Systematic Treatise on Arithmetic (1592) - Page 20

Dividend	Dividend				Dividend	Tail
◯	Start ◯ Begin	◯	◯	◯	◯	Mace
	Encountering six, carry twenty This is an abacus mnemonic for $6 \div 3 = 2$. The "twenty" refers to placing the quotient 2 in the column to the left.	Encountering three, carry ten	Encountering three, carry ten	Encountering three, carry ten	Encountering nine, carry thirty
	Three-two is sixty-two	Three-three, carry twelve	Three-six, carry twenty-one	Three-one is thirty-one
	Remove six from the base position, carry twenty to the left	Remove three from the base position, carry ten to the left	Change two to six, add two to the position on the right	Remove six from the base position, carry twenty to the left	Remove the value from the base position, carry thirty to the left
		Change two to six, add two to the position on the right		Change one to three, add one to the base position

			Positioning
Hundreds ◯ Begin	Tens	Taels	Above the divisor is the tael; where it meets the tael, stop
Call the hundreds and move backward	Combined to obtain		One position ahead yields the dan dan (shi); a unit of volume for grain, often translated as a "stone"
Count in order down to the taels			This is the dan; mark backward in reverse

Restoration: Using Multiplication by Three

original: Huanyuan yong sanyin; "Restoration" is the mathematical practice of checking a division result by multiplying it back by the original divisor.

Three times six is eighteen; three times eight is twenty-four; three times seven is twenty-one.

Two times three is six.

Suppose now we have 735 catties catty (jin); a unit of weight, roughly 600 grams of hemp. For every 4 catties of hemp, the selling price is 1 mace mace (qian); a unit of silver currency weight, one-tenth of a tael of silver. Question: How much silver should be received?

This problem requires dividing the total weight of hemp (735) by the rate (4) to find the total mace of silver. The answer, not explicitly written in the OCR but implied by the surrounding context, would be 183.75 mace.