# Dividing Fractions: Fitting the Divisor

In this lesson we solve fraction divisions by thinking how many times the divisor "fits" or "goes into" the dividend. For example, the fraction 1/4 goes into five 20 times, so 5 ÷ (1/4) = 20. The lesson has lots of exercises with visual models and many word problems. The previous lesson had to do with dividing fractions by whole numbers.

In the video, I explain two different division situations where we don't have to use the "rule" or shortcut for fraction division, but instead can use mental math. The first is when a fraction is divided by a whole number. The second is when the answer to a fraction division is a whole number.

How many times does one thing fit into another? You can always write a division from this situation. Think: “How many times does the divisor go into the dividend?”
 How many times does go into ?
 Eight times. We can write a division:  2 ÷ 1 4 = 8.
 Then check the division:  8 × 1 4 = 8 4 = 2.
 How many times does 1 2 go into 3?
 Six times. We can write a division:  3 ÷ 1 2 = 6.
 Then check the division:  6 × 1 2 = 6 2 = 3.

1. Solve. Write a division. Then write a multiplication that checks your division.

 a. How many times does go into ?
 2 ÷ 1 3 = _____
 Check: ____ × 1 3 =
 b. How many times does go into ?
 1 ÷ 1 4 = _____
 Check: ____ × 1 4 =
 c. How many times does go into ?
 6 ÷ 1 3 = ____

Check:

 d. How many times does go into ?
 5 ÷ 1 4 = ____

Check:

Now you write the division. Be careful: the divisor is the number that “goes into” the dividend.

 e. How many times does go into ?
 ____ ÷ =

Check:

 f.  How many times does go into ?
 ____ ÷ ____ =

Check:

 g. How many times does 1 6 go into 2?
 ____ ÷ ____ =
 h. How many times does 1 5 go into 3?
 ____ ÷ ____ =

2. Divide. Think, “How many times does the divisor go into the dividend?” Use the pictures to help.

 a.   3 ÷ 1 6 =

 b.   4 ÷ 1 9 =
 c.  4 ÷ 1 8 =
 d.   3 ÷ 1 2 =
 e.   3 ÷ 1 7 =
 f.   4 ÷ 1 5 =
 g.  2 ÷ 1 3 =

Did you notice a pattern? There is a shortcut to dividing a whole number by a unit fraction!

 5 ÷ 1 4 ↓ ↓ 5 × 4 = 20
 3 ÷ 1 8 ↓ ↓ 3 × 8 = 24
 9 ÷ 1 7 ↓ ↓ 9 × 7 = 63

Why does it work that way? For example, consider the problem 5 ÷ (1/4). Since 1/4 goes
into 1 exactly 4 times, it must go into 5 exactly 5 × 4 = 20 times.

3. Solve. Use the shortcut.

 a.   3 ÷ 1 6 =
 b.   4 ÷ 1 5 =
 c.   3 ÷ 1 10 =
 d.   5 ÷ 1 10 =
 e.   7 ÷ 1 4 =
 f.    4 ÷ 1 8 =
 g.   4 ÷ 1 10 =
 h.   9 ÷ 1 8 =

4. Write a division for each word problem, and solve. Do not write just the answer.

 a. How many 1/2-meter pieces can you cut from     a roll of string that is 6 meters long? b. How many 1/4-cup servings can you get from 2 cups of almonds? c. Ben has small weights that weigh 1/10 kg each. How many of those would he need to make 5 kg? d. An eraser is 1/8 inches thick. How many erasers can be stacked into a 4-inch tall box?

5. Write a story problem to match each division, and solve.

 a.   2 ÷ 1 2 =

 b.   5 ÷ 1 3 =

6. These divisions are not as easy as the previous ones, but they are not difficult either. Again, think
how many times the divisor goes into the dividend. The pictures can help.

 a.   4 ÷ 2 3 =

 b.   4 ÷ 4 5 =
 c.   2 5 6 ÷ 1 6 =
 d.   3 ÷ 6 10 =

 e.   3 5 9 ÷ 4 9 =
 f.   2 4 8 ÷ 5 8 =

7. Write a division and solve. Write also a multiplication to check your division.

 a. How many times does go into ?
 ____ ÷ ____ = ____
 ____ × ____ = ____
 b. How many times does go into ?
 ____ ÷ ____ = ____
 ____ × ____ = ____

 c. How many times does go into ?
 ____ ÷ ____ = ____
 ____ × ____ = ____
 d. How many times does go into ?
 ____ ÷ ____ = ____
 ____ × ____ = ____

8. A recipe calls for 1/2 cup of butter, among other ingredients.
Alison had plenty of all of the other ingredients except the butter.
How many batches of the recipe can she make if she has ...

a. 3 cups of butter?

b. 2 ½ cups of butter?

9. Jackie made three apple pies and divided them into twelfths.
She plans on serving two slices to each guest. How many
servings will she get out of the three pies?
Hint: Draw a picture.

10. How many 2/10-liter servings do you get from 1 liter of juice?

Out of 4 liters of juice?

11. When Natalie goes jogging, she jogs for 1/4 mile, then walks for 1/4 mile,
then again jogs for 1/4 mile, and so on. How many such stretches are there
for her in a jogging track that is 2 1/2 miles long?

12. Jill makes bead necklaces that must be exactly 24 inches long.  She has size
SS beads, which are 1/8-inch thick, and size S beads, which are 1/4-inch thick.

 Bead Width SS 1/8 in S 1/4 in

a. How many beads would be in a necklace

b. How many beads would be in a necklace

c. She also makes a necklace with the pattern SS-S-SS-S.
How many of each kind of bead does she need?

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.