The Crest of the Peacock by George Gheverghese Joseph

Author:George Gheverghese Joseph
Language: eng
Format: epub
Publisher: Princeton University Press

EXAMPLE 7.11 If cockerels cost 5 qians each, hens cost 3 qians each, and 3 chickens cost 1 qian, and if 100 fowls are bought for 100 qians, how many cockerels, hens, and chickens are there [the qian is a copper coin]?

Various alternative answers are offered:

• Answer: 4 cockerels, 18 hens, and 78 chickens costing 20, 54, and 26 qians respectively

• Answer: 8 cockerels, 11 hens, and 81 chickens costing 40, 33, and 27 qians respectively

• Answer: 12 cockerels, 4 hens, and 84 chickens costing 60, 12, and 28 qians respectively

Given our powerful tool of symbolic algebra, this problem is easy to solve by specifying the following set of equations, given that x, y, and z are the number of cockerels, hens, and chickens respectively:

5x + 3y + 1/3 z = 100,

x + y + z = 100.

Substitute z = 100 – x – y in the first equation to obtain 7x + 4y = 100 or y = 25 – (7/4)x. Since y must be an integer, it would follow that x must be a multiple of 4. In other words, the solutions are obtained from the equations

x = 4t, y = 25 – 4t, z = 100 – 4t – [25 – 4t ] = 75 + 3t.

For t = 1, 2, 3, we get solution sets (x, y, z): (4, 18, 78), (8, 11, 81), and (12, 4, 84), which are the solution sets given above. For t = 4 or greater, y become negative, which is plainly absurd. However, the solution set corresponding to t = 0 (or no cockerels) was not offered. Neither is a plausible explanation or justification of how the answers were obtained.18

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