Topics
1. Operations with Fractions
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United States of AmericaVirginia
Grade 6

1.03 Multiply fractions and mixed numbers

Lesson
Practice

Multiply fractions and whole numbers

Fractions describe parts of a whole, but they can also describe parts of a quantity.

Find \dfrac{1}{12} of 36.

This image shows a square divided into 36 smaller squares

Let's start by drawing a grid of 36.

This image shows a 12 vertical rectangles that consists 3 squares each.

To find one twelfth, we split this grid into 12 equal parts.

Looking at the pieces, each piece has 3 squares. So \dfrac{1}{12} of 36 is 3.

We can also work this out using arithmetic. We know that \dfrac{1}{12} of 36 can be written using multiplication, \dfrac{1}{12} \cdot 36.

This is the same as \dfrac{1}{12} \cdot \dfrac{36}{1} because the whole number 36 can be written as a fraction over 1.

First, if we evaluate the multiplication of the numerators we get 36. And if we evaluate the multiplication in of the denominators we get 12.

Next we can simplify the fraction by factoring out the greatest common factor, which is 12. This gives us \dfrac{3}{1} which is the same as 3.

We can check this answer by multiplying back. 12 \cdot 3 = 36, so we know that 3 is \dfrac{1}{12} of 36.

Exploration

Use the dropdown menus to choose a whole number and fraction to be multiplied.

Drag the slider to combine the models.

Check the 'Arrange' box to reorganize the model.

Loading interactive...

Use the applet to model a multiplication problem then answer the following questions:

  1. What is happening when you slide the slider? How does this represent the multiplication?
  2. What does checking the 'Arrange' checkbox do? How might this help with understanding the result?
  3. Use the applet with a few more multiplication problems. What patterns do you notice between the numbers being multiplied and the result?
  4. What can you say about multiplying a whole number by a fraction between 0 and 1?

When a whole number is multiplied by a fraction between 0 and 1, the result is smaller than the original whole number.

Examples

Example 1

Each rectangle represents 1 whole.

Three rectangles divided into six parts. Each rectangle has 4 of the squares shaded
a

Write the product that is represented by the model.

Worked Solution
Create a strategy

Look at the shading in each rectangle to figure out what fraction is shaded. Multiply the shaded fraction of a rectangle by the total number of large rectangles to find the product represented by the model.

Apply the idea

In each rectangle, there are 6 equal parts and 4 of them are shaded. We could say "4 out of 6 parts of each rectangle are shaded, \dfrac{4}{6} of each rectangle is shaded. This simplifies to \dfrac{2}{3}. There are 3 rectangles, so the product represented by the model is: 3 \cdot \dfrac{4}{6} \text{ or }3 \cdot \dfrac{2}{3}.

b

Evaluate the product using the model.

Worked Solution
Apply the idea
\displaystyle 3 \cdot \dfrac{2}{3}\displaystyle =\displaystyle \dfrac{3}{1} \cdot \dfrac{2}{3}Rewrite the whole number as a fraction
\displaystyle =\displaystyle \dfrac{6}{3}Multiply the numerators and denominators
\displaystyle =\displaystyle 2Divide

Another way to evaluate this is we divide out the common factors in the numerator and denominator:

\displaystyle 3 \cdot \dfrac{2}{3}\displaystyle =\displaystyle \dfrac{3}{1} \cdot \dfrac{2}{3}Rewrite the whole number as a fraction
\displaystyle =\displaystyle \dfrac{\cancel{3}_{1}}{1} \cdot \dfrac{2}{\cancel{3}_{1}}Divide out the common factors
\displaystyle =\displaystyle 2Simplify

So, the product represented by the model is 2.

Reflect and check

We can further visualize the result by rearranging the shaded boxes into smaller rectangles and comparing them with two whole rectangles that have 6 parts each.

Four vertical rectangular bars each divided into four squares by dashed lines, arranged side by side with an arrow pointing right to another set of identical four vertical rectangular bars. The layout suggests a transformation or rearrangement process between the two sets of rectangles. Each rectangle is filled with a light teal color, and the dashed lines are white, emphasizing segmentation within the rectangles.

In the image, we can see that the 12 shaded boxes from the original three rectangles can be rearranged to fill two whole rectangles with 6 parts each. There is one rectangle without any shaded parts. This shows that the shading in the model represents the product 2, as the shaded boxes are equivalent to two whole rectangles.

Example 2

Evaluate \dfrac{2}{5}\cdot 35

Worked Solution
Create a strategy

Multiply numerators and denominators separately. The denominator of a whole number is always 1.

