A farmer died and left 17 cows to be divided among three
workers. The first worker was to receive of the cows, the second worker was to
receive of the cows, and the third worker was to receive of the cows. The
executor of the farmer’s estate realized that 17 cows could not be divided into
halves, thirds, or ninths and so added a neighbor’s cow to the farmer’s. With 18
cows, the executor gave 9 cows to the first worker, 6 cows to the second worker,
and 2 cows to the third worker. This accounted for the 17 cows, so the executor
returned the borrowed cow to the neighbor. Explain why this works.
what is one half, one third, and one ninth added? Is it a whole?
The problem seems a bit confusing at first, but let's break it down step by step to understand why it works:
1. The farmer left 17 cows to be divided among three workers, and their respective shares were specified as follows:
- The first worker gets 1/2 (half) of the cows.
- The second worker gets 1/3 (one-third) of the cows.
- The third worker gets 1/9 (one-ninth) of the cows.
2. However, it is stated that 17 cows cannot be divided into halves, thirds, or ninths evenly. This means that dividing the cows as per the original plan would not work mathematically.
3. To address this problem, the executor of the farmer's estate adds a neighbor's cow to the original 17 cows, making the total count 18 cows.
4. Now, with 18 cows in total, the executor redistributes them as follows:
- The first worker receives 9 cows (which is 1/2 of 18).
- The second worker receives 6 cows (which is 1/3 of 18).
- The third worker receives 2 cows (which is 1/9 of 18).
5. If we add up the shares received by the workers in the new distribution, we get a total of 9 + 6 + 2 = 17 cows. This accounts for all the cows, leaving one cow unaccounted for.
6. The unaccounted cow is the one that was borrowed from the neighbor. Since it was originally added to make the division work, it is now returned to the neighbor, leaving the final distribution with the correct number of cows.
In summary, by adding one extra cow to the original count and redistributing them accordingly, the executor was able to divide the cows among the workers in a mathematically feasible way, and the addition and subsequent removal of the extra cow balanced out the distribution.