By: Tao Steven Zheng (鄭濤)

【Problem】

Zhang Qiujian Suanjing Chapter 3 Problem 38

Suppose roosters costs?5?coins each, hens?3?coins each and three chickens cost?1?coin. If?100?fowls are bought for?100?coins, how many roosters, hens and chickens are there?

【Solution】

Let ?represent the number of roosters, hens, chickens respectively. The problem is a system of two linear equations in three unknowns:

?

Multiply equation (2) by 3 to get

Subtract equation (1) from equation (2) to obtain

Hence,

Substitute ?into equation (1) to obtain

?Therefore,

By the nature of the problem, the solutions for must be non-negative integers; hence, ?must be divisible by 4. Substitute ? into the above solution set to obtain

where .

?

Solution 1: 0 roosters, 25 hens, and 75 chickens. [1]

Solution 2: 4 roosters, 18 hens, and 78 chickens.

Solution 3: 8 roosters, 11 hens, and 81 chickens.

Solution 4: 12 roosters, 4 hens, and 84 chickens.

?

[1] Zhang Qiujian only gave positive integer solutions, so the first solution (0 roosters, 25 hens, and 75 chickens) was not included in the “Zhang Qiujian Suanjing”.