Apply the idea
\displaystyle \dfrac{2}{5}\cdot 35\displaystyle =\displaystyle \dfrac{2\cdot35}{5\cdot1}Multiply numerators and denominators
\displaystyle =\displaystyle \dfrac{70}{5}Evaluate
\displaystyle =\displaystyle 14Simplify
Idea summary

Finding a fraction of a quantity is the same as multiplying a fraction by a whole number.

To multiply a fraction by a whole number, multiply the numerator by the whole number.

When multiplying any whole number by a fraction between 0 and 1 the result is smaller than the original whole number.

Multiply fractions by fractions

To multiply two fractions together, we'll start by thinking of the fractions as multiples of unit fractions, and work towards a more efficient strategy.

Let's take an example of \,\dfrac{2}{3} \cdot \dfrac{4}{5} . We can rewrite these fractions as

\dfrac{2}{3} = 2 \cdot \dfrac{1}{3} \quad\text{and}\quad \dfrac{4}{5} = 4 \cdot \dfrac{1}{5}

We can then multiply the whole parts together:

\begin{aligned} \dfrac{2}{3} \cdot \dfrac{4}{5} &=2 \cdot \dfrac{1}{3} \cdot 4 \cdot \dfrac{1}{5}\\\\ &=8 \cdot \dfrac{1}{3} \cdot \dfrac{1}{5} \end{aligned}

What can we do with the product of the unit fractions \dfrac{1}{3} and \dfrac{1}{5}?

Square divided into 3 vertical parts where the 1 part is shaded.

Well, this is like taking one whole, dividing it into 3 pieces to get thirds.

Square divided into 15 parts where 1 part is shaded.

Then dividing each of those thirds into 5 pieces.

The result is that the whole has been divided into 15 pieces where we only want 1 piece.

This image represents the fraction \dfrac{1}{15}.

We can now finish our multiplication:

\begin{aligned} \dfrac{2}{3} \cdot \dfrac{4}{5} &=8 \cdot \dfrac{1}{3} \cdot \dfrac{1}{5}\\\\ &=8 \cdot \dfrac{1}{15}\\\\ &=\dfrac{8}{15} \end{aligned}

Do you notice the pattern that has happened here?

In a fraction, the denominator tells us the size of the pieces, and the numerator tells us how many pieces there are. When we multiply two fractions, the denominators multiply together to tell us the new size of the pieces, and the numerators also multiply together to tell us how many of the new pieces there are.

That is:

\begin{aligned} \dfrac{2}{3} \cdot \dfrac{4}{5} &=\dfrac{2\cdot4}{3\cdot5}\\\\ &=\dfrac{8}{15} \end{aligned}

We can multiply with mixed numbers as well, because they're really just fractions. We just have the added step of converting the mixed number to a fraction first.

Exploration

Use the dropdown menus to choose the fractions to be multiplied.

Drag the slider to combine the models.

Loading interactive...

Use the applet to model a multiplication problem then answer the following questions:

  1. What is happening when you slide the slider? How does this represent the multiplication?
  2. Use the applet with a few more multiplication problems. What patterns do you notice between the numbers being multiplied and the result?
  3. What can you say about multiplying a fraction by a fraction between 0 and 1?

When a fraction is multiplied by a fraction between 0 and 1, the result is smaller than the original fraction.

Examples

Example 3

Demonstrate how to multiply 1\dfrac{1}{4} and \dfrac{2}{3} using a number line. Then find the product.

Worked Solution
Create a strategy

To multiply 1\dfrac{1}{4} and \dfrac{2}{3} using a number line, start by drawing a number line from 0 to 1\dfrac{1}{4}. Then, divide each part between 0 and 1\dfrac{1}{4} into 3 equal parts. Finally, shade 2 of the 3 parts in each of the sections.

Apply the idea

Draw a number line from 0 to 1\dfrac{1}{4}.

0\frac{1}{4}\frac{1}{2}\frac{3}{4}11\frac{1}{4}

Divide each part between 0 and 1\dfrac{1}{4} into 3 equal parts.

0\frac{1}{4}\frac{1}{2}\frac{3}{4}11\frac{1}{4}

Shade 2 of the 3 parts in each of the sections.

A number line ranging from 0 to 1 1/4 with label markings at 0, 1/4, 1/2, 3/4, and 1 1/4. Each step is divided into 3 portions. Each 2 of 3 is shaded.

Since 10 pieces are shaded and 12 pieces make up one whole, 1\dfrac{1}{4} \cdot \dfrac{2}{3} = \dfrac{10}{12}.

Example 4

Find the value of the following:

a

\dfrac{1}{3}\cdot\dfrac{7}{10}

Worked Solution
Create a strategy

Multiply the numerators and denominators together.

