Systematic Treatise on Arithmetic: Volume 2

Tael: The divisor is silver; when it reaches the 'tael' position, stop. The place immediately ahead is the gold 'tael' position. Advancing forward, the result is Mace. Tael: Five-three doubled makes six; change three to six This is a mnemonic for dividing by five: since $3 \div 5 = 0.6$, the operator replaces the 3 on the abacus with a 6 and shifts the decimal.. Thousand: Five-two doubled makes four; change two to four. Dividend Hundred: Start here; Five-one doubled makes two; change one to two. Ten. Original Hundred: Start here; call out the hundreds going backward; count forward to the taels; the result is Thousand. Dividend Hundred.

Five-nine is forty-five; five-six gives thirty; four-five gives twenty.

Two-five gives ten.

Problem: Suppose you now have 20 dan dan (石); a unit of dry measure often translated as a "picul," roughly 100 liters of rice to be divided among 50,000 people. How much rice does each person receive?

Answer: 4 shao shao (勺); a small unit of volume, 1/100,000th of a dan.

Method: Set the rice as the dividend shi (實); the total quantity to be divided and the number of people 50,000 as the divisor fa (法); the number of units to divide by. Perform the division (gui) to find the answer.

The Positioning Method: For cases where the divisor is large and the dividend is small: Start from the original position of the first digit of the dividend. Count backward [upward on the abacus] until you reach the position of the first digit of the divisor; this determines the unit of the result.

**Technical