}{4} + \frac{2}{5}) \cdot (6 \frac{2}{3} + \frac{2}{5})

▼

Write mixed number as a fraction

MixedToFrac

$a \frac{b}{c} = \frac{a \cdot c + b}{c}$

(\frac{8 \cdot 4 + 3}{4} + \frac{2}{5}) \cdot (\frac{6 \cdot 3 + 2}{3} + \frac{2}{5})

Multiply

(\frac{3 2 + 3}{4} + \frac{2}{5}) \cdot (\frac{1 8 + 2}{3} + \frac{2}{5})

AddTerms

Add terms

(\frac{3 5}{4} + \frac{2}{5}) \cdot (\frac{2 0}{3} + \frac{2}{5})

▼

Add fractions

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 5}{b \cdot 5}$

(\frac{3 5 \cdot 5}{4 \cdot 5} + \frac{2}{5}) \cdot (\frac{2 0}{3} + \frac{2}{5})

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 4}{b \cdot 4}$

(\frac{3 5 \cdot 5}{4 \cdot 5} + \frac{2 \cdot 4}{5 \cdot 4}) \cdot (\frac{2 0}{3} + \frac{2}{5})

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 5}{b \cdot 5}$

(\frac{3 5 \cdot 5}{4 \cdot 5} + \frac{2 \cdot 4}{5 \cdot 4}) \cdot (\frac{2 0 \cdot 5}{3 \cdot 5} + \frac{2}{5})

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

(\frac{3 5 \cdot 5}{4 \cdot 5} + \frac{2 \cdot 4}{5 \cdot 4}) \cdot (\frac{2 0 \cdot 5}{3 \cdot 5} + \frac{2 \cdot 3}{5 \cdot 3})

Multiply

(\frac{1 7 5}{2 0} + \frac{8}{2 0}) \cdot (\frac{1 0 0}{1 5} + \frac{6}{1 5})

AddFrac

Add fractions

\frac{1 7 5 + 8}{2 0} \cdot \frac{1 0 0 + 6}{1 5}

AddTerms

Add terms

\frac{1 8 3}{2 0} \cdot \frac{1 0 6}{1 5}

When multiplying fractions, the product is equal to the product of the numerators divided by the product of the denominators.

\frac{1 8 3}{2 0} \cdot \frac{1 0 6}{1 5}

MultFrac

Multiply fractions

\frac{1 8 3 \cdot 1 0 6}{2 0 \cdot 1 5}

Multiply

\frac{1 9 3 9 8}{3 0 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 6}{b / 6}$

\frac{1 9 3 9 8 / 6}{3 0 0 / 6}

CalcQuot

Calculate quotient

\frac{3 2 3 3}{5 0}

▼

Write fraction as a mixed number

Rewrite

Rewrite $3233$ as $3200 + 33$

\frac{3 2 0 0 + 3 3}{5 0}

WriteSumFrac

Write as a sum of fractions

\frac{3 2 0 0}{5 0} + \frac{3 3}{5 0}

CalcQuot

Calculate quotient

64 + \frac{3 3}{5 0}

Rewrite

Rewrite $64 + \frac{3 3}{5 0}$ as $64 \frac{3 3}{5 0}$

64 \frac{3 3}{5 0}

The area of the photo with the frame is

64 \frac{3 3}{5 0}

square inches.

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.

$product of random fractions$

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 \frac{2}{6} \times 2 = \frac{8}{6} \times \frac{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 \frac{8}{6} 3 \times \frac{2}{1} = \frac{4}{3} \times \frac{2}{1} = \frac{8}{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?

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 .

\frac{2}{6} \times 5

Notice that the fraction can be simplified to its lowest terms.

\frac{2}{6} \times 5

▼

Reduce by

2

ReduceFrac

$\frac{a}{b} = \frac{a / 2}{b / 2}$

\frac{2 / 2}{6 / 2} \times 5

CalcQuot

Calculate quotient

\frac{1}{3} \times 5

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

\frac{5}{3}

Paulina runs for

\frac{5}{3}

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.

4 \times \frac{5}{3}

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

\frac{4 \times 5}{3}

Multiply

\frac{2 0}{3}

This final fraction can also be written as a mixed number.

\frac{2 0}{3}

▼

Write fraction as a mixed number

Rewrite

Rewrite $20$ as $18 + 2$

\frac{1 8 + 2}{3}

WriteSumFrac

Write as a sum of fractions

\frac{1 8}{3} + \frac{2}{3}

CalcQuot

Calculate quotient

6 + \frac{2}{3}

Rewrite

Rewrite $6 + \frac{2}{3}$ as $6 \frac{2}{3}$

6 \frac{2}{3}

4

days, Paulina runs

6 \frac{2}{3}

hours.

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Catch-Up and Review

Hint

Solution

Extra

Multiplying Fractions By Whole Numbers

Multiplying Fractions By Mixed Numbers

Hint

Solution

Hint

Solution

Hint

Solution