July 2022
Problem of the Month
Cattle Problem
by Benjamin Banneker
A gentleman sent his friend with £100 to buy 100 cattle, with orders to give £5 for each Bullock, 20 Shillings for cows, and one Shilling
for each sheep.
How many of each did his friend buy?
Note: Banneker lived in the eighteenth century when the word cattle was used to describe domesticated animals, including sheep, goats, cows, and bulls.
A shilling was 1/20 of a pound (£).
Solution:
His friend bought 19 bulls, 1 cow, and 80 sheep.
Let b = # of bulls
Let c = # of cows
Let s = # of sheep
Then we can write two Diophantine equations (they have integer solutions):
b + c + s = 100
5b + c + s/20 = 100
c = 100 - b - s
So, 5b + (100 - b - s) + s/20 = 100
Then 80b = 19s.
The first integral solution is b = 19 and s = 80.
Substituting back in the first equation yields c = 1.