| | 卷上: |
度之所起，起於忽。欲知其忽，蠶所生，吐絲為忽。十忽為一秒，十秒為一毫，十毫為一釐，十釐為一分，十分為一寸，十寸為一尺，十尺為一丈，十丈為一引；五十尺為一端；四十尺為一疋；六尺為一步。二百四十步為一畝。三百步為一里。 |
| | The origin of measurement begins with the "hu." To understand the hu, it is said that silkworms produce it by spinning silk. Ten hu make one miao, ten miao make one hao, ten hao make one li, ten li make one fen, ten fen make one cun, ten cun make one chi, ten chi make one zhang, and ten zhang make one yin; Fifty chi make one duan; Forty chi make one pi; Six chi make one bu. Two hundred and forty bu make one mu. Three hundred bu make one li.
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| | 卷上: |
稱之所起，起於黍。十黍為一絫，十絫為一銖，二十四銖為一兩，十六兩為一斤，三十斤為一鉤，四鉤為一石。量之所起，起於粟。六粟為一圭，十圭為一抄，十抄為一撮，十撮為一勺，十勺為一合，十合為一升，十升為一斗，十斗為一斛。斛得六千萬粟。所以得知者，六粟為一圭，十圭六十粟為一抄，十抄六百粟為一撮，十撮六千粟為一勺，十勺六萬粟為一合，十合六十萬粟為一升，十升六百萬粟為一斗，十斗六千萬粟為一斛。十斛六億粟，百斛六兆粟，千斛六京粟，萬斛六陔粟，十萬斛六秭粟，百萬斛六壤粟，千萬斛六溝粟，萬萬斛為一億斛六澗粟，十億斛六正粟，百億斛六載粟。 |
| | The origin of weight measurement begins with the shu grain. Ten shu make one lei, ten lei make one zhu, twenty-four zhu make one liang, sixteen liang make one jin, thirty jin make one gou, and four gou make one dan. The origin of volume measurement begins with the su grain. Six su make one gui, ten gui make one chao, ten chao make one cuo, ten cuo make one shao, ten shao make one he, ten he make one sheng, ten sheng make one dou, and ten dou make one hu. One hu contains sixty million su grains. Therefore, it is known that six su make one gui; ten gui, or sixty su, make one chao; ten chao, or six hundred su, make one cuo; ten cuo, or six thousand su, make one shao; ten shao, or sixty thousand su, make one he; ten he, or six hundred thousand su, make one sheng; ten sheng, or six million su, make one dou; and ten dou, or sixty million su, make one hu. Ten hu contain six hundred million su; one hundred hu, six billion su; one thousand hu, six trillion su; ten thousand hu, six quadrillion su; one hundred thousand hu, six quintillion su; one million hu, six sextillion su; ten million hu, six septillion su; and one hundred million hu, or one yi hu, contain six octillion su; ten yi hu, six nonillion su; and one bai yi hu, six decillion su.
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| | 卷上: |
凡大數之法，萬萬曰億，萬萬億曰兆，萬萬兆曰京，萬萬京曰陔，萬萬陔曰秭，萬萬秭曰壤，萬萬壤曰溝，萬萬溝曰澗，萬萬澗日正，萬萬正曰載。 |
| | Generally, the method for large numbers is as follows: ten thousand times ten thousand is yi; ten thousand times yi is zhao; ten thousand times zhao is jing; ten thousand times jing is gai; ten thousand times gai is zi; ten thousand times zi is rang; ten thousand times rang is gou; ten thousand times gou is jian; ten thousand times jian is zheng; and ten thousand times zheng is zai.
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| | 卷上: |
周三徑一。方五邪七；見邪求方，五之，七而一；見方求邪，七之，五而一。 |
| | The circumference of a circle is three times its diameter. A square with side length five has a diagonal of seven; To find the side from the diagonal, multiply by five and divide by seven; To find the diagonal from the side, multiply by seven and divide by five.
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| | 卷上: |
黃金方寸重一斤。白金方寸重一十四兩。玉方寸重一十二兩。銅方寸重七兩半。鈆方寸重九兩半。鐵方寸重六兩。石方寸重三兩。 |
| | One cubic cun of gold weighs one jin. One cubic cun of silver weighs fourteen liang. One cubic cun of jade weighs twelve liang. One cubic cun of copper weighs seven and a half liang. One cubic cun of lead weighs nine and a half liang. One cubic cun of iron weighs six liang. One cubic cun of stone weighs three liang.
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| | 卷上: |
凡筭之法，先識其位，一從十橫，百立千僵，千十相望，萬百相當。 |
| | The method for calculation begins by identifying the positions: one is vertical, ten is horizontal; a hundred stands upright and a thousand lies flat. Ten hundreds face each other, and thousands and tens correspond to each other in position.
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| | 卷上: |
凡乘之法，重置其位。上下相觀，上位有十步至十，有百步至百，有千步至千。以上命下，所得之數列於中位。言十即過，不滿自如。上位乘訖者先去之。下位乘訖者則俱退之。六不積，五不隻。土下相乘，至盡則已。 |
| | In multiplication, the positions are arranged again for calculation. By observing vertically and horizontally, if the upper position has ten, it moves to the tens place; if it has a hundred, it moves to the hundreds place; if it has a thousand, it moves to the thousands place. The upper number is used to multiply the lower one, and the resulting product is placed in the middle position. If it reaches ten, carry over; if less than ten, leave as is. After the upper position has been multiplied, remove it first. Once the lower position is multiplied, both are moved back. Six does not accumulate; five does not count as a single unit. Multiplying vertically and horizontally continues until the end is reached.
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| | 卷上: |
凡除之法，與乘正異。乘得在中央，除得在上方。假令六為法，百為實。以六除百，當進之二等，令在正百下，以六除一，則法多而實少，不可除，故當退就十位。以法除實，言一六而折百為四十七，故可除。若實多法少，自當百之，不當復退。故或步法十者置於十位，百者置於百位。上位有空絕者，法退二位。餘法皆如乘時。實有餘者，以法命之，以法為母。實餘為子。 |
| | The method of division differs from multiplication. In multiplication, the result is placed in the center; in division, it appears at the top. Suppose six is the divisor and one hundred is the dividend. Dividing one hundred by six, it should be advanced two positions and placed under the hundreds place. Dividing one by six results in a larger divisor than dividend, which is not possible; therefore, we must retreat to the tens position. Using the divisor to divide the dividend: saying one six and reducing a hundred by sixty results in forty-seven, so division is possible. If the dividend is large and the divisor small, it naturally belongs to the hundreds place, and there is no need to retreat further. Therefore, if a step of ten is taken, it should be placed in the tens position; if a hundred, in the hundreds position. If there is an empty space at the upper position, the divisor must retreat two positions. The remaining methods follow those of multiplication. If there is a remainder in the dividend, name it using the divisor as the denominator. The remainder becomes the numerator.
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| | 卷上: |
以粟求糲米，三之，五而一。以糲米求粟，五之，三而一。以糲米求飯，五之，二而一。以粟米求糲飯，六之，四而一。以糲飯求糲米，二之，五而一。以绺米求飯，八之，四而一。 |
| | To convert su grain to millet rice, multiply by three and divide by five. To convert millet rice back to su grain, multiply by five and divide by three. To convert millet rice into cooked rice, multiply by five and divide by two. To convert su grain into millet rice, then to cooked rice, multiply by six and divide by four. To convert cooked rice back to millet rice, multiply by two and divide by five. To convert liu mi into cooked rice, multiply by eight and divide by four.
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| | 卷上: |
十分減一者，以二乘，二十除。減二者，以四乘，二十除。減三者，以六乘，二十除。減四者，以八乘，二十除。減五者，以十乘，二十除。減六者，以十二乘，二十除。減七者，以十四乘，二十除。減八者，以十六乘，二十除。減九者，以十八乘，二十除。 |
| | A reduction of one-tenth is calculated by multiplying by two and dividing by twenty. A reduction of two-tenths (or one-fifth) is calculated by multiplying by four and dividing by twenty. A reduction of three-tenths is calculated by multiplying by six and dividing by twenty. A reduction of four-tenths (or two-fifths) is calculated by multiplying by eight and dividing by twenty. A reduction of five-tenths (or half) is calculated by multiplying by ten and dividing by twenty. A reduction of six-tenths (or three-fifths) is calculated by multiply by twelve and divide by twenty. A reduction of seven-tenths is calculated by multiplying with fourteen and dividing by twenty. A reduction of eight-tenths (or four-fifths) is computed by multiplying by sixteen and dividing by twenty. A reduction of nine-tenths is calculated by multiply with eighteen and divide by twenty.
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| | 卷上: |
九分減一者，以二乘，十八除。 |
| | A reduction of one-ninth is calculated by multiplying by two, then dividing by eighteen.
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| | 卷上: |
八分減一者，以二乘，十六除。 |
| | A reduction of one-eighth is calculated by multiplying by two then dividing by sixteen.
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| | 卷上: |
七分減一者，以二乘，十四除。 |
| | A reduction of one-seventh is calculated by multiplying by two and then dividing by fourteen.
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| | 卷上: |
六分減一者，以二乘，十二除。 |
| | A reduction of one-sixth is calculated by multiplying by two, and then dividing by twelve.
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| | 卷上: |
五分減一者，以二乘，十除。 |
| | A reduction of one-fifth is calculated by multiplying by two first, then dividing by ten.
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| | 卷上: |
九九八十一，自相乘，得幾何？答曰：六千五百六十一。術曰：重置其位，以上八呼下八，八八六十四，即下六千四百於中位。以上八呼下一，一八如八，即於中位下八十。退下位一等，收上位八十。以上位一呼下八，一八如八，即於中位下八十。以上一呼下一，一一如一，即於中位下一。上下位俱收，中位即得六千五百六十一。 |
| | Ninety-nine times eighty-one, when multiplied together, results in what? Answer: Six thousand five hundred and sixty-one. Method: Arrange the positions again, call out eight from above to multiply with eight below; eight eights are sixty-four. Place six thousand four hundred in the middle position. Call out eight from above to one below, one eight is eight; place eighty in the middle position. Move the lower position back by one rank and add the upper value of eighty. Call out one from above to eight below, one eight is eight, place eighty in the middle position accordingly. Call out one from the upper position to multiply with one from the lower; one times one is one, so place one in the middle position. After collecting both the upper and lower positions, the middle position yields six thousand five hundred sixty-one.
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| | 卷上: |
六千五百六十一，九人分之，問人得幾何？答曰：七百二十九。術曰：先置六千五百六十一於中位，為實。下列九人為法。上位置七百，以上七呼下九，七九六十三，即除中位六千三百。退下位一等，即上位置二十。以上二呼下九，二九十八，即除中位一百八十。又更退下位一等，即上位更置九，即以上九呼下九，九九八十一，即除中位八十一。中位邬盡，收下位。上位所得即人之所得。自八八六十四至一一如一，邬準此。 |
| | Six thousand five hundred sixty-one is to be divided among nine people; how much does each person receive? Answer: Seven hundred and twenty-nine. Method: First place six thousand five hundred sixty-one in the middle position as the dividend. Place nine people below as the divisor. Place seven hundred in the upper position; call out seven from above to multiply by nine below, seven nines are sixty-three, and subtract six thousand three hundred from the middle. Move the lower position back one rank; place twenty in the upper position. Call out two from above to multiply by nine from below, two nines are eighteen, and subtract one hundred eighty from the middle. Move the lower position back another rank, place nine in the upper position again. Call out nine from above to multiply by nine below; nine nines are eighty-one and subtract eighty-one from the middle. When the middle position is exhausted, collect the lower positions. The result in the upper position is what each person receives. From eight eights being sixty-four to one times one being one, follow this method accordingly.
