九章算術·方田（分數諸術）
The Nine Chapters: The Arts of Fractions
Anonymous (Han dynasty) 佚名 (compiled by the 1st century CE)
Qin–Han · compiled by the 1st century CE

Cold Library · Open Stacks · S-007006 · https://coldlibrary.com/s/007006
Rights: A classical Chinese text about two thousand years old; public domain everywhere. Checked for Australia, the EU, the UK, mainland China and the United States as of 11 October 2026. Guidance, not legal advice.
Source: Wikisource, https://zh.wikisource.org/wiki/九章算術 (retrieved 2026-10-11)
English translation: Claude (an AI), 2026 · CC0 1.0
Generated 2026-10-14 from https://coldlibrary.com/en/open-stacks/nine-chapters-the-arts-of-fractions

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卷第一　方田

今有十八分之十二。問：約之得幾何？

荅曰：三分之二。

又有九十一分之四十九。問：約之得幾何？

荅曰：十三分之七。

約分術曰：可半者半之；不可半者，副置分母、子之數，以少減多，更相減損，求其等也。以等數約之。

今有三分之一，五分之二。問：合之得幾何？

荅曰：十五分之十一。

又有三分之二，七分之四，九分之五。問：合之得幾何？

荅曰：得一、六十三分之五十。

又有二分之一，三分之二，四分之三，五分之四。問：合之得幾何？

荅曰：得二、六十分之四十三。

合分術曰：母互乘子，并以為實，母相乘為法。實如法而一。不滿法者，以法命之。其母同者，直相從之。

今有九分之八，減其五分之一。問：餘幾何？

荅曰：四十五分之三十一。

又有四分之三，減其三分之一。問：餘幾何？

荅曰：十二分之五。

減分術曰：母互乘子，以少減多，餘為實。母相乘為法。實如法而一。

今有八分之五，二十五分之十六。問：孰多？多幾何？

荅曰：二十五分之十六多，多二百分之三。

又有九分之八，七分之六。問：孰多？多幾何？

荅曰：九分之八多，多六十三分之二。

又有二十一分之八，五十分之十七。問：孰多？多幾何？

荅曰：二十一分之八多，多一千五十分之四十三。

課分術曰：母互乘子，以少減多，餘為實。母相乘為法。實如法而一，即相多也。

今有三分之一，三分之二，四分之三。問：減多益少，各幾何而平？

荅曰：減四分之三者二，三分之二者一，并，以益三分之一，而各平於十二分之七。

又有二分之一，三分之二，四分之三。問：減多益少，各幾何而平？

荅曰：減三分之二者一，四分之三者四，并以益二分之一，而各平於三十六分之二十三。

平分術曰：母互乘子，副并為平實。母相乘為法。以列數乘未并者各自為列實。亦以列數乘法。以平實減列實，餘，約之為所減。并所減以益於少。以法命平實，各得其平。

今有七人，分八錢三分錢之一。問：人得幾何？

荅曰：人得一錢、二十一分錢之四。

又有三人，三分人之一，分六錢三分錢之一，四分錢之三。問：人得幾何？

荅曰：人得二錢、八分錢之一。

經分術曰：以人數為法，錢數為實，實如法而一。有分者通之；重有分者同而通之。

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卷第一　方田

今有十八分之十二。问：约之得几何？

答曰：三分之二。

又有九十一分之四十九。问：约之得几何？

答曰：十三分之七。

约分术曰：可半者半之；不可半者，副置分母、子之数，以少减多，更相减损，求其等也。以等数约之。

今有三分之一，五分之二。问：合之得几何？

答曰：十五分之十一。

又有三分之二，七分之四，九分之五。问：合之得几何？

答曰：得一、六十三分之五十。

又有二分之一，三分之二，四分之三，五分之四。问：合之得几何？

答曰：得二、六十分之四十三。

合分术曰：母互乘子，并以为实，母相乘为法。实如法而一。不满法者，以法命之。其母同者，直相从之。

今有九分之八，减其五分之一。问：余几何？

答曰：四十五分之三十一。

又有四分之三，减其三分之一。问：余几何？

答曰：十二分之五。

减分术曰：母互乘子，以少减多，余为实。母相乘为法。实如法而一。

今有八分之五，二十五分之十六。问：孰多？多几何？

答曰：二十五分之十六多，多二百分之三。

又有九分之八，七分之六。问：孰多？多几何？

答曰：九分之八多，多六十三分之二。

又有二十一分之八，五十分之十七。问：孰多？多几何？

答曰：二十一分之八多，多一千五十分之四十三。

课分术曰：母互乘子，以少减多，余为实。母相乘为法。实如法而一，即相多也。

今有三分之一，三分之二，四分之三。问：减多益少，各几何而平？

答曰：减四分之三者二，三分之二者一，并，以益三分之一，而各平于十二分之七。

又有二分之一，三分之二，四分之三。问：减多益少，各几何而平？

答曰：减三分之二者一，四分之三者四，并以益二分之一，而各平于三十六分之二十三。

平分术曰：母互乘子，副并为平实。母相乘为法。以列数乘未并者各自为列实。亦以列数乘法。以平实减列实，余，约之为所减。并所减以益于少。以法命平实，各得其平。

今有七人，分八钱三分钱之一。问：人得几何？

答曰：人得一钱、二十一分钱之四。

又有三人，三分人之一，分六钱三分钱之一，四分钱之三。问：人得几何？

答曰：人得二钱、八分钱之一。

经分术曰：以人数为法，钱数为实，实如法而一。有分者通之；重有分者同而通之。

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English translation by Claude (an AI)

Chapter One: Field Measurement

Now there are twelve eighteenths. Question: reduced, what does it make?

Answer: two thirds.

Again there are forty-nine ninety-firsts. Question: reduced, what does it make?

Answer: seven thirteenths.

The art of reducing fractions: if both can be halved, halve them. If they cannot be halved, set down the denominator and the numerator side by side; take the lesser from the greater, and go on subtracting each from the other in turn, to find the number in which they become equal. Reduce both by this equal number.

Now there are one third and two fifths. Question: put together, what do they make?

Answer: eleven fifteenths.

Again there are two thirds, four sevenths and five ninths. Question: put together, what do they make?

Answer: one, and fifty sixty-thirds.

Again there are one half, two thirds, three quarters and four fifths. Question: put together, what do they make?

