九章算術·方田（分數諸術）九章算术·方田（分数诸术）

今有十八分之十二。問：約之得幾何？

今有十八分之十二。问：约之得几何？

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

荅曰：三分之二。

答曰：三分之二。

Answer: two thirds.

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。

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。

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.

今有三分之一，五分之二。問：合之得幾何？

今有三分之一，五分之二。问：合之得几何？

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

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。

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。

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.

今有八分之五，二十五分之十六。問：孰多？多幾何？

今有八分之五，二十五分之十六。问：孰多？多几何？

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

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。

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。

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.

今有三分之一，三分之二，四分之三。問：減多益少，各幾何而平？

今有三分之一，三分之二，四分之三。问：减多益少，各几何而平？

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?

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。

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。

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.

今有七人，分八錢三分錢之一。問：人得幾何？

今有七人，分八钱三分钱之一。问：人得几何？

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

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？

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？

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?

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。

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。

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.