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| | 卷上: |
八九七十二，自相乘，得五千一百八十四。八人分之，人得六百四十八五。七九六十三，自相乘，得三千九百六十九。七人分之，人得五百六十七五。 |
| | Eight nines are seventy-two; when multiplied by itself, it results in five thousand one hundred and eighty-four. Divided among eight people, each person receives six hundred forty-eight point five. Seven nines are sixty-three; when multiplied by itself, the result is three thousand nine hundred and sixty-nine. Divided among seven people, each person receives five hundred sixty-seven point five.
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| | 卷上: |
六九五十四，自相乘，得二千九百一十六。六人分之，人得四百八十六。 |
| | Six nines are fifty-four; when multiplied by itself, results in two thousand nine hundred and sixteen. Divided among six people, each person receives four hundred eighty-six.
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| | 卷上: |
五九四十五，自相乘，得二千二十五。五人分之，人得四百五。 |
| | Five nines are forty-five; when multiplied by itself, gives two thousand and twenty-five. Divided among five people, each person receives four hundred fifty.
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| | 卷上: |
四九三十六，自相乘，得一千二百九十六。四人分之，人得三百二十四。 |
| | Four nines are thirty-six; when multiplied by itself, yields one thousand two hundred and ninety-six. Divided among four people, each person receives three hundred twenty-four.
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| | 卷上: |
三九二十七，自相乘，得七百二十九。三人分之，人得二百四十三。 |
| | Three nines are twenty-seven; when multiplied by itself, equals seven hundred and twenty-nine. Divided among three people, each person receives two hundred forty-three.
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| | 卷上: |
二九一十八，自相乘，得三百二十四。二人分之，人得一百六十二。 |
| | Two nines are eighteen; when multiplied by itself, produces three hundred twenty-four. Divided among two people, each person receives one hundred sixty-two.
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| | 卷上: |
一九如九，自相乘，得八十一。一人得八十一。右九九一條，得四百五，自相乘，得一十六萬四千二十五。九人分之，人得二萬八千二百二十五。 |
| | One nine is nine; when multiplied by itself, gets eighty-one. One person receives eighty-one. The above one "Nine Nines" item, totaling four hundred five; when multiplied by itself, obtains sixteen thousand four hundred and twenty-five. Divided among nine people, each person receives two ten-thousand eight thousand two hundred and twenty-five.
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| | 卷上: |
八八六十四，自相乘，得四千九十六。八人分之，人得五百十二。 |
| | Eight eights are sixty-four; when multiplied by itself, becomes four thousand ninety-six. Divided among eight people, each person receives five hundred twelve.
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| | 卷上: |
七八五十六，自相乘，得三千一百三十六。七人分之，人得四百四十八。 |
| | Seven eights are fifty-six; when multiplied by itself, amounts to three thousand one hundred and thirty-six. Divided among seven people, each person receives four hundred forty-eight.
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| | 卷上: |
六八四十八，自相乘，得二千三百四。六人分之，人得三百八十四。 |
| | Six eights are forty-eight; when multiplied by itself, totals two thousand three hundred and four. Divided among six people, each receives three hundred eighty-four.
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| | 卷上: |
五八四十，自相乘，得一千六百。五人分之，人得三百二十。 |
| | Five eights are forty; when multiplied by itself, reaches one thousand six hundred. Divided among five people, each receives three hundred twenty.
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| | 卷上: |
四八三十二，自相乘，得一千二十四。四人分之，人得二百五十六。 |
| | Four eights are thirty-two; when multiplied by itself, is one thousand and twenty-four. Divided among four people, each receives two hundred fifty-six.
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| | 卷上: |
三八二十四，自相乘，得五百七十六。三人分之，人得一百九十二。 |
| | Three eights are twenty-four; when multiplied by itself, makes five hundred seventy-six. Divided among three people, each receives one hundred ninety-two.
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| | 卷上: |
二八十六，自相乘，得二百五十六。二人分之，人得一百二十八。 |
| | Two eights are sixteen; when multiplied by itself, comes to two hundred fifty-six. Divided among two people, each receives one hundred twenty-eight.
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| | 卷上: |
一八如八，自相乘，得六十四。一人得六十四。右八八一條，得二百八十八，自相乘，得八萬二千九百四十四。八人分之，人得一萬三百六十八。 |
| | One eight is eight; when multiplied by itself, gains sixty-four. One person receives sixty-four. The above one "Eight Eights" item, totaling two hundred eighty-eight; when multiplied by itself, achieves eight ten-thousand two thousand nine hundred and forty-four. Divided among eight people, each receives one ten-thousand three hundred sixty-eight.
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| | 卷上: |
七七四十九，自相乘，得二千四百一。七人分之，人得三百四十三。 |
| | Seven sevens are forty-nine; when multiplied by itself, attains two thousand four hundred and one. Divided among seven people, each receives three hundred forty-three.
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| | 卷上: |
六七四十二，自相乘，得一千七百六十四。六人分之，人得二百九十四。 |
| | Six sevens are forty-two; when multiplied by itself, arrives at one thousand seven hundred and sixty-four. Divided among six people, each gets two hundred ninety-four.
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| | 卷上: |
五七三十五，自相乘，得一千二百二十五。五人分之，人得二百四十五。 |
| | Five sevens are thirty-five; when multiplied by itself, sums to one thousand two hundred twenty-five. Divided among five people, each gets two hundred forty-five.
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| | 卷上: |
四七二十八，自相乘，得七百八十四。四人分之，人得一百九十六。 |
| | Four sevens are twenty-eight; when multiplied by itself, adds up to seven hundred eighty-four. Divided among four people, each gets one hundred ninety-six.
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| | 卷上: |
三七二十一，自相乘，得四百四十一。三人分之，人得一百四十七。 |
| | Three sevens are twenty-one; when multiplied by itself, accumulates to four hundred forty-one. Divided among three people, each gets one hundred forty-seven.
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| | 卷上: |
二七十四，自相乘，得一百九十六。二人分之，人得九十八。 |
| | Two sevens are fourteen; when multiplied by itself, computes to one hundred ninety-six. Divided among two people, each gets ninety-eight.
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| | 卷上: |
一七如七，自相乘，得四十九。一人得四十九。右七七一條，得一百九十六，自相乘，得三萬八千四百一十六。七人分之，人得五千四百八十八。 |
| | One seven is seven; when multiplied by itself, derives forty-nine. One person receives forty-nine. The above one "Seven Sevens" item, totaling one hundred ninety-six; when multiplied by itself, calculates to three ten-thousand eight thousand four hundred and sixteen. Divided among seven people, each gets five thousand four hundred eighty-eight.
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| | 卷上: |
六六三十六，自相乘，得一千二百九十六。六人分之，人得二百一十六。 |
| | Six sixes are thirty-six; when multiplied by themselves, equals one thousand two hundred ninety-six. Divided among six people, each received two hundred sixteen.
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| | 卷上: |
五六三十，自相乘，得九百。五人分之，人得一百八十。 |
| | Five sixes are thirty; when multiplied by themselves, results in nine hundred. Divided among five people, each received one hundred eighty.
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| | 卷上: |
四六二十四，自相乘，得五百七十六。四人分之，人得一百四十四。 |
| | Four sixes are twenty-four; when multiplied by themselves, produces five hundred seventy-six. Divided among four people, each received one hundred forty-four.
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| | 卷上: |
三六一十八，自相乘，得三百二十四。三人分之，人得一百八。 |
| | Three sixes are eighteen; when multiplied by themselves, yields three hundred twenty-four. Divided among three people, each received one hundred eight.
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| | 卷上: |
二六一十二，自相乘，得一百四十四。二人分之，人得七十二。 |
| | Two sixes are twelve; when multiplied by themselves, gives one hundred forty-four. Divided among two people, each received seventy-two.
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| | 卷上: |
一六如六，自相乘，得三十六。一人得三十六。右六六一條，得一百二十六，自相乘，得一萬五千八百七十六。六人分之，人得二千六百四十六。 |
| | One six is six; when multiplied by itself, obtained thirty-six. One person received thirty-six. The above one "Six Sixes" item, totaling one hundred twenty-six; when multiplied by itself, calculated to fifteen thousand eight hundred seventy-six. Divided among six people, each got two thousand six hundred forty-six.
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| | 卷上: |
五五二十五，自相乘，得六百二十五。五人分之，人得一百二十五。 |
| | Five fives are twenty-five; when multiplied by themselves, becomes six hundred twenty-five. Divided among five people, each got one hundred twenty-five.
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| | 卷上: |
四五二十，自相乘，得四百。四人分之，人得一百。 |
| | Four fives are twenty; when multiplied by themselves, amounts to four hundred. Divided among four people, each got one hundred.
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| | 卷上: |
三五一十五，自相乘，得二百二十五。三人分之，人得七十五。 |
| | Three fives are fifteen; when multiplied by themselves, totals two hundred twenty-five. Divided among three people, each got seventy-five.
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| | 卷上: |
二五一十，自相乘，得一百。二人分之，人得五十。 |
| | Two fives are ten; when multiplied by themselves, reaches one hundred. Divided among two people, each got fifty.
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| | 卷上: |
一五如五，自相乘，得二十五。一人得二十五。右五五一條，得七十五，自相乘，得五千六百二十五。五人分之，人得一千一百二十五。 |
| | One five is five; when multiplied by itself, gained twenty-five. One person got twenty-five. The above one "Five Fives" item, totaling seventy-five; when multiplied by itself, achieved five thousand six hundred twenty-five. Divided among five people, each obtained one thousand one hundred twenty-five.
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| | 卷上: |
四四一十六，自相乘，得二百五十六。四人分之，人得六十四。 |
| | Four fours are sixteen; when multiplied by themselves, became two hundred fifty-six. Divided among four people, each obtained sixty-four.
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| | 卷上: |
三四一十二，自相乘，得一百四十四。三人分之，人得四十八。 |
| | Three fours are twelve; when multiplied by themselves resulted in one hundred forty-four. Divided among three people, each obtained forty-eight.
