Dividing fractions by whole numbers

This lesson teaches how to divide fractions by whole numbers (sharing divisions) using mental math. We use the analogy of dividing pie pieces evenly among a certain number of people.

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.

First, let’s divide pieces of pie evenly among a certain number of people. This means that we divide a fraction by a whole number.
4 5  of a pie is divided among two people. Each person gets   2 5   of the pie.       4 5  ÷ 2 =  2 5      Check:   2 5  × 2  =  4 5	9 10   is divided among three people. Each person gets   3 10   of the pie.       9 10  ÷ 3 =  3 10      Check:  3 10  × 3  =  9 10
Note how we can check each division by multiplication!

1.  Color each person’s share with a different color, and write a division sentence.

2. Write a division sentence for each problem and solve it. 

Next, we divide unit fractions—fractions like 1/2, 1/3, 1/5, 1/8, 1/12, and so on (of the form 1/n).
One-half is divided equally among four people. Each person gets 1/8 of it. Can you see why? 1 2  ÷ 4 =  1 8      Check:     1 8  × 4  =  4 8   =  1 2	One-fifth is divided among three people. Each person gets 1/15. To see that, divide each fifth (colored and uncolored) into three new parts. 1 5  ÷ 3 =  1 15      Check:     1 15  × 3  =  3 15   =  1 5

3. Split the unit fraction equally among the people. Write a division sentence. Write a multiplication
    sentence to check your division.

a. Divide between two people.      1 2  ÷ 2 = Check:    × 2 =	b. Divide between two people.      1 3  ÷ 2 = Check:    × 2 =	c. Divide between two people.      1 5  ÷ 2 = Check:    × ____ =
d. Divide between two people.       ÷ 2 =	e. Divide among five people.     ÷ 5 =	f. Divide among four people.
g. Divide among four people.	h. Divide among three people.	i. Divide among three people.
Here is a shortcut for dividing a unit fraction 1/n by a whole number m:    1 n  ÷ m  =  1 m × n  . Example.   1 8  ÷ 7  =  1 56  . Multiply the denominator of the unit fraction by the divisor to get the new denominator.

4. Solve.

5. Three children share 1/4 lb of chocolate equally.

    a. How much does each one get, in pounds?

    b. In ounces?

6. A half a liter of juice is poured evenly into five glasses.

    a. How much juice is in each glass, measured in liters?

    b. How many milliliters of juice is in each glass?

7. There are 12 beakers with various amounts of oil in them. The line plot shows how much oil
    each beaker has, in cups.

    If all the oil in the beakers was poured together, and then distributed evenly into the 12 beakers,
    how much oil would be in each beaker?

8. Solve.

9. Solve the opposite problem: if each person got this much pie, how much was there originally?

a.   ÷ 3  =  1 4

b.   ÷ 2  =  2 5

c.   ÷ 6  =  1 7

d.   ÷ 3  =  3 10

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

11. One morning, Joshua's gasoline container was only 1/8 full.
      He poured half of it into his lawnmower.

      a. How full is the gasoline container now?
 

      b. If the container holds 3 gallons, what is the amount of gasoline left, in gallons?

          (Challenge) In quarts?

Lastly, we will divide multiple leftover pie pieces among a certain number of people. This is a bit trickier, but I think you can do it!

When 3/4 is divided equally between two people, one fourth piece must be split into two. Each person gets 1/4 and 1/8.      3 4  ÷ 2 =  1 4  +  1 8  =   3 8

Another way of solving the same problem is to split each fourth piece into 2. This means we first change the 3/4 into 6/8, and then we can divide evenly by 2.     3 4  ÷ 2   ↓     6 8  ÷ 2  =  3 8

12. The leftover pie is divided equally. How much does each person get? Write a division sentence.

a. Divide 5/6 between two people.     First, split each piece into 2 new ones.	b. Divide 2/3 among three people.     First, split each piece into 3 new ones.
c. Divide 2/3 among four people.	d. Divide 3/4 among four people.
e. Divide 2/5 among three people.     First, split each piece into 3.	f. Divide 4/5 among three people.

This lesson is taken from Maria Miller's book Math Mammoth Fractions 2, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.