Apply the idea
\displaystyle \dfrac{1}{3}\cdot\dfrac{7}{10}\displaystyle =\displaystyle \dfrac{1\cdot7}{3\cdot10}Multiply the numerators and denominators
\displaystyle =\displaystyle \dfrac{7}{30}Evaluate
Reflect and check

We can use an area model to verify our answer.

For an area model, we can think of the product as the area of a rectangle with a width of \dfrac{1}{3} and a length of \dfrac{7}{10}.

A grid comprised of five rows and six columns, each cell distinctly colored, forming a rectangular block. The first row contains five purple squares and one blue square. The second and third rows each consist of six red squares. The fourth row features three red squares followed by three white squares, and the fifth row contains six white squares. Each square is bordered by thin black lines that define the grid structure.

The first row of the rectangle represents \dfrac{1}{3} of its total area and is shaded blue. We further divide each of these thirds into 10 equal pieces, resulting in a total of 30 small rectangles. We then shade 7 columns, each containing three small rectangles, in red. The overlapping purple-shaded area represents the product of \dfrac{1}{3} and \dfrac{7}{10}.

There are 7 purple small rectangles out of 30 small rectangles, which shows that the product of \dfrac{1}{3} and \dfrac{7}{10} is \dfrac{7}{30}.

b

\dfrac{5}{3}\cdot\dfrac{21}{2}

Worked Solution
Create a strategy

Multiply numerators and denominators separately.

Apply the idea
\displaystyle \dfrac53\cdot\dfrac{21}{2}\displaystyle =\displaystyle \dfrac{5\cdot21}{3\cdot2}Multiply numerators and denominators
\displaystyle =\displaystyle \dfrac{105}{6}Evaluate
\displaystyle =\displaystyle \dfrac{35}{2}Simplify
c

5 \dfrac{4}{7} \cdot 4 \dfrac{2}{3}

Worked Solution
Apply the idea
\displaystyle 5 \dfrac{4}{7} \cdot 4 \dfrac{2}{3} \displaystyle =\displaystyle \dfrac{39}{7} \cdot \dfrac{14}{3}Rewrite mixed numbers as improper fractions
\displaystyle =\displaystyle \dfrac{\cancel{39}^{13}}{7} \cdot \dfrac{14}{\cancel{3}_{1}}Simplify \dfrac{39}{3}
\displaystyle =\displaystyle \dfrac{13}{\cancel{7}_{1}} \cdot \dfrac{\cancel{14}^{2}}{1}Simplify \dfrac{14}{7}
\displaystyle =\displaystyle 26Evaluate
Reflect and check

We can use estimation as a quick way to check if our answer is reasonable. Start by rounding the mixed numbers to numbers that are more familiar and easier to work with.

5\dfrac{4}{7} is close to 5\dfrac{1}{2} because \dfrac{4}{7} is close to \dfrac{4}{8}=\dfrac{1}{2}.

Next, we can approximate 4\dfrac{2}{3}as 5 because it is just \dfrac{1}{3} less than 5.

For our estimation we are calculating: 5 \dfrac{1}{2} \cdot 5.

We can easily find that 5 \cdot 5=25 and 5 \cdot \dfrac{1}{2}=\dfrac{5}{2}=2.5 so 5 \dfrac{1}{2} \cdot 5=25+2.5=27.5

This is close to our actual answer of 26 so we know our answer is reasonable. However, our estimation is a little too high. This is because we rounded 4\dfrac{2}{3} up to 5.

Example 5

Danielle takes 4 \dfrac {7}{9} minutes to drive from her home to the local shopping center. She spends \dfrac {1}{9} of this time waiting at traffic lights.

Find the number of minutes she spends waiting.

Worked Solution
Create a strategy

Multiply the given numbers.

Apply the idea
\displaystyle \text{Time}\displaystyle =\displaystyle 4 \dfrac{7}{9} \cdot \dfrac{1}{9}Multiply 4 \dfrac{7}{9} by \dfrac{1}{9}
\displaystyle =\displaystyle \dfrac {43}{9} \cdot \dfrac{1}{9}Rewrite 4 \dfrac{7}{9} as improper fraction
\displaystyle =\displaystyle \dfrac {43}{81} Evaluate
Idea summary

To multiply two fractions, multiply the numerators and the denominators separately.

To multiply mixed numbers by a fraction or by another mixed number, convert the mixed number to an improper fraction first. Then, multiply the numerators and denominators separately.

When multiplying any number by a fraction between 0 and 1 the result is smaller than the original number.

Outcomes

6.CE.1

The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context.

6.CE.1a

Demonstrate/model multiplication and division of fractions (proper or improper) and mixed numbers using multiple representations.*

6.CE.1b

Multiply and divide fractions (proper or improper) and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form.*

6.CE.1c

Investigate and explain the effect of multiplying or dividing a fraction, whole number, or mixed number by a number between zero and one.*

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