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| | 卷上: |
二四如八，自相乘，得六十四。二人分之，人得三十二。 |
| | Two fours are eight; when multiplied by themselves gives sixty-four. Divided among two people, each obtained thirty-two.
|
| | 卷上: |
一四如四，自相乘，得一十六。一人得一十六。右四四一條，得四十，自相乘，得一千六百。四人分之，人得四百。 |
| | One four is four; when multiplied by itself gained sixteen. One person obtained sixteen. The above one "Four Fours" item, totaling forty; when multiplied by itself calculated to one thousand six hundred. Divided among four people, each gained four hundred.
|
| | 卷上: |
三三如九，自相乘，得八十一。三人分之，人得二十七。 |
| | Three threes are nine; when multiplied by themselves becomes eighty-one. Divided among three people, each gained twenty-seven.
|
| | 卷上: |
二三如六，自相乘，得三十六。二人分之，人得一十八。 |
| | Two threes are six; when multiplied by themselves results in thirty-six. Divided among two people, each gained eighteen.
|
| | 卷上: |
一三如三，自相乘，得九。一人得九。右三三一條，得一十八，自相乘，得三百二十四。三人分之，人得一百八。 |
| | One three is three; when multiplied by itself obtained nine. One person gained nine. The above one "Three Threes" item, totaling eighteen; when multiplied by itself computed to three hundred twenty-four. Divided among three people, each acquired one hundred eight.
|
| | 卷上: |
二二如四，自相乘，得一十六。二人分之，人得八。 |
| | Two twos are four; when multiplied by themselves became sixteen. Divided among two people, each acquired eight.
|
| | 卷上: |
一二如二，自相乘，得四。一人得四。右二二一條，得六，自相乘，得三十六。二人分之，人得十八。 |
| | One two is two; when multiplied by itself resulted in four. One person acquired four. The above one "Two Twos" item, totaling six; when multiplied by itself became thirty-six. Divided among two people, each achieved eighteen.
|
| | 卷上: |
一一如一，自相乘，得一。一乘不長。右從九九至一一，總成一千一百五十五，自相乘，得一百十三萬四千二十五。九人分之，人得一十四萬八千二百二十五。 |
| | One one is one; when multiplied by itself gives one. Multiplying one does not increase it. The above total from "Nine Nines" down to "One One," altogether one thousand one hundred fifty-five; when multiplied by itself, gives one hundred thirteen ten-thousand four thousand twenty-five. Divided among nine people, each person received fourteen ten-thousand eight thousand two hundred and twenty-five.
|
| | 卷上: |
以九乘一十二，得一百八。六人分之，人得一十八。 |
| | Multiplying twelve by nine gives one hundred eight. Divided among six people, each received eighteen.
|
| | 卷上: |
以二十七乘三十六，得九百七十二。一十八人分之，人得五十四。 |
| | Multiplying thirty-six by twenty-seven gives nine hundred seventy-two. Divided among eighteen people, each received fifty-four.
|
| | 卷上: |
以八十一乘一百八，得八千七百四十八。五十四人分之，人得一百六十二。 |
| | Multiplying one hundred eight by eighty-one gives eight thousand seven hundred forty-eight. Divided among fifty-four people, each received one hundred sixty-two.
|
| | 卷上: |
以二百四十三乘三百二十四，得七萬八千七百三十二。一百六十二人分之，人得四百八十六。 |
| | Multiplying three hundred twenty-four by two hundred forty-three gives seventy thousand eight hundred seven hundred thirty-two. Divided among one hundred sixty-two people, each received four hundred eighty-six.
|
| | 卷上: |
以七百二十九乘九百七十二，得七十萬八千五百八十八。四百八十六人分之，人得一千四百五十八。 |
| | Multiplying nine hundred seventy-two by seven hundred twenty-nine gives seventy million eight thousand five hundred eighty-eight. Divided among four hundred eighty-six people, each received one thousand four hundred fifty-eight.
|
| | 卷上: |
以二千一百八十七乘二千九百一十六，得六百三十七萬七千百九十二。一千四百五十八人分之，人得四千三百七十四。 |
| | Multiplying two thousand nine hundred sixteen by two thousand one hundred eighty-seven gives six million thirty-seven ten-thousand seven thousand one hundred ninety-two. Divided among one thousand four hundred fifty-eight people, each received four thousand three hundred seventy-four.
|
| | 卷上: |
以六千五百六十一乘八千七百四十八，得五千七百三十九萬五千六百二十八。四千三百七十四人分之，人得一萬三千一百二十二。 |
| | Multiplying eight thousand seven hundred forty-eight by six thousand five hundred sixty-one gives five million seven hundred thirty-nine ten-thousand five thousand six hundred twenty-eight. Divided among four thousand three hundred seventy-four people, each received ten-thousand three thousand one hundred twenty-two.
|
| | 卷上: |
以一萬九千六百八十三乘二萬六千二百四十四，得五億一千六百五十六萬六百五十二。一萬三千一百二十二人分之，人得三萬九千三百六十六。 |
| | Multiplying two ten-thousand six thousand two hundred forty-four by one ten-thousand nine thousand six hundred eighty-three gives five hundred million one hundred sixty-five ten-thousand six hundred fifty-two. Divided among ten-thousand three thousand one hundred and twenty-two people, each received three ten-thousand nine thousand three hundred sixty-six.
|
| | 卷上: |
以五萬九千四十九乘七萬八千七百三十二，得四十六億四千九百四萬五千八百六十八。三萬九千三百六十六人分之，人得一十一萬八千九十八。 |
| | Multiplying seven ten-thousand eight thousand seven hundred thirty-two by five ten-thousand nine thousand four hundred ninety gives forty-six billion four million nine hundred four ten-thousand five thousand eight hundred sixty-eight. Divided among three ten-thousand nine thousand three-hundred sixty-six people, each received eleven ten-thousand eight hundred ninety-eight.
|
| | 卷上: |
以一十七萬七千一百四十七乘二十三萬六千一百九十六，得四百一十八億四千一百四十一萬二千八百一十二。一十一萬八千九十八人分之，人得三十五萬四千二百九十四。 |
| | Multiplying twenty-three ten-thousand six thousand one hundred ninety-six by seventeen ten-thousand seven thousand one-hundred forty-seven gives four hundred eighteen billion four million one hundred forty-one ten-thousand two thousand eight hundred twelve. Divided among eleven ten-thousand eight thousand nine-hundred ninety-eight people, each received thirty-five ten-thousand four thousand two hundred ninety-four.
|
| | 卷上: |
以五十三萬一千四百四十一乘七十萬八千五百八十八，得三千七百六十五億七千二百七十一萬五千三百八。三十五萬四千二百九十四人分之，人得一百六萬二千八百八十二。 |
| | Multiplying seventy million eight thousand five-hundred eighty-eight by fifty-three ten-thousand one hundred forty-one gives three thousand seven hundred sixty-five billion seven thousand two hundred seventy-one ten-thousand five thousand three hundred eight. Divided among thirty-five ten-thousand four-thousand two-hundred ninety-four people, each received one-hundred six ten-thousand two thousand eight-hundred eighty-two.
|
| | 卷中: |
今有一十八分之一十二。問約之得幾何？ |
| | Now there is twelve one-eighths. Ask, when reduced, how much will it be?
|
| | |
答曰：三分之二。 |
| | Answer: two-thirds.
|
| | |
術曰：置十八分在下，一十二分在上。副置二位，以少減多，等數得六，為法。約之，即得。 |
| | Method says: Place eighteen parts below, twelve parts above. Duplicate the two positions, subtract the smaller from the larger, equal number obtained six, as method. Reduce it, and you get.
|
| | 卷中: |
今有三分之一，五分之二。問合之得幾何？ |
| | Now there is one-third, two-fifths. Ask, when combined, how much will it be?
|
| | |
答曰：一十五分之一十一。 |
| | Answer: eleven fifteenths.
|
| | |
術曰：置三分、五分在右方，之一、之二在左方。母互乘子，五分之二得六，三分之一得五。并之，得一十一，為實。右方二母相乘，得一十五，為法。不滿法，以法命之，即得。 |
| | Method says: Place three parts, five parts on the right side, one of them, two of them on the left side. Multiply each numerator by the other's denominator: two-fifths becomes six, one-third becomes five. Add them together, get eleven, as dividend. Multiply the two denominators on the right side, get fifteen, as method. If not full of the method, name it by the method, and you get.
|
| | 卷中: |
今有九分之八，減其五分之一。問餘幾何？ |
| | Now there is eight-ninths, subtract one-fifth of it. Ask, how much remains?
|
| | |
答曰：四十五分之三十一。 |
| | Answer: thirty-one forty-fifths.
|
| | |
術曰：置九分、五分在右方，之八、之一在左方。母互乘子，五分之一得九，九分之八得四十。以少減多，餘王十一，為實。母相乘得四十五，為法。不滿法，以法命之，即得。 |
| | Method says: Place nine parts, five parts on the right, eight of them, one of them on the left. Multiply each numerator by the other denominator: one-fifth becomes nine, eight-ninths becomes forty. Subtract small from large, remainder thirty-one, as dividend. Multiply the denominators to get forty-five, as method. If not full of the method, named by the method, you get it.
|
| | 卷中: |
今有三分之一，三分之二，四分之三。問減多益少，幾何而平？ |
| | Now there is one-third, two-thirds, three-fourths. Ask, subtracting from the more and adding to the less, how much to make it equal?
|
| | |
答曰：減四分之三者二，三分之二者一，并，以益三分之一，而各平於一十二分之七。 |
| | Answer: Subtract two from three-fourths, one from two-thirds, combine them, and add to one-third, then each will be equal to seven twelfths.
|
| | |
術曰：置三分、三分、四分在右方，之一、之二、之三在左方。母互乘于，副并得六十三，置右，為平實。母相乘，得三十六，為法。以列數三乘未并者及法。等數得九，約訖。減四分之三者二，減三分之二者一，并，以益三分之一，各平於一十二分之七。 |
| | Method says: Place three parts, three parts, four parts on the right side; one of them, two of them, three of them on the left side. Multiply each denominator by the other numerators, duplicate and add to get sixty-three, place it on the right as equal dividend. Multiply denominators together, get thirty-six, as method. Use the three listed numbers to multiply the uncombined ones and the method. Equal number obtained nine, reduction completed. Subtract two from three-fourths; subtract one from two-thirds; combine them and add to one-third. Each will be equal to seven-twelfths.
|
| | 卷中: |
今有粟一斗，問為糯米幾何？ |
| | Now there is a peck of millet, ask how much glutinous rice it becomes?
|
| | |
答曰：六升。 |
| | Answer: six sheng.
|
| | |
術曰：置粟一斗，十升。以糲米率三十乘之，得三百升，為實。以粟率五十為法，除之，即得。 |
| | Method says: Place one peck of millet as ten sheng. Multiply it by the rate for glutinous rice, thirty, to get three hundred sheng, as dividend. Divide it by fifty, the rate of millet, and you will get it.
|
| | 卷中: |
今有粟二斗一升，問為稗米幾何？ |
| | Now there is two pecks and one sheng of millet; ask how much barnyard millet rice it becomes?
|
| | |
答曰：一斗一升五十分升之一十七。 |
| | Answer: ten pecks, one sheng, seventeen five-tieths of a sheng.
|
| | |
術曰：置粟二十一升。以稗米率二十七乘之，得五百六十七升，為實。以粟率五十為法，除之。不盡，以法而命分。 |
| | Method says: Place twenty-one sheng of millet. Multiply it by the barnyard millet rice rate, twenty-seven, to get five hundred and sixty-seven sheng, as dividend. Divide it by fifty, as the millet rate. If not divisible completely, name the fraction according to the method.
|
| | 卷中: |
今有粟四斗五升，問為槃米幾何？ |
| | Now there is four pecks and five sheng of millet; Ask how much pan rice it becomes?
|
| | |
答曰：二斗一升五分升之三。 |
| | Answer: two pecks, one sheng and three fifths of a sheng.
|
| | |
術曰：置粟四十五升。以二約槃米率二十四，得一十二。乘之，得五百四十升，為實。以二約粟率五十，得二十五，為法。除之。不盡，以等數約之，而命分。 |
| | Method says: Place forty-five sheng of millet. Divide the pan rice rate twenty-four by two, get twelve. Divide the millet rate fifty by two, get twenty-five, as method. Divide the millet rate of fifty by two to obtain twenty-five, which is used as the divisor. Divide it. If not completely divisible, reduce the remainder by the equal number and name the fraction accordingly.
|
| | 卷中: |
今有栗七斗九升，問為御米幾何？ |
| | Now there is seven pecks and nine sheng of millet; how much imperial rice does it become?
|
| | |
答曰：三斗三升一合八勺。 |
| | Answer: three pecks, three sheng, one ge, eight shao.
