Systematic Treatise on Arithmetic (1592) - Page 22

Division Rhymes for Four and Three

Four-five gives twenty	Four-seven is twenty-eight	Three-four is twelve
Four-eight is thirty-two	One-four is like four

Problem: Suppose you now have 123 taels, 4 mace, and 5 candareens of silver These are traditional units: 1 Tael (兩) = 10 Mace (錢); 1 Mace = 10 Candareens (分).. If every 5 taels of silver is exchanged for 1 tael of gold, how much gold should be received?

Answer: 24 taels, 6 mace, 9 candareens

Method: Set the total silver as the dividend dividend (shi); the amount being divided. Use five as the divisor divisor (fa); the number of units per one of the target, and perform the division (gui). The result matches the question.

The Positioning Method: Simply recognize that the position one place ahead of the silver "taels" digit is the "taels" digit for the gold. Advance the positions in reverse order to obtain the result This refers to the decimal shift; since we are dividing by 5, the "tens" place of the silver becomes the "tens" place of the gold, effectively shifting the units..

Division by Five Diagram

Divisor	Dividend	Remainder/Units
Circle containing the character 歸 (Division)	Circle containing the character 逢 (Meeting) Meeting five, carry ten Remove from the current rod and carry 1 to the left This represents $5 \div 5 = 1$.	Circle containing the character 錢 (Mace)
Circle containing the character 五 (Five)	Five-four, make eight; change 4 to 8 This represents $4 \div 5 = 0.8$. On the abacus, a 4 is replaced by an 8 on the same rod.	Circle containing the character 分 (Candareen)

Positioning