Answer: two, and forty-three sixtieths.

The art of joining fractions: let each denominator multiply the other numerators, and add the results together to make the dividend; multiply the denominators together to make the divisor. Divide the dividend by the divisor. What falls short of the divisor is named as a part of the divisor. Where the denominators are the same, simply add the numerators.

Now there are eight ninths, from which one fifth is taken. Question: what remains?

Answer: thirty-one forty-fifths.

Again there are three quarters, from which one third is taken. Question: what remains?

Answer: five twelfths.

The art of subtracting fractions: let each denominator multiply the other numerator, and take the lesser from the greater; the remainder is the dividend. Multiply the denominators together to make the divisor. Divide the dividend by the divisor.

Now there are five eighths and sixteen twenty-fifths. Question: which is more, and by how much?

Answer: sixteen twenty-fifths is more, by three two-hundredths.

Again there are eight ninths and six sevenths. Question: which is more, and by how much?

Answer: eight ninths is more, by two sixty-thirds.

Again there are eight twenty-firsts and seventeen fiftieths. Question: which is more, and by how much?

Answer: eight twenty-firsts is more, by forty-three thousand-and-fiftieths.

The art of comparing fractions: let each denominator multiply the other numerator, and take the lesser from the greater; the remainder is the dividend. Multiply the denominators together to make the divisor. Divide the dividend by the divisor; this is how much one exceeds the other.

Now there are one third, two thirds and three quarters. Question: taking from the more to add to the less, how much of each makes them level?

Answer: take two twelfths from the three quarters and one twelfth from the two thirds, put them together and add them to the one third, and each is level at seven twelfths.

Again there are one half, two thirds and three quarters. Question: taking from the more to add to the less, how much of each makes them level?

Answer: take one thirty-sixth from the two thirds and four from the three quarters, put them together and add them to the one half, and each is level at twenty-three thirty-sixths.

The art of levelling fractions: let each denominator multiply the other numerators, and add the results aside to make the level dividend. Multiply the denominators together to make the divisor. Multiply each of the products not yet added by the number of terms, to make the dividend of each term; multiply the divisor, too, by the number of terms. Take the level dividend from the dividend of each term; reduce the remainder, and this is what is taken away. Put together what is taken away and add it to the lesser. Name the level dividend as parts of the divisor, and each comes to its level.

Now seven people share eight coins and one third of a coin. Question: how much does each person get?

Answer: each person gets one coin and four twenty-firsts of a coin.

Again three people and one third of a person share six coins, one third of a coin and three quarters of a coin. Question: how much does each person get?

Answer: each person gets two coins and one eighth of a coin.

The art of dividing fractions: take the number of people as the divisor and the number of coins as the dividend, and divide the dividend by the divisor. Where there are fractions, bring them through to a single kind; where both have fractions, make their denominators the same and bring them through.