This unit is suitable for upper elementary and middle school.

THE LESSON

Many problems which deal with sharing when the division is into unequal parts lead to computing a sum of unit fractions (fractions with 1 in the numerator) such as:

1/2 + 1/3 + 1/8 + … + … = ?

You have probably heard the problem of the last will of an old shepherd:

“My oldest child will get ½ of my herd, the second oldest will get ¼, the third oldest will get 1/8, and the youngest will get 1/16.”

The shepherd left 15 cows. How can we divide them?

The solution: Borrow one cow, making a herd of 16 cows. Now give one half (8 cows) to the oldest child, ¼ (4 cows) to the second, 1/8 (2 cows) to the third, and 1/16 (one cow) to the youngest. 8 + 4 + 2 + 1 = 15, so we have one cow left, which we return to the lender.

Of course, this strange solution is due to a lack of arithmetical skills on the part of the old shepherd. ½ + ¼ + 1/8 + 1/16 = 15/16, so he willed less than he had. (He willed only 15/16 of his herd.)

      1/2 of 16 =   8

      1/4 of 16 =   4

      1/8 of 16 =   2

    1/16 of 16 =   1

  15/16 of 16 = 15

You can use different sets of numbers for similar problems. Three good sets (for a shepherd with 3 children) are:

• 1/2 + 1/3 + 1/7 = 41/42 (herd of 41 cows) 1/2 of 42 = 21
                                                                     1/3 of 42 = 14
                                                                     1/7 of 42 =   6
                                                                 41/42 of 42 = 41


• 1/2 + 1/3 + 1/8 = 23/24 (flock of 23 sheep) 1/2 of 24 = 12
                                                                       1/3 of 24 =   8
                                                                       1/8 of 24 =   3
                                                                   23/24 of 24 = 23


• 1/2 + 1/3 + 1/9 = 17/18 (herd of 17 camels) 1/2 of 18 =  9
                                                                        1/3 of 18 =  6
                                                                        1/9 of 18 =  2
                                                                    17/18 of 18 = 17

Here is one more:

• 1/2 + 1/4 + 1/5 = 19/20 (19 partridges)  1/2 of 20 = 10
                                                                  1/4 of 20 =   5
                                                                  1/5 of 20 =   4
                                                              19/20 of 20 = 19