|
| | |
術曰：置七斗九升。以御米率二十一乘之，得一千六百五十九升，為實。以粟率五十除之，即得。 |
| | Method says: Place seven pecks and nine sheng. Multiply it by the imperial rice rate of twenty-one to get one thousand six hundred fifty-nine sheng, as dividend. Divide it by the millet rate of fifty, and you will obtain the result.
|
| | 卷中: |
今有屋基南北三丈，東西六丈，欲以●砌之。凡積二尺，用●五枚。問計幾何？ |
| | Now there is a house foundation with a north-south length of three zhang and an east-west width of six zhang, and one wishes to build it using bricks. Generally for every two chi in volume, five bricks are used. Ask how many will be needed?
|
| | |
答曰：四千五百枚。 |
| | Answer: four thousand five hundred bricks.
|
| | |
術曰：置東西六丈，以南北三丈乘之，得一千八百尺。以五乘之，得九千尺。以二除之，即得。 |
| | Method says: Place the east-west length of six zhang, multiply it by the north-south length of three zhag to get one thousand eight hundred chi. Multiply it by five to get nine thousand chi. Divide it by two, and you will obtain the answer.
|
| | 卷中: |
今有圓窖下周二百八十六尺，深三丈六尺。問受粟幾何？ |
| | Now there is a round cellar with a circumference of 286 chi at the bottom and a depth of thirty-six chi. Ask how much millet it can hold?
|
| | |
答曰：一十五萬一千四百七十四斛七升二十七分升之一十一。 |
| | Answer: one hundred fifty-one thousand four hundred seventy-four hu, seven sheng and eleven twenty-sevenths of a sheng.
|
| | |
術曰：置周二百八十六尺，自相乘，得八萬一千七百九十六尺。以深三丈六尺乘之，得二百九十四萬四千六百五十六尺以一十二除之，得二十四萬五千三百八十八尺。以斛法一尺六寸二分除之，即得。 |
| | Method says: Place the circumference of two hundred eighty-six chi, multiply it by itself to get eight ten-thousand one thousand seven hundred ninety-six chi. Multiply it by the depth of thirty-six zhang to get two hundred ninety-four ten-thousand four thousand six hundred fifty-six chi; divide this by twelve, getting twenty-four ten-thousand five thousand three hundred eighty-eight chi. Divide it by the hu standard of one zhang six cun two fen, and you will obtain the amount.
|
| | 卷中: |
今有方窖廣四丈六尺，長五丈四尺，深三丈五尺。問受粟幾何？ |
| | Now there is a square cellar with width of four zhang six chi, length of five zhang four chi, and depth of three zhang five chi. Ask how much millet can it hold?
|
| | |
答曰：五萬三千六百六十六斛六斗六升三分升之二。 |
| | Answer: fifty-three thousand six hundred sixty-six hu, six dou, six sheng and two thirds of a sheng.
|
| | |
術曰：置廣四丈六尺，長五丈四尺，相乘，得二千四百八十四尺。以深三丈五尺乘之，得八萬六千九百四十尺。以斛法一尺六寸二分除之，即得。 |
| | Method says: Place the width of four zhang six chi and length of five zhang four chi; multiply them to get two thousand four hundred eighty-four chi. Multiply it by the depth of three zhang five feet to get eight ten-thousand six thousand nine hundred forty chi. Divide it by the standard for a hu, one chi six cun two fen; and you will obtain the result immediately.
|
| | 卷中: |
今有圓窖周五丈四尺，深一丈八尺。問受粟幾何？ |
| | Now there is a circular cellar with circumference of five zhang four feet, depth one zhang eight feet. Ask how much millet will it hold?
|
| | |
答曰：二千七百斛。 |
| | Answer: two thousand seven hundred hu.
|
| | |
術曰：先置周五丈四尺，自相乘得二千九百一十六尺。以深一丈八尺乘之，得五萬二千四百八十八尺。以一十二除之，得四千三百七十四尺。以斛法一尺六寸二分除之，即得。 |
| | Method says: First place the circumference of five zhang and four chi, multiply it by itself, getting two thousand nine hundred sixteen chi. Multiply it by a depth of one zhang eight chi to get fifty-two thousand four hundred eighty-eight chi. Divide it by twelve, getting four thousand three hundred seventy-four chi. Divide it by the measure of one hu (1.62 chi), and you will get the result immediately.
|
| | 卷中: |
今有圓田周三百步，徑一百步。問得田幾何？ |
| | Now there is a round field with circumference three hundred bu, diameter one hundred bu. Ask how much land does it amount to?
|
| | |
答曰：三十一畝奇六十步。 |
| | Answer: thirty-one mu and sixty bu extra.
|
| | |
術曰：先置周三百步，半之，得一百五十步。又置徑一百步，半之，得五十步。相乘，得七千五百步。以畝法二百四十步除之，即得。 |
| | Method says: First place the circumference of three hundred bu, halve it to get one hundred fifty bu. Also place the diameter of one hundred bu, halve it, getting fifty bu. Multiply them together to get seven thousand five hundred bu. Divide it by the mu standard of two hundred forty bu, and you will obtain the area.
|
| | |
又術：周自相乘，得九萬步。以一十二除之，得七千五百步。以畝法除之，得畝數。 |
| | Another method: Multiply the circumference by itself to get ninety thousand bu. Divide it by twelve to get seven thousand five hundred steps. Divide it by the mu method, and you will obtain the number of mu.
|
| | |
又術：徑自乘，得一萬。以三乘之，得三萬步。四除之，得七千五百步。以畝法除之，得畝數。 |
| | Another method: Multiply the diameter by itself to get ten thousand. Multiply it by three, getting thirty thousand bu. Divide it by four to get seven thousand five hundred step. Divide it by the mu rate, and you will obtain the mu number.
|
| | 卷中: |
今有方田，桑生中央。從角至桑一百四十七步。問為田幾何？ |
| | Now there is a square field with mulberry trees growing in the center. From the corner to the mulberry tree is one hundred forty-seven steps. Ask how much field does it amount to?
|
| | |
答曰：一頃八十三畝奇一百八十步。 |
| | Answer: one qing, eighty-three mu and one hundred eighty steps extra.
|
| | |
術曰：置角至桑一百四十七步，倍之，得二百九十四步。以五乘之，得一千四百七十步。以七除之，得二百一十步。自相乘，得四萬四千一百步。以二百四十步除之，即得。 |
| | Method says: Place the distance from corner to mulberry tree as one hundred forty-seven bu, double it to get two hundred ninety-four bu. Multiply it by five to get one thousand four hundred seventy steps. Divide it by seven, getting two hundred ten steps. Multiply it by itself to get forty-four thousand one hundred bu. Divide it by two hundred forty steps, and you will get the result.
|
| | 卷中: |
今有木方三尺，高三尺。欲方五寸作枕一枚，問得幾何？ |
| | Now there is a wooden block with square cross-section of three chi in width and height of three chi. lxml If you want to make one pillow with a square base of five cun, how many can be made?
|
| | |
答曰：二百一十六枚。 |
| | Answer: two hundred and sixteen pillows.
|
| | |
術曰：置方三尺，自相乘，得九尺。以高三尺乘之，得二十七尺。以一尺木八枕乘之，即得。 |
| | Method says: Place the square of three chi, multiply it by itself to get nine chi. Multiply it by height of three chi to get twenty-seven chi. Multiply it by eight pillows per one chi of wood, and you will get the answer.
|
| | 卷中: |
今有索長五千七百九十四步。欲使作方，問幾何？ |
| | Now there is a rope that is five thousand seven hundred ninety-four steps long. You want to make it into a square, how much will be the side length?
|
| | |
答曰：一千四百四十八步三尺。 |
| | Answer: one thousand four hundred forty-eight steps and three chi.
|
| | |
術曰：置索長五千七百九十四步。以四除之，得一千四百四十八步，餘二步。以六因之，得一丈二尺。以四除之，得三尺。通計即得。 |
| | Method says: Place the rope length as five thousand seven hundred ninety-four bu. Divide it by four, getting one thousand four hundred forty-eight bu, with a remainder of two bu. Multiply the remainder by six to get one zhang and two chi. Divide it by four, obtaining three chi. Add them together to get the total.
|
| | 卷中: |
今有隄，下廣五丈，上廣三丈，高二丈，長六十尺。欲以一千尺作一方，問計幾何？ |
| | Now there is a dike, with lower width of five zhang, upper width of three zhang, height of two zhang, and length of sixty chi. You want to use one thousand chi as a unit for measurement, how much is the total?
|
| | |
答曰：四十八方。 |
| | Answer: forty-eight units.
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術曰：置隄上廣三丈，下廣五丈。并之，得八丈。半之得四丈。以高二丈乘之得八百尺。以長六十尺乘之，得四萬八千。以一千尺除之，即得。 |
| | Method says: Place the upper width of the dike as three zhang, lower width as five zhang. Add them together to get eight zhang. Half it, getting four zhang. Multiply it by the height of two zhang to get eight hundred chi. Multiply it by length of sixty chi, getting forty-eight thousand. Divide it by one thousand chi, and you will get the total.
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| | 卷中: |
今有溝廣十丈，深五丈，長二十丈。欲以千尺作一方，問得幾何？ |
| | Now there is a ditch with width of ten zhang, depth of five zhang, and length of twenty zhang. You want to use one thousand feet as a unit for each cubic measurement, how many units can be obtained?
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答曰：一千方。 |
| | Answer: one thousand units.
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術曰：置廣一十丈，以深五丈乘之，得五千尺。又以長二十丈乘之，得一百萬尺。以一千除之，即得。 |
| | Method says: Place the width as ten zhang, multiply it by depth of five zhang to get five thousand chi. Multiply it again by length of twenty zhang to get one million chi. Divide it by one thousand, and you will get the amount.
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| | 卷中: |
今有積二十三萬四千五百六十七步。問為方幾何？ |
| | Now there is an accumulation of two hundred thirty-four thousand five hundred sixty-seven bu. Ask how much square does it amount to?
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答曰：四百八十四步九百六十八分步之三百一十一。 |
| | Answer: four hundred eighty-four bu and three hundred eleven parts of nine hundred sixty-eight parts per bu.
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術曰：置積二十三萬四千五百六十七步，為實。次借一算，為下法。步之，超一位，至百而止。商置四百於實之上。副置四萬於實之下，下法之上，名為方法。命上商四百，除實。除訖，倍方法，一退，下法再退。復置上商八十，以次前商。副置八百於方法之下，下法之上，名為廉法。方、廉各命上商八十，以除。訖，倍廉法，上從方法。方法一退，下法再退。復置上商四，以次前。副置四於方法之下，下法之上，名曰隅法。方、廉、隅各命上商四，以除實除訖，倍隅法，從方法。上商得四百八十四，下法得九百六十八，不盡三百一十一。是為方四百八十四步九百六十八分步之三百一十一。 |
| | Method says: Place the accumulation as two hundred thirty-four thousand five hundred six-seven bu, as the dividend. Next borrow one calculation, as the lower method. Step through it, skip one position, until reaching hundreds and stop. Place four hundred as the quotient above the dividend. Place a copy of forty thousand below the dividend, above the lower method, named as the square method. Assign four hundred to the upper quotient and subtract from the dividend. After subtraction, double the square method, move one position back, lower method moves two positions back. Place again upper quotient as eighty, following the previous quotient. Place a copy of eight hundred below the square method and above the lower method, called the Lianfa (Lian Method). Assign eighty to the upper quotient for both the Fangfa (Square Method) and Lianfa (Lian method), in order to subtract. After completion, double the Lianfa and add it to the Fangfa. Move the Fangfa one position back, move the lower method two positions back. Place again upper quotient as four, following the previous. Place a copy of four below the Fangfa and above the lower method; it is called Yufa (Corner Method). Assign four to the upper quotient for Fangfa, Lianfa, and Yufa respectively, in order to subtract from the dividend. After subtraction is completed, double the Yufa and add it to the method of Fangfa. The upper quotient obtained four hundred eighty-four, the lower method got nine hundred sixty-eight, with a remainder of three hundred eleven. This is equivalent to four hundred and eighty-four bu plus three hundred and eleven parts out of nine hundred and sixty-eight parts per bu as the side length.
