九章算術·方田（分數諸術）九章算术·方田（分数诸术）

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.