# Multiply Fractions by Whole Numbers

This lesson teaches you how to multiply fractions by whole numbers, based on visual models. We simply find the total number of pieces by multiplication, which means you multiply the whole number and the top number (numerator) of the fraction. The lesson also has many word problems.

In the video below, I teach multiplying fractions by whole numbers, which is a fairly easy concept. You just need to remember that 4 x (2/3) is not calculated as (4 x 2) / (4 x 3). In the visual model, you can color two thirds, four times, to get the answer. I also show an interesting connection between (1/3) x 5 or one-third of five pies, and 5 x (1/3), or five copies of 1/3.

 3 × 4 5 is three copies of 4 5 . (Look at the picture.)

How many fifths are there in total?

 There are 12 fifths. So,  3 × 4 5 = 12 5 .
Lastly we give the answer as a mixed number:
12/5 is 2 2/5.
=

 3 × 4 5
=
 12 5 =   2 2 5

1. Repeatedly color in the parts to solve the multiplications. Give your answer as a mixed number.

 a.  4 × 7 9 =
 b.  3 × 5 8 =
 c.  5 × 11 12 =
 d.  6 × 7 10 =

2. Fill in.

a.

 2 4 5 =  2 ×

b.

 25 9 =  5 ×

c.

 2 2 8 =  3 ×

Solve, for example by drawing.

3. Erica’s tall drinking glasses each hold 3/8 liters.
How much water does she need to fill four of them?

4. Marlene wants to triple this recipe (make it three times).
How much of each ingredient will she need?

 Brownies 3/4 cup butter 1 1/2 cups brown sugar 4 eggs 1 1/4 cups cocoa powder 1/2 cup flour 2 tsp vanilla

To multiply a whole number by a fraction, find the total number of “pieces” (by multiplication). This means you multiply the whole number and the top number (numerator) of the fraction.
 Example 1.  8 × 3 4 means 8 × 3 pieces, or 24 pieces. Each piece is a fourth. So, we get 24 4 .
 Lastly, we write the answer as a mixed number. This time, 24 4 happens to be the whole number 6.

Example 2. Multiplication can be done in either order. (In other words, multiplication is commutative.)

 So, 3 10 × 5  is the same as  5 × 3 10 .  They both equal 5 × 3 10 = 15 10 . This simplifies to 3 2 , which is 1 1 2 .

5. Solve. Give your answer in lowest terms (simplified) and as a mixed number. Study the example.

 a.  6 × 4 9 = 24 9 = 8 3 =  2 2 3
 b.  4 × 7 10 =
 c.  2 × 11 20 =
 d.  9 × 2 15 =
 e. 15 6 × 2  =
 f.  6 × 7 100 =
 g. 1 12 × 16  =
 h.  2 × 35 100 =
 i. 9 20 × 10  =
 j. 7 15 × 7  =

6. William asked 20 fifth graders how much time they spent on housework/chores the day before. He
then rounded the answers to the nearest 1/8 hour. The line plot shows his results. Each x-mark

a. Exclude the three students who did the least housework and the three who did the most, and fill in:

Most students used between ___________ and __________ hours for housework and chores.

b. The average for this data is 7/8 hours. Use this to calculate how
many hours these 20 fifth graders used for housework in total.

A REMINDER

A fraction of a number means that fraction TIMES the number.
In other words, the word “of”  translates into multiplication. For example

 3 10 of \$120 ↓ 3 10 × \$120

Now, you have previously learned how to find 3/10 of \$120 using division:

• First, divide \$120 by 10 to find 1/10 of it. It is \$12.
• Then, multiply that by 3 to get 3/10 of \$120. You get \$36.

 We also get the same answer with fraction multiplication: 3 10 × \$120  = 3 × \$120 10 = \$360 10 = \$36.

Both methods are essentially the same: you divide by 10 and multiply by 3, just in two different orders.

7. Find the following quantities.

a. 2/5 of 35 lb

b. 4/9 of 180 km

8. Dad is building a shelf that is 4 meters long. He wants to use
2/5 of it for gardening supplies and the rest for tools.
How long are these two parts of the shelf?
(Hint: Using centimeters can help.)

9. a. Janet and Sandy earned \$81 for doing yard work. They divided
the money unequally so that Janet got 2/3 of it and Sandy got
the rest. How much money did each girl get?

b. What happens if the amount they earned is \$80 instead?

10. Andy drew a 5 in. by 4 in. rectangle on paper. Then he drew
a second rectangle that was 3/4 as long and wide as the first one.

a. How long and how wide was Andy’s second rectangle?

b. Draw both rectangles (on separate paper).

Epilogue: There is something interesting about multiplying “a fraction times a whole number”
or multiplying “a whole number times a fraction.” Let’s compare.
 1 4 × 12 means a fourth part of 12, which is 3.
 =

 12 × 1 4 means 12 copies of 1/4,

which makes 3 whole pies.

 Notice: Both 1 4 × 12  and  12 × 1 4 equal 3. That makes sense, because multiplication can be

done in any order. But they mean different things (a fourth part of 12, and 12 copies of 1/4).

11. Fill in the missing parts.

a. A two-fifth part of 10 10 copies of 2/5
 × 10 means a two-fifth part of 10,

which is equal to        .

 =
 10 × means 10 copies of ,

which is equal to        .

b. A ______________ part of 5 5 copies of 1/3
 1 3 × 5 means a                                   part of 5,

which is equal to        .

 =
 5 × means 5 copies of ,

which is equal to        .

c. ____________________  of 7 7 copies of _______
 × 7 means                                            of 7,

which is equal to        .

 =
 7 × means 7 copies of ,

which is equal to        .

This lesson is taken from my book Math Mammoth Fractions 2.

#### Math Mammoth Fractions 2

A self-teaching worktext that teaches fractions using visual models, a sequel to Math Mammoth Fractions 1. The book covers simplifying fractions, multiplication and division of fractions and mixed numbers, converting fractions to decimals, and ratios.