第一章   方田

现在有一块田地，宽15步，长16步。问这块田地有多大？
回答是：一亩。

There is a field that is 15 steps wide and 16 steps long. How large is the field? The answer is: One mu (a traditional unit of area, often translated as "acre").

又有一块田地，宽12步，长14步。问这块田地有多大？
回答是：一百六十八步。 方田术（一种古代的测量土地的方法）说：将田地的宽和长的步数相乘得到积步。用亩法二百四十步去除这个积步，就可以得到亩数。百亩为一顷。

There is another field that is 12 steps wide and 14 steps long. How large is this field? The answer is: One hundred and sixty-eight steps. According to the ancient method of measuring land, one multiplies the width and length of the field in steps to get the total number of steps. Dividing this by two hundred and forty steps, which is the traditional method for measuring mu, yields the number of mu. A hundred mu make up one qing (a traditional unit of area).

现在有一块田地，宽一里，长一里。问这块田地有多大？
回答是：三顷七十五亩。

There is a field that is one li (a traditional Chinese unit of distance) wide and one li long. How large is the field? The answer is: Three qing and seventy-five mu (traditional Chinese units of area).

又有一块田地，宽二里，长三里。问这块田地有多大？
回答是：二十二顷五十亩。 里田术（一种古代的测量土地的方法）说：将田地的宽和长的里数相乘得到积里。用三百七十五乘这个积里，就可以得到亩数。

There is another field that is two li wide and three li long. How large is this field?The answer is: Twenty-two qing and fifty mu.According to the ancient method of measuring land by li, one multiplies the width and length of the field in li to get the total area in li. Multiplying this by three hundred and seventy-five gives the number of mu.

现在有十八分之十二。问约简之后得到多少？
回答是：三分之二。

We have eighteen twelfths. What is it simplified to?The answer is: Two thirds.

又有九十一分之四十九。问约简之后得到多少？
回答是：十三分之七。 约分术（一种古代的数学方法）说：可以减半的就减半，不能减半的，将分子和分母的数字放在一起，用较小的数减去较大的数，再相互减损，寻求它们的等数。然后用这个等数来约分。

We also have ninety-one forty-ninths. What is it simplified to? The answer is: Seven thirteenths. The method of reducing fractions says: Halve if possible, and if not, lay out the numbers of both the numerator and the denominator, subtract the smaller from the larger, and continue to reduce them alternately until you find a common number. Then use this common number to simplify the fraction.

现在有三分之一，五分之二。问合并起来得到多少？
回答是：十五分之十一。

We have one third and two fifths. What is the sum? The answer is: Eleven fifteenths.

又有三分之二，七分之四，九分之五。问合并起来得到多少？
回答是：得到一、六十三分之五十。

We also have two thirds, four sevenths, and five ninths. What is the sum? The answer is: Five hundred sixty-thirds.

又有二分之一，三分之二，四分之三，五分之四。问合并起来得到多少？
回答是：得到二、六十分之四十三。 合分术（一种古代的数学方法）说：将不同分数的分子相乘，并将结果相加作为新的分子，将不同分数的分母也相乘作为新的分母，然后用新的分子除以新的分母得到一个数。如果分子小于分母，就用分母来表示这个数。如果分数的分母相同，直接将分子相加即可。

We also have one half, two thirds, three fourths, and four fifths. What is the sum? The answer is: Forty-three sixty-seconds. The method for combining fractions states: Multiply the numerators of different fractions, add them together to form the new numerator, multiply the denominators to form the new denominator, and then divide the new numerator by the new denominator to get a number. If the numerator is less than the denominator, represent it with the denominator. If the fractions have the same denominator, simply add the numerators together.

现在有九分之八，减去它的五分之一。问剩下多少？
回答是：四十五分之三十一。

We have eight ninths, from which we subtract one fifth. What is the remainder? The answer is: Thirty-one forty-fifths.

又有四分之三，减去它的三分之一。问剩下多少？
回答是：十二分之五。 减分术（一种古代的数学方法）说：将不同分数的分子相乘，用较大的数减去较小的数，剩下的数作为新的分子，将不同分数的分母也相乘作为新的分母，然后用新的分子除以新的分母得到一个数。

We also have three fourths, from which we subtract one third. What is the remainder? The answer is: Five twelfths. The method for subtracting fractions states: Multiply the numerators of different fractions, subtract the smaller from the larger, and the remainder becomes the new numerator. Multiply the denominators to form the new denominator, and then divide the new numerator by the new denominator to get a number.