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| | 卷中: |
今有積三萬五千步。問為圓幾何？ |
| | Now there is an accumulation of thirty-five thousand bu. Ask how much circle does it amount to?
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答曰：六百四十八步一千二百九十六分步之九十六 |
| | Answer: six hundred forty-eight bu and ninety-six parts of one thousand two hundred ninety-six parts per bu.
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術曰：置積三萬五千步，以一十二乘之，得四十二萬步，為實。次借一算，為下法。步之，超一位，至百而止。上商置六百於實之上。副置六萬於實之下，下法之上，名為方法。命上商六百，除實。除訖，倍方法。方法一退，下法再退。復置上商四十，以次前商。副置四百於方法之下，下法之上，名為廉法。方、廉各命上商，以除實。除訖，倍廉法，從方法。方法一退，下法再退。復置上商八，次前商。副置八於方法之下，下法之上，名為隅法。方、廉、隅各命上前，以除實。除訖，倍隅法，從方法。上商得六百四十八，下法得一千二百九十六，不盡九十六。是為方六百四十八步一千二百九十六分步之九十六。 |
| | Method says: Place the accumulation as thirty-five thousand bu, multiply it by twelve to get four hundred twenty thousand bu, as dividend. Next borrow one calculation, for lower method. Step through it, skip a position, until reaching the hundreds and stop. Place six hundred as upper quotient above the dividend. Place a copy of sixty thousand below the dividend and above the lower method, it is called Fangfa (Square Method). Assign six hundred to upper quotient, subtract from dividend. After subtraction is completed, double Fangfa (Square Method). Move the square method one position backward, and the lower method two positions backward. Place again forty as upper quotient, following previous quotient. Place a copy of four hundred below the square method and on top of the lower method; this is called Lianfa (Side Method). Assign the upper quotient to Fangfa (Square Method), Lianfa (Side Method) respectively, and use them to subtract from dividend. After subtraction is completed, double Lianfa (Side Method), add it to Fangfa (Square Method). Move the square method one step back, and move the lower method two steps back. Place again eight as upper quotient, following previous. Place a copy of eight below the square method and above lower method; it is named Yufa (Angle Method). Assign to Fangfa, Lianfa and Yufa according to upper quotient in order to subtract the dividend. After completing the subtraction, double the Yufa (corner method) and add it to the square method. The upper quotient is six hundred forty-eight, the lower method is one thousand two hundred ninety-six, with a remainder of ninety-six. This means a circle with diameter six hundred forty-eight bu plus ninety-six parts out of one thousand two hundred and ninety-six parts per bu.
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| | 卷中: |
今有丘田周六百三十九步，徑三百八十步。問為田幾何？ |
| | Now there is a hill field with circumference of six hundred thirty-nine steps, diameter of three hundred eighty steps. Ask how many mu of farmland are there?
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答曰：二頃五十二畝二百二十五步。 |
| | Answer: two qing, fifty-two mu and two hundred twenty-five steps.
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術曰：半周得三百一十九步五分，半徑得一百九十步，二位相乘，六萬七百五步。以畝法除之，即得。 |
| | Method says: Half the circumference is three hundred nineteen bu and five fen, half the diameter is one hundred ninety bu. Multiply these two numbers to get six thousand seven hundred fifty bu. Divide it by mu rate, you will obtain the result.
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| | 卷中: |
今有築城，上廣二丈，下廣五丈四尺，高三丈八尺，長五千五百五十尺。秋程人功三百尺。問須功幾何？ |
| | Now there is a city wall to be built, with upper width of two zhang, lower width of five zhang four chi, height of three zhang eight chi, and length of five thousand five hundred fifty chi. The autumn quota for one person's work is three hundred chi. Ask how much labor force is needed?
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答曰：二萬六千一十一功。 |
| | Answer: twenty-six thousand and eleven person-days.
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術曰：并上、下廣，得七十四尺，半之，得三十七尺。以高乘之，得一千四百六尺。又以長乘之，得積七百八十萬三千三百尺。以秋程人功三百尺除之，即得。 |
| | Method says: Add the upper and lower widths, getting seventy-four chi; half it to get thirty-seven chi. Multiply it by the height, getting one thousand and forty-six chi. Multiply it again by the length, getting an accumulation of 7803300 chi. Divide it by autumn quota per person's work of three hundred chi, you will get the result.
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| | 卷中: |
今有穿渠，長二十九里一百四步，上廣一丈二尺六寸，下廣八尺，深一丈八尺。秋程人功三百尺。問須功幾何？ |
| | Now there is a canal to be dug, with length of twenty-nine li and one hundred four bu, upper width of one zhang two chi six cun, lower width of eight chi, depth of one zhang eight chi. The autumn quota for a person's work is three hundred bu. Note: The term "bu" here might be used instead of "chi" in the original text, but it was translated as "bu" to maintain accuracy. If there are any inconsistencies or if further clarification is needed, please let me know. Ask how much labor force is required?
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答曰：三萬二千六百四十五人，不盡六十九尺六寸。 |
| | Answer: thirty-two thousand six hundred forty-five persons, with a remainder of sixty-nine chi six cun.
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術曰：置里數，以三百步乘之，內零步，六之，得五萬二千八百二十四尺。并上、下廣，得二丈六寸。半之，以深乘之，得一百八十五尺四寸。以長乘，得九百七十九萬三千五百六十九尺六寸。以人功三百尺除之，即得。 |
| | Method says: Set the li number, multiply it by three hundred bu, add in the remaining bu, multiply by six, getting fifty-two thousand eight hundred twenty-four chi. Add upper and lower widths together to get two zhang six cun. Halve it, multiply by depth to get one hundred eighty-five chi four cun. Multiply by length to get nine hundred seventy-nine million three thousand five hundred sixty-nine chi six cun. Divide it by the per-person quota of 300 chi, and you will obtain the result.
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| | 卷中: |
今有錢六千九百三十，欲令二百一十六人作九分分之，八十一人，人與二分；七十二人，人與三分；六十三人，人與四分。問三種各得幾何？ |
| | Now there is 6930 coins, wanting to let two hundred and sixteen people divide it into nine parts, eighty-one people, each person gets two shares; 72 people, each receives 3 shares; sixty-three people, each person receives four shares. Ask how much does each of the three groups get?
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答曰：二分，人得錢二十二。三分，人得錢三十三。四分，人得錢四十四。 |
| | Answer: Two shares, each person receives twenty-two coins. Three shares, each person receives thirty-three coins. Four shares, each person receives forty-four coins.
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術曰：先置八十一人於上，七十二人次之，六十三人在下。上位以二乘之，得一百六十二；次位以三乘之，得二百一十六；下位以四乘之，得二百五十二。副并三位，得六百三十，為法。又置錢六千九百三十為三位。上位以一百六十二乘之四，得一百一十二萬二千六百六十，又以二百十六乘中位，得一百四十九萬六千八百八十；又以二百五十二乘下位，得一百七十四萬六千三百六十；各為實。以法六百三十各除之，上位得一千七百八十二，中位得二千三百七十六，下位得二千七百七十二。各以人數除之，即得。 |
| | Method says: First place the eighty-one people on top, then the seventy-two people next, and the sixty-three people at the bottom. Multiply the upper position by two to get one hundred and sixty-two; Multiply the next level by three, getting 216; Multiply the lower position by four, getting two hundred fifty-two. Copy and add the three positions together, obtaining 630, which is used as the denominator. Also set up the money amount of six thousand nine hundred thirty into three positions. The upper position is multiplied by one hundred and sixty-two times four, resulting in one million twelve thousand two hundred and sixty. Then multiply the middle position by two hundred sixteen to get one million forty-nine thousand six hundred and eighty; Multiply the lower position again by 252, getting one million seventy-four thousand six hundred thirty-six; Each of these becomes the dividend. Divide each by the divisor 630: the upper position gets one thousand seven hundred eighty-two, the middle position gets two thousand three hundred seventy-six, and the lower position gets two thousand seven hundred seventy-two. Divide each by the number of people respectively, you will get the answer.
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| | 卷中: |
今有五等諸侯，共分橘子六十顆。人別加三顆。問五人各得幾何？ |
| | Now there are five ranks of feudal lords who share sixty tangerines together. Each person adds 3 more. Ask how many does each of the five people get?
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答曰：公一十八顆。侯一十五顆。伯一十二顆。子九顆。男六顆。 |
| | Answer: The first person receives 18 oranges. The Marquis receives 15 oranges. The Bo receives 12 oranges. The Zi receives 9 oranges. The Nan receives 6 oranges.
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術曰：先置人數，別加三顆於下，次六顆，次九顆，次一十二顆，上十五顆。副并之，得四十五。以減六十顆，餘，人數除之，人得三顆。各加不并者，上得一十八，為公分；次得一十五，為侯分；次得十二，為伯分；次得九，為子分；下得六，為男分。 |
| | Method says: First set the number of people, add three tangerines to the lowest one separately, then six, nine, twelve, and fifteen on top. Copy and sum them up, getting forty-five. Subtract it from sixty tangerines, the remainder is divided by the number of people to get three tangerines per person. Add separately to each: the top gets eighteen, which is the Gong's share; The second receives 15, as the Hou's portion; Next comes 12, which is the Bo's portion; The next one receives 9, as the Zi's portion; the bottom gets six, for Nan's share.
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| | 卷中: |
今有甲、乙、丙三人持錢。甲語乙、丙：「各將公等所持錢半以益我錢，成九十。」乙復語甲、丙：「各將公等所持錢半以益我錢，成七十。」丙復語甲、乙：「各將公等所持錢半以益我錢，成五十六。」問三人元持錢各幾何？ |
| | Now there are three people named Jia, Yi and Bing holding money. Jia said to Yi and Bing: "Each of you take half of the money that you hold, add it to my money, then I will have ninety." Yi in turn said to Jia and Bing: "Each of you take half the money that you hold, give it to me, then I will have seventy." Bing in turn said to Jia and Yi: "Each of you take half the money you hold, give it me, then I will have fifty-six." Ask how much money did each of the three originally hold?
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答曰：甲七十二。乙三十二。丙四。 |
| | Answer: Jia had seventy-two. Yi had thirty-two. Bing had four.
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術曰：先置三人所語為位，以三乘之，各為積，甲得二百七十，乙得二百一十，丙得一百六十八。各半之，甲得一百三十五，乙得一百五，丙得八十四。又置甲九十、乙七十、丙五十六，各半之。以甲、乙減丙，以甲、丙減乙，以乙、丙減甲，即各得元數。 |
| | Method says: First set the three people's statements as positions, multiply each by three to get accumulations respectively; Jia gets two hundred seventy, Yi gets two hundred ten, Bing gets one hundred sixty-eight. Halve each of them: Jia gets 135, Yi gets 105, Bing gets 84. Also set Jia's ninety, Yi's seventy, and Bing's fifty-six, each halved. Subtract Jia and Yi from Bing, subtract Jia and Bing from Yi, subtract Yi and Bing from Jia; then each will get the original amount.
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| | 卷中: |
今有女子善織，日自倍。五日織通五尺扣問日織幾何？ |
| | Now there is a woman who is good at weaving, her daily output doubles. In five days she wove a total of five chi. Ask how much did she weave each day?
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答曰：初日織一寸三十一分寸之一十九次日織三寸三十一分寸之七次日織六寸三十一分寸之一十四次日織一尺二寸三十一分寸之二十八次日織二尺五寸三十一分寸之二十五 |
| | Answer: On the first day, she wove one cun and 19/31 of a cun; on the second day, three cuns and seven thirty-firsts of a cun; next day, six cuns and fourteen thirty-firsts of a cun ； next day, twelve cuns and twenty-eight thirty-firsts of a cunc; on the fifth day, two chi five cuns and twenty-five thirty-firsts of a cunic.
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術曰：各置列衰，副并，得三十一，為法。以五尺乘未并者，各自為實。實如法而一，即得。 |
| | Method says: Set up each as a sequence of ratios, copy and sum them to get 31, which is the divisor. Multiply five chi by those not yet summed, each becomes the dividend. Divide the dividend by the divisor to get the result.
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| | 卷中: |
今有人盜庫絹，不知所失幾何。但聞草中分絹，人得六匹，盈六匹；人得七匹，不足七匹。問人、絹各幾何？ |
| | Now there is someone who stole silk from a warehouse, but it's unknown how much was lost. But I heard that in the grass they divided up the silk; each person got six bolts and had a surplus of six bolts; each person gets seven bolts, it is short by seven bolts. Ask how many people and how much silk are there?
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答曰：賊一十三人。絹八十四匹。 |
| | Answer: thirteen thieves. Silk, eighty-four bolts.
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術曰：先置人得六匹於右上，盈六匹於右下；後置人得七匹於左上，不足七匹於左下。維乘之，所得，并之，為絹。并下盈、不足，為人。 |
| | Method says: First place six bolts per person at the upper right, and a surplus of six bolts at the lower right; Then place seven bolts each for people in the upper left, and seven bolts short in the lower left. Multiply diagonally to get the products, add them together to obtain the amount of silk. Add up the surplus and shortage below to get the number of people.
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| | 卷下: |
今有甲、乙、丙、丁、戊、己、庚、辛、壬九家共翰租。甲出三十五斛，乙出四十六斛，丙出五十七斛，丁出六十八斛，戊出七十九斛，己出八十斛，庚出一百斛，辛出二百一十斛，壬出三百二十五斛。凡九家共翰租一千斛。僦運直折二百斛外，問家各幾何？ |
| | Now there are Jia, Yi, Bing, Ding, Wu, Ji, Geng, Xin, and Ren nine families jointly paying rent. Jia contributed 35 hu, Yi contributed 46 hu, Bing contributed 57 hu, Ding contributed 68 hu, Wu contributed 79 hu, Ji contributed 80 hu, Geng contributed 100 hu, Xin contributed 210 hu, and Ren contributed 325 hu. In total, the nine families jointly paid a rent of 1,000 hu. After deducting the cost of hiring transport, which amounted to 200 hu, how much should each family contribute?
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答曰：甲二十八斛。乙三十六斛八蚪。丙四十五斛六蚪。丁五十四斛四蚪。戊六十三斛二蚪。己六十四斛。庚八十斛。辛一百六十八斛。壬二百六十斛。 |
| | Answer: Jia contributed 28 hu. Yi contributed 36 hu and 8 dou. Bing contributed 45 hu and 6 dou. Ding contributed 54 hu and 4 dou. Wu contributed 63 hu and 2 dou. Ji contributed 64 hu. Geng contributed 80 hu. Xin contributed 168 hu. Ren contributed 260 hu.
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術曰：置甲出三十五斛，以四乘之，得一百四十斛。以五除之，得二十八斛。乙出四十六斛，以四乘之，得一百八十四斛。以五除之，得三十六斛八蚪。丙出五十七斛，以四乘之，得二百二十八斛。以五除之，得四十五斛六蚪。丁出六十八斛，以四乘之，得二百七十二斛。以五除之，得五十四斛四斟。戊出七十九斛，以四乘之，得三百一十六斛。以五除之，得六十三斛二斟。己出八十斛，以四乘之，得三百二十斛。以五除之，得六十四斛。庚出一百斛，以四乘之，得四百斛。以五除之，得八十斛。辛出二百一十斛，以四乘之，得八百四十斛。以五除之，得一百六十八斛。壬出三百二十五斛，以四乘之，得一千三百斛。以五除之，得二百六十斛。 |
| | Method: Take Jia's contribution of 35 hu and multiply it by four, resulting in 140 hu. Divide this by five, yielding 28 hu. Yi's contribution of 46 hu is multiplied by four, resulting in 184 hu. Divide this by five, obtaining 36 hu and 8 dǒu. Bing's contribution of 57 hu is multiplied by four, yielding 228 hu. Divide this by five, resulting in 45 hu and 6 dǒu. Ding's contribution of 68 hu is multiplied by four, giving 272 hu. Divide this by five, getting 54 hu and four zhen. Wu's contribution of 79 hu is multiplied by four, obtaining 316 hu. Divide this by five, giving 63 hu and two zhen. Ji's contribution of 80 hu is multiplied by four, getting 320 hu. Divide this by five, which results in 64 hu. Geng's contribution of one hundred hu is multiplied by four, producing four hundred hu. Divide this by five to get eighty hu. Xin's contribution of 210 hu is multiplied by four, amounting to 840 hu. Divide this by five and obtain 168 hu. Ren's contribution of 325 hu is multiplied by four, totaling one thousand three hundred hu. Divide this by five and get two hundred sixty hu.
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| | 卷下: |
今有丁一千五百萬，出兵四十萬。問幾丁科一兵？ |
| | Now there are 15 million households, contributing 400,000 soldiers. The question is: how many households contribute one soldier?
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答曰：三十七丁五分。 |
| | Answer: 37.5 households.
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術曰：置丁一千五百萬，為實。以兵四十萬為法。實如法，即得。 |
| | Method: Take the number of households, one thousand five hundred million, as the dividend. Use the 400,00 soldiers as the divisor. Divide the dividend by the divisor, and the result is obtained directly.
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| | 卷下: |
今有平地聚粟，下周三丈六尺，高四尺五寸。問粟幾何？ |
| | Now there is a pile of millet on flat ground, with a base circumference of three zhang six chi and a height of four chi five cun. How much millet is there?
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答曰：一百斛。 |
| | Answer: 100 hu.
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術曰：置周三丈六尺，自相乘，得一千二百九十六尺。以高四尺五寸乘之，得五千八百三十二尺。以三十六除之，得一百六十二尺。以斛法一尺六寸二分除之，即得。 |
| | Method: Take the base circumference of three zhang and six chi, multiply it by itself, resulting in 1296 square chi. Multiply this by the height of four chi and five cun, obtaining 5832 cubic chi. Divide this by thirty-six, resulting in one hundred sixty-two cubic chi. Divide this by the hu standard of 1 chi six cun two fen to obtain the result directly.
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| | 卷下: |
今有佛書凡二十九章，章六十三字。問字幾何？ |
| | Now there is a Buddhist scripture with a total of twenty-nine chapters, each chapter containing sixty-three characters. How many characters are there in total?
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答曰：一千八百二十七。 |
| | Answer: 1,827.
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術曰：置二十九章，以六十三字乘之，即得。 |
| | Method: Take the twenty-nine chapters and multiply them by sixty-three characters per chapter; this gives the total directly.
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| | 卷下: |
今有諪局方一十九道。問用諪幾何？ |
| | Now there is a chessboard with a square pattern of nineteen lines on each side. How many pieces are used?
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答曰：三百六十一。 |
| | Answer: 361.
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術曰：置一十九道，自相乘之，即得。 |
| | Method: Take the nineteen lines, multiply them by themselves, and you get the result directly.
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| | 卷下: |
今有租九萬八千七百六十二斛，欲以一車載五十斛，問用車幾何？ |
| | Now there is a total of 98,762 hu of rent, and it is desired to transport one cartload of fifty hu per vehicle. How many carts are needed?
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答曰：一千九百七十五乘，奇一十二斛。 |
| | Answer: 1975 carts will be used, with a remainder of twelve hu.
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術曰：置租九萬八千七百六十二斛，為實。以一車所載五十斛為法。實如法，即得。 |
| | Method: Take the total rent of ninety-eight thousand seven hundred and sixty-two hu as the dividend. Use fifty hu, the amount one cart can carry, as the divisor. Divide the dividend by the divisor to obtain the result directly.
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| | 卷下: |
今有丁九萬八千七百六十六，凡二十五丁出一兵。問兵幾何？ |
| | Now there are 98,766 households, and for every twenty-five households, one soldier is contributed. How many soldiers are there?
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答曰：三千九百五十人，奇一十六丁。 |
| | Answer: 3,950 soldiers, with a remainder of sixteen households.
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術曰：置丁九萬八千七百六十六，為實。以二十五為法。實如法，即得。 |
| | Method: Take the number of households as ninety-eight thousand seven hundred sixty-six and set it as the dividend. Use twenty-five as the divisor. Divide the dividend by the divisor and you will obtain the result directly.
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| | 卷下: |
今有絹七萬八千七百三十二匹，令一百六十二人分之。問人得幾何？ |
| | Now there are 78,732 bolts of silk to be divided among 162 people. How many bolts does each person get?
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答曰：四百八十六匹。 |
| | Answer: 486 bolts.
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術曰：置絹七萬八千七百三十二匹，為實。以一百六十二人為法。實如法，即得。 |
| | Method: Take the silk of seventy-eight thousand seven hundred thirty-two bolts as the dividend. Use 162 people as the divisor. Divide the dividend by the divisor directly to obtain the result.
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| | 卷下: |
今有三萬六千四百五十四戶，戶輸綿二斤八兩。問計幾何？ |
| | Now there are 36,454 households, each household is required to contribute two jin and eight liang of cotton. What is the total amount?
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| | |
答曰：九萬一千一百三十五斤。 |
| | Answer: 91,135 jin.
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術曰：置三萬六千四百五十四戶，上十之，得三十六萬四千五百四十。以四乘之，得一百四十五萬八千一百六十兩。以十六除之，即得。 |
| | Method: Take the number of households (36,454), multiply it by ten, resulting in 364,540. Multiply this by four to get 1,458,160 liang. Divide this by sixteen, and the result is directly obtained.
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| | 卷下: |
今有綿九萬一千一百三十五斤，給與三萬六千四百五十四戶。問戶得幾何？ |
| | Now there are 91,135 catties of cotton to be distributed among 36,454 families. How many catties does each family receive?
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答曰：二斤八兩。 |
| | Answer: 2 jin and 8 liang.
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術曰：置九萬一千一百三十五斤，為實。以三萬六千四百五十四戶為法。除之，得二斤，不盡一萬八千二百二十七斤。以一十六乘之，得二十九萬一千六百三十二兩。以戶除之，即得。 |
| | Method: Take the 91,135 pounds of cotton as the dividend. Use the number of families, 36,454, as the divisor. Divide it and get two jin, with a remainder of eighteen thousand two hundred and twenty-seven jin. Multiply the remainder by sixteen to obtain 291,632 liang. Divide this amount by the number of families and you will get the result directly.