现在有八分之五，二十五分之十六。问哪个分数更大？大多少？
回答是：二十五分之十六更大，多二百分之三。

We have five eighths and sixteen twenty-fifths. Which is greater, and by how much? The answer is: Sixteen twenty-fifths are greater, by three two hundredths.

又有九分之八，七分之六。问哪个分数更大？大多少？
回答是：九分之八更大，多六十三分之二。

We have eight ninths and six sevenths. Which is greater, and by how much?The answer is: Eight ninths are greater, by sixty-three halves.

又有二十一分之八，五十分之十七。问哪个分数更大？大多少？
回答是：二十一分之八更大，多一千五十分之四十三。 课分术（一种古代的数学方法）说：将不同分数的分子相乘，用较大的数减去较小的数，剩下的数作为新的分子，将不同分数的分母也相乘作为新的分母，然后用新的分子除以新的分母得到一个数，这样就可以看出它们相差多少。

We have eight twenty-firsts and seventeen fiftieths. Which is greater, and by how much? The answer is: Eight twenty-firsts are greater, by forty-three one thousand and fifty-hundredths. The method for comparing fractions states: Multiply the numerators of different fractions, subtract the smaller from the larger, and the remainder becomes the new numerator. Multiply the denominators to form the new denominator, and then divide the new numerator by the new denominator to get a number, which shows the difference between them.

现在有三分之一，三分之二，四分之三。问减少多少和增加多少才能使它们相等？
回答是：从四分之三者中减去两份，从三分之二者中减去一份，并将这些减去的部分加到三分之一上，这样每个分数都等于十二分之七。

We have one third, two thirds, and three fourths. How much should be subtracted from the larger ones and added to the smaller one to make them equal? The answer is: Subtract two parts from the three fourths, subtract one part from the two thirds, and add these subtracted parts to the one third, so that each fraction becomes seven twelfths.

又有二分之一，三分之二，四分之三。问减少多少和增加多少才能使它们相等？
回答是：从三分之二者中减去一份，从四分之三者中减去四份，并将这些减去的部分加到二分之一上，这样每个分数都等于三十六分之二十三。 平分术（一种古代的数学方法）说：将不同分数的分子相乘，然后将所有分子相加作为新的分子（平实），将不同分数的分母也相乘作为新的分母（法）。用列数乘以没有相加的分子各自得到新的分子（列实）。同样，用列数乘以分母，用平实减去列实，剩下的数经过约分后就是需要减去的数。将这些减去的数加到分子最小的分数上，用分母来衡量平实，这样就能得到它们相等时的分数。

We have one half, two thirds, and three fourths. How much should be subtracted from the larger ones and added to the smaller one to make them equal? The answer is: Subtract one part from the two thirds, subtract four parts from the three fourths, and add these subtracted parts to the one half, so that each fraction becomes twenty-three thirty-sixths. The method for balancing fractions states: Multiply the numerators of different fractions, add all the numerators together to form the new numerator (ping shi), multiply the denominators to form the new denominator (fa). Multiply the number of sets by the numerators that have not been added to get the new numerators (lie shi). Also, multiply the number of sets by the denominators, subtract the lie shi from the ping shi, the remainder, after being reduced, is what should be subtracted. Add the subtracted amounts to the fraction with the smallest numerator, and use the denominator to measure the ping shi, thus obtaining the equalized fractions.

现在有七人，分八钱三分之一钱。问每人得到多少？
回答是：每人得到一钱、二十一分钱之四。

There are seven people who are to divide eight and one-third of a qian. How much does each person get? The answer is: Each person gets one qian and four out of twenty-one fen.