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| | 卷下: |
今有粟三千九百九十九斛九蚪六升，凡栗九蚪易豆一斛。問計豆幾何？ |
| | Now there are 3,999 hu nine dǒu six sheng of millet. For every nine dǒu of millet, it can be exchanged for one hu of broad beans. How many hu of broad beans are there in total?
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答曰：四千四百四十四斛四斟。 |
| | Answer: 4,444 hu and four zhen (of broad beans).
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術曰：置粟三千九百九十九斛九斗六升，為實。以九斗為法。實如法。即得。 |
| | Method: Take the millet of three thousand nine hundred ninety-nine hu, nine dǒu and six sheng as the dividend. Use nine dǒu as the divisor. Divide the dividend by the divisor. The result is obtained directly.
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| | 卷下: |
今有粟二千三百七十四斛，斛加三升。問共粟幾何？ |
| | Now there are 2,374 hu of millet, with an additional three sheng per hu. What is the total amount of millet?
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答曰：二千四百四十五斛二斗二升。 |
| | Answer: 2,445 hu two dǒu and two sheng.
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術曰：置粟二千三百七十四斛，以一斛三升乘之，即得。 |
| | Method: Take the millet amount of 2,374 hu, multiply it by one hu and three sheng, and you will directly obtain the result.
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| | 卷下: |
今有粟三十六萬九千九百八十斛七斗，在倉九年，年斛耗三升。問一年、九年各耗幾何？ |
| | Now there are 369,980 hu seven dǒu of millet in the granary, stored for nine years, with an annual loss of three sheng per hu. Now there are 369,980 hu and seven dǒu of millet stored in the granary for nine years, with an annual loss of three sheng per hu. How much is lost in one year, and how much in nine years?
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答曰：一年耗一萬一千九十九斛四斗二升一合。九年耗九萬九千八百九十四斛七斗八升九合。 |
| | Answer: In one year, the loss amounts to 11,994 hu four dǒu two sheng and one ge. In nine years, the total loss is 99,894 hu seven dǒu eight sheng and nine ge.
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術曰：置三十六萬九千九百八十斛七斗，以三升乘之得一年之耗。又以九乘之，即九年之耗。 |
| | Method: Take 369,987 hu seven dǒu, multiply it by three sheng to obtain the loss for one year. Multiply this result by nine again, and you will get the total loss over nine years.
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| | 卷下: |
今有貸與人絲五十七斤，限歲出息一十六斤。問斤息幾何？ |
| | Now there is a loan of 57 jin of silk to someone, with an interest requirement of 16 jin per year. What is the interest rate per jin?
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答曰：四兩五十七分兩之二十八。 |
| | Answer: Four liang and 28/57 of a liang.
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術曰：列限息絲一十六斤，以一十六兩乘之，得二百五十六兩。以貸絲五十七斤除之⑤。不盡，約之，即得。 |
| | Method: Take the interest silk of sixteen jin, multiply it by sixteen liang to get 256 liang. Divide this by the loaned silk of fifty-seven jin. If it does not divide evenly, approximate it; thus you obtain the result.
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| | 卷下: |
今有三人共車，二車空；二人共車，九人步。問人與車各幾何？ |
| | Now there are three people sharing a cart, and two carts remain empty; two people share another cart, while nine people walk. What are the numbers of people and carts?
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答曰：一十五車。三十九人。 |
| | Answer: Fifteen carts. Thirty-nine people.
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術曰：置二車，以三乘之，得六。加步者九人，得車一十五。欲知人者，以二乘車，加九人，即得。 |
| | Method: Take two carts, multiply by three to get six. Add the nine people walking, resulting in fifteen carts. To find out the number of people, multiply two by the number of carts and add nine people; thus you get the result.
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| | 卷下: |
今有粟一十二萬八千九百四十斛九斗三合，出與人買絹，一匹直粟三斛五斗七升。問絹幾何？ |
| | Now there are 128,940 hu nine dǒu three ge of millet given to someone for buying silk, with one bolt of silk valued at three hu five dǒu seven sheng of millet. What is the amount of silk?
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| | |
答曰：三萬六千一百一十七匹三丈六尺。 |
| | Answer: 36,117 bolts three zhang six chi.
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術曰：置粟一十二萬八千九百四十斛九斗三合，為實。以三斛五斗七升為法。除之，得匹。餘四十之，所得，又以法除之，即得。 |
| | Method: Take the millet of 128,949 hu nine dǒu and three ge as the dividend. łoż Use three hu five dǒu and seven sheng as the divisor. Divide it, and you get the number of bolts. Take the remainder, multiply it by forty, then divide again using the same divisor; thus you obtain the final result.
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| | 卷下: |
今有婦人河上蕩桮。津吏問曰：「桮何以多？」婦人曰：「家有客。」津吏曰：「客幾何？」婦人曰：「二人共飯，三人共羹，四人共肉，凡用聣六十五。不知客幾何？」 |
| | Now there is a woman washing cups at the riverbank. The ferry official asked, "Why are there so many cups?" The woman replied, "There are guests at my home." The ferry official asked, "How many guests?" The woman said: "Two people share a meal, three people share soup, four people share meat, and in total sixty-five cups are used. I do not know how many guests there are."
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答曰：六十人。 |
| | Answer: Sixty people.
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術曰：置六十五桮，以一十二乘之，得七百八十，以十三除之，即得。 |
| | Method: Take the 65 cups, multiply by twelve to get 780, then divide by thirteen; thus you obtain the answer.
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| | 卷下: |
今有木，不知長短。引繩度之，餘繩四尺五寸。屈繩量之，不足一尺。問木長幾何？ |
| | Now there is a piece of wood whose length is unknown. Using a rope to measure it, the rope has 4 chi and 5 cun left over. Folding the rope to measure it results in one chi short. What is the length of the wood?
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答曰：六尺五寸。 |
| | Answer: Six chi and five cun.
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術曰：置餘繩四尺五寸，加不足一尺，共五尺五寸。倍之，得一丈一尺。減餘四尺五寸，即得。 |
| | Method: Take the remaining rope of four chi and five cun, add to it the one chi that was insufficient, making a total of five chi and five cun. Double this amount, resulting in eleven chi. Subtract the remaining four chi and five cun; thus you obtain the length of the wood.
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| | 卷下: |
今有器中米，不知其數。前人取半，中人三分取一，後人四分取一，餘米一斗五升。問本米幾何？ |
| | Now there is rice in a vessel, but its quantity is unknown. The first person took half, the second person took one-third of what remained, the third person took one-fourth of what was left after that, and there were 1 dǒu five sheng of rice remaining. What was the original amount of rice?
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答曰：六斗。 |
| | Answer: Six dǒu.
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術曰：置餘米一斗五升，以六乘之，得九斗。以二除之，得四斗五升。以四乘之，得一斛八斗。以三除之，即得。 |
| | Method: Take the remaining rice of one dǒu five sheng, multiply it by six to get nine dǒu. Divide this amount by two, resulting in four dǒu and five sheng. Multiply this result by four to get one hu eight dǒu. Divide this amount by three; thus you obtain the original quantity of rice.
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| | 卷下: |
今有黃金一斤，直錢一十萬。問兩直幾何？ |
| | Now there is one jin of gold, valued at 100,000 qian. What is the value per liang?
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答曰：六千二百五十錢。 |
| | Answer: 6,250 qian.
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術曰：置錢一十萬，以一十六兩除之，即得。 |
| | Method: Take the money amount of one hundred thousand qian and divide it by sixteen liang; thus you obtain the value per liang.
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| | 卷下: |
今有錦一疋，直錢一萬八千。問丈、尺、寸各直幾何？ |
| | Now there is one bolt of brocade, valued at 18,000 qian. Now there is one bolt of brocade, valued at 18,000 qian. What are the values per zhang, chi, and cun?
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答曰：丈，四千五百錢。尺，四百五十錢。寸，四十五錢。 |
| | Answer: One zhang is worth 4,500 qian. One chi is worth 450 qian. One cun is worth forty-five qian.
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術曰：置錢一萬八千，以四除之得尺、寸之直。得一丈之直。一退、再退，得尺、寸之直。 |
| | Method: Take the money amount of 18,000 and divide it by four to obtain the value per chi and cun. This gives you the value for one zhang. One step back, then another step back, to obtain the values per chi and cun.
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| | 卷下: |
今有地長一千步，廣五百步。尺有鶉，寸有鷃。問鶉、鷃各幾何？ |
| | Now there is a field that is 1,000 bu long and 500 bu wide. There are quail per chi and yàn birds per cun. What are the numbers of quail and yàn?
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答曰：鶉一千八百萬。鷃一億八千萬。 |
| | Answer: 18 million quail. Yàn birds, one hundred and eighty million.
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術曰：置長一千步，以廣五百步乘之，得五十萬步。以三十六乘之，得一千八百萬尺，即得鶉數。上十之，即得赉數。 |
| | Method: Take the length of 1,000 bu, multiply it by the width of 500 bu to get fifty thousand bu. Multiply this by thirty-six to get eighteen million chi, which is the number of quail. Increase it tenfold, and you obtain the number of yàn birds.
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| | 卷下: |
今有六萬口，上口三萬人，日食九升；中口二萬人，日食七升；下口一萬人，日食五升。問上、中、下口共食幾何？ |
| | Now there are 60,000 people, with 30,000 being adults who consume nine sheng of food per day; 20,000 are middle-aged individuals who consume seven sheng per day; 10,000 are children who consume five sheng per day. What is the total daily food consumption for adults, middle-aged individuals, and children combined?
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答曰：四千六百斛。 |
| | Answer: 4,600 hu.
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術曰：各置口數，以日食之數乘之。所得，并之，即得。 |
| | Method: Take the number of each category of people and multiply it by their daily food consumption. Add up all the results, and you will obtain the total amount.
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| | 卷下: |
今有方物一束，外周一匝有三十二枚。問積幾何？ |
| | Now there is a bundle of square-shaped items; one full layer around the outside contains thirty-two pieces. What is the total number?
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答曰：八十一枚。 |
| | Answer: Eighty-one pieces.
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術曰：重置二位。上位減八，餘加下位。至盡虛加一，即得。 |
| | Method: Set up two positions again. Subtract eight from the upper position, and add the remainder to the lower position. Continue until it is exhausted; then add one more at the end, and you will obtain the result.
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| | 卷下: |
今有竿不知長短，度其影得一丈五尺。別立一表，長一尺五寸，影得五寸。問竿長幾何？ |
| | Now there is a pole of unknown length; measuring its shadow gives 1 zhang five chi. Set up another gnomon one chi and five cun tall, whose shadow measures five cun. What is the length of the pole?
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答曰：四丈五尺。 |
| | Answer: Four zhang and five chi.
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術曰：置竿影一丈五尺，以表長一尺五寸乘之，上十之，得二十二丈五尺。以表影五寸除之，即得。 |
| | Method: Take the pole's shadow of one zhang and five chi, multiply it by the gnomon height of one chi and five cun, then increase this result tenfold to get twenty-two zhang and five chi. Divide this amount by the gnomon's shadow of five cun; thus you obtain its length.
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| | 卷下: |
今有物，不知其數。三、三數之，賸二；五、五數之，賸三；七、七數之，賸二。問物幾何？ |
| | Now there is an object, but we do not know its quantity. Counting it in groups of three leaves a remainder of two; counting it in groups of five leaves a remainder of three; Counting it in groups of seven leaves a remainder of two. What is the quantity of the object?
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答曰：二十三。 |
| | Answer: Twenty-three.
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術曰：「三、三數之，賸二」，置一百四十；「五、五數之，賸三」，置六十三；「七、七數之，賸二」，置三十。并之，得二百三十三。以二百一十減之，即得。凡三、三數之，賸一，則置七十；五，五數之，賸一，則置二十一；七、七數之，賸一，則置十五。一百六以上，以一百五減之，即得。 |
| | Method: "Counting in groups of three, remainder two," set down one hundred and forty; "Counting in groups of five, remainder three," set down sixty-three; "Counting in groups of seven, remainder two," set down thirty. Add them together to get two hundred and thirty-three. Subtract two hundred and ten from it, and you will obtain the answer. Generally, if counting in groups of three leaves remainder one, set down seventy; if counting in groups of five leaves remainder one, set down twenty-one; if counting in groups of seven leaves remainder one, then set down fifteen. If the total is more than 160, subtract 150 from it; thus you obtain the answer.
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| | 卷下: |
今有獸六首四足，禽四首二足。上有七十六首，下有四十六足。問篱、獸各幾何？ |
| | Now there are beasts with six heads and four feet, and birds with four heads and two feet. Above there are seventy-six heads, below there are forty-six feet. Ask: How many birds and beasts are there?
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答曰：八獸，七禽。 |
| | Answer: Eight beasts, seven birds.
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術曰：倍足以減首，餘，半之之，即禽。即獸。以四乘獸，減足，餘，半 |
| | Method: Double the number of feet and subtract it from the number of heads; half the remainder is the number of birds. The rest are beasts. Multiply the number of beasts by four, subtract it from the total feet; half
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| | 卷下: |
今有甲、乙二人，持錢各不知數。甲得乙中半，可滿四十八。乙得甲太半，亦滿四十八。問甲、乙二人元持錢各幾何？ |
| | Now there are two people, Jia and Yi, each holding an unknown amount of money. If Jia receives half of what Yi has, he will have a total of forty-eight. If Yi receives two-thirds of what Jia has, he too will have a total of fortyeight. Ask: Originally, how much money did Jia and Yi each hold?
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答曰：甲持錢三十六。乙持錢二十四。 |
| | Answer: Jia originally held thirty-six units of money. Yi originally held twenty-four units of money.
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術曰：如方程求之。置二甲、一乙、錢九十六，於右方。置二甲、三乙、錢一百四十四，於左方。以右方二乘左方，上得四，中得六，下得二百八十八錢。以右行再減左行，左上空，中餘四乙，為法；下餘九十六錢，為實。上法，下實，得二十四錢，為乙錢。以減右下九十六，餘七十二，為實；以右上二甲為法。上法、下實，得三十六，為甲錢也。 |
| | Method: Solve it using the method of equations. Place two Jia, one Yi, and ninety-six units of money on the right side. Place two Jia, three Yi, and one hundred forty-four units of money on the left side. Multiply the right side by two times the left side, resulting in four at the top, six in the middle, and 288 units of money at the bottom. Subtract twice the right row from the left row; the top on the left is empty, and the middle has four Yi remaining as the divisor; the bottom has ninety-six units of money remaining as the dividend. Apply the divisor to the dividend, resulting in twenty-four units of money, which is Yi's amount. Subtract this from ninety-six at the bottom on the right; the remainder is seventy-two as the dividend; take two Jia at the top of the right side as the divisor. Apply the divisor to the dividend and obtain thirty-six, which is Jia's amount.
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| | 卷下: |
今有百鹿入城，家取一鹿，不盡；又三家共一鹿，適盡。問城中家幾何？ |
| | Now there are one hundred deer entering a city; each household takes one deer, but they do not finish; then three households share one deer and it is just finished. Ask: How many households are in the city?
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答曰：七十五家。 |
| | Answer: Seventy-five households.
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術曰：以盈不足取之。假令七十二家，鹿不盡四。令之九十家，鹿不足二十。置七十二於右上，盈四於右下。置九十於左上，不足二十於左下。為維乘之凹，所得，并為實。并盈不足為法。除之，即得。 |
| | Method: Solve it using the principle of surplus and deficit. Assume there are seventy-two households, then four deer remain unclaimed. If we assume ninety households, twenty fewer deer would be needed. Place seventy-two at the upper right and four surplus at the lower right. Place ninety at the upper left and twenty deficit at the lower left. Multiply crosswise, add the results to obtain the dividend. Add the surplus and deficit together as the divisor. Divide, and you obtain the answer.
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| | 卷下: |
今有三雞共啄粟一千一粒。雛啄一，母啄粟。三雞主各償幾何？ |
| | Now there are three chickens pecking at one thousand and one grains of millet. A chick pecks one grain, a hen pecks ten grains. How many grains should each of the three chickens compensate?
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答曰：雞雛主一百四十三。雞母主二百八十六。雞翁主五百七十二。翁啄四。主責本 |
| | Answer: The chick's owner compensates 143. The hen's owner compensates 286. The rooster's owner compensates 572. A rooster pecks four grains. The owner is responsible for the original amount.
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術曰：置粟一千一粒，為實。副并三雞所啄粟七粒，為法。除之，得一百四十三粒，為雞雛主所償之數。遞倍之，即得母、翁主所償之數。 |
| | Method: Place one thousand and one grains as the dividend. Add up the total number of grains pecked by three chickens, which is seven grains, as the divisor. Divide it to get 143 grains, which is the amount that the chick's owner should compensate. Multiply this by two successively, and you will obtain the amounts that the hen and rooster owners should compensate.
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| | 卷下: |
今有雉、兔同籠，上有三十五頭，下九十四足。問雉、兔各幾何？ |
| | Now there are pheasants and rabbits in a cage, with thirty-five heads above and ninety-four feet below. Ask: How many pheasants and how many rabbits are there?
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答曰：雉二十三。兔一十二。 |
| | Answer: Twenty-three pheasants. Twelve rabbits.
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術曰：上置三十五頭，下置九十四足。半其足，得四十七。以少減多，再命之，上三除下三，上五除下五。下有一除上一，下有二除上二，即得。 |
| | Method: Place thirty-five heads at the top and ninety-four feet at the bottom. Halve the number of feet, getting forty-seven. Subtract the smaller from the larger and then reassign: for every three at the top, subtract three at the bottom; for every five at the top, subtract five at the bottom. For each one below, subtract one above; for each two below, subtract two above; thus you obtain the result.
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又術曰：上置頭，下置足。半其足，以頭除足，以足除頭，即得。 |
| | Another method: Place the heads at the top and feet at the bottom. Halve the number of feet; subtract the heads from the feet, then subtract the feet from the heads, thus obtaining the result.
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今有九里渠，三寸魚，頭頭相次。問魚得幾何？ |
| | Now there is a ditch nine li long and three cun wide; fish are lined up head to tail. Ask: How many fish can fit?
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答曰：五萬四千。 |
| | Answer: Fifty-four thousand.
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術曰：置九里，以三百步乘之，得二千七百步。又以六尺乘之，得一萬六千二百尺。上十之，得一十六萬二千寸。以魚三寸除之，即得。 |
| | Method: Place nine li, multiply by three hundred bu to get two thousand seven hundred bu. Then multiply by six chi to get one hundred sixty-two thousand chi. Multiply this result by ten, getting 1620 thousand cun. Divide it by three cun for each fish; thus you obtain the number of fish.
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今有長安、洛陽相去九百里。車輪一匝一丈八尺。欲自洛陽至長安，問輪匝幾何？ |
| | Now there are Chang'an and Luoyang, 900 li apart. A cart wheel turns one zha (turn) for every one zhang eight chi. If one wishes to travel from Luoyang to Chang'an, how many wheel turns are needed?
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答曰：九萬匝。 |
| | Answer: Ninety thousand turns.
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術曰：置九百里，以三百步乘之，得二十七萬步。又以六尺乘之，得一百六十二萬尺。以車輪一丈八尺為法。除之，即得。 |
| | Method: Place nine hundred li, multiply by three hundred steps to get two hundred seventy thousand steps. Then multiply by six chi, getting one million sixty-two thousand chi. Use the cart wheel's length of one zhang eight chi as the divisor. Divide, and you will get the result.
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今有出門望見九隄。隄有九木，木有九枝，枝有九巢，巢有九禽，禽有九雛，雛有九毛，毛有九色。問各幾何？ |
| | Now there is someone who goes out of the gate and sees nine dikes. Each dike has nine trees, each tree has nine branches, each branch has nine nests, each nest has nine birds, each bird has nine chicks, each chick has nine feathers, and each feather has nine colors. Ask: How many of each are there?
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答曰：木八十一。枝七百二十九。巢六千五百六十一。禽五萬九千四十九。雛五十三萬一千四百四十一。毛四百七十八萬二千九百六十九。色四千三百四萬六千七百二十一。 |
| | Answer: Eighty-one trees. Seven hundred and twenty-nine branches. Six thousand five hundred and sixty-one nests. Fifty-nine thousand four hundred and ninety birds. Five hundred thirty-one thousand four hundred forty-one chicks. Four million seven hundred eighty-two thousand nine hundred sixty-nine feathers. Forty-three million four hundred six thousand seven hundred twenty-one colors.
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術曰：置九隄，以九乘之，得木之數。又以九乘之，得枝之數。又以九乘之，得巢之數。又以九乘之，得禽之數。又以九乘之，得雛之數。又以九乘之，得毛之敷。又以九乘之，得色之數。 |
| | Method says: Set nine dikes, multiply by nine, to get the number of trees. Multiply again by nine, to get the number của branches. Multiply again by nine, to obtain the number of nests. Multiply again by nine, and you get the number of birds. Multiply again by nine to obtain the number of chicks. Multiply again by nine, get the number of feathers. Multiply again by nine to get the number of colors.
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今有三女，長女五日一歸，中女四日一歸，少女三日一歸。問三女幾何日相會？ |
| | Now there are three daughters: the eldest daughter returns home every five days, the middle daughter every four days, and the youngest daughter every three days. Ask: After how many days will the three daughters meet again?
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答曰：六十日。 |
| | Answer: Sixty days.
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術曰：置長女五日、中女四日、少女三日，於右方。各列一算於左方。維乘之，各得所到數：長女十二到，中女十五到，少女二十到。又各以歸日乘到數，即得。 |
| | Method says: Place the eldest daughter's five days, the middle daughter's four days, and the youngest daughter's three days on the right side. Place one counter for each on the left side. Multiply them, and you get their respective numbers of arrivals: the eldest daughter arrives twelve times, the middle daughter fifteen times, and the youngest daughter twenty times. Then multiply each return day by the number of arrivals, and you will get it.
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今有孕婦行年二十九，難九月。未知所生？ |
| | Now there is a pregnant woman whose age is twenty-nine, giving birth in the ninth month. It is unknown what will be born?
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答曰：生男。 |
| | Answer: A boy will be born.
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術曰：置四十九，加難月，減行年。所餘，以天除一，地除二，人除三，四時除四，五行除五，六律除六，七星除七，八風除八，九州除九。其不盡者，奇則為男，耦則為女。 |
| | Method says: Set forty-nine, add the month of childbirth, subtract the age. The remainder is divided by: Heaven subtracts one, Earth subtracts two, Humanity subtracts three, Four Seasons subtract four, Five Elements subtract five, Six Laws subtract six, Seven Stars subtract seven, Eight Winds subtract eight, Nine Provinces subtract nine. If the remainder is not exhausted, an odd number means a boy will be born, and an even number means a girl will be born.
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