又有三人，其中三分之一是分钱者，要分六钱三分之一和四钱三分之一。问每人得到多少？
回答是：每人得到二钱、八分钱之一。 经分术（一种古代的数学方法）说：以人数作为分母，钱数作为分子，进行除法运算。如果有分数，则将其转换为通分后的形式，如果有多个分数，则统一分母后进行通分。

There are three people, one-third of whom are to share six qian and one-third of a qian, and three qian and one-third of a qian. How much does each person get? The answer is: Each person gets two qian and one-eighth of a qian. The method for sharing among fractions states: Use the number of people as the denominator and the amount of money as the numerator, and perform division. If there are fractions, convert them to their equivalent forms with a common denominator, and if there are multiple fractions, find a common denominator before conversion.

现在有田地的宽度是七分步之四，长度是五分步之三。问田地的面积是多少？
回答是：三十五分步之十二。

There is a field with a width of four-sevenths of a bu and a length of three-fifths of a bu. What is the area of the field? The answer is: twelve over thirty-five bu squared.

又有田地的宽度是九分之七步，长度是十一分之九步。问田地的面积是多少？
回答是：十一分之七步。

There is a field with a width of seven-ninths of a bu and a length of nine-elevens of a bu. What is the area of the field? The answer is: seven over eleven bu squared.

又有田地的宽度是五分之四步，长度是九分之五步。问田地的面积是多少？
回答是：九分之四步。 乘分术（一种古代的数学方法）说：分母相乘得到新的分母（法），分子相乘得到新的分子（实），用实除以法得到结果。

There is a field with a width of four-fifths of a bu and a length of five-ninths of a bu. What is the area of the field? The answer is: four over nine bu squared. The method for multiplying fractions states: Multiply the denominators to get the new denominator (fa), multiply the numerators to get the new numerator (shi), and divide shi by fa to get the result.

现在有田地的宽度是三步又三分之一步，长度是五步又五分之二步。问田地的面积是多少？
回答是：十八步。

There is a field with a width of three bu and one-third of a bu, and a length of five bu and two-fifths of a bu. What is the area of the field? The answer is: eighteen bu squared.

又有田地的宽度是七步又四分之三步，长度是十五步又九分之五步。问田地的面积是多少？
回答是：一百二十步又九分之五步。

There is a field with a width of seven bu and three-quarters of a bu, and a length of fifteen bu and five-ninths of a bu. What is the area of the field? The answer is: one hundred and twenty bu and five-ninths of a bu squared.

又有田地的宽度是十八步又七分之五步，长度是二十三步又十一分之六步。问田地的面积是多少？
回答是：一亩二百步又十一分之七步。 大广田术（一种古代的数学方法）说：将分母各自乘以整数部分，分子也相应相乘，得到的结果是实际的分子。分母相乘得到新的分母。用实际的分子除以新的分母得到结果。

There is a field with a width of eighteen bu and five-sevenths of a bu, and a length of twenty-three bu and six-elevenths of a bu. What is the area of the field? The answer is: one mu and two hundred bu, plus seven over eleven bu squared. The method for calculating the area of a large field states: Multiply each denominator by its whole number part, and multiply the numerators accordingly to get the actual numerator. Multiply the denominators to get the new denominator (fa). Divide the actual numerator by the new denominator to get the result.

现在有一块三角形田地（圭田），底边宽度是十二步，高是二十一步。问田地的面积是多少？
回答是：一百二十六步。

There is a triangular field (gui tian) with a base width of twelve bu and a height of twenty-one bu. What is the area of the field? The answer is: one hundred and twenty-six bu squared.

又有一块三角形田地（圭田），底边宽度是五步又二分之一步，高是八步又三分之二步。问田地的面积是多少？
回答是：二十三步又六分之五步。 术（一种古代的数学方法）说：将底边宽度除以二，然后乘以高。

There is a triangular field (gui tian) with a base width of five bu and one-half of a bu, and a height of eight bu and two-thirds of a bu. What is the area of the field? The answer is: twenty-three bu and five over six bu squared. The method says: Half the base width multiplied by the height.

现在有一块梯形田地（邪田），上底宽度是三十步，下底宽度是四十二步，高是六十四步。问田地的面积是多少？
回答是：九亩一百四十四步。

There is a trapezoidal field (xie tian) with a top width of thirty bu, a bottom width of forty-two bu, and a height of sixty-four bu. What is the area of the field? The answer is: nine mu and one hundred forty-four bu squared.

又有一块梯形田地（邪田），高是六十五步，一边长一百步，另一边长七十二步。问田地的面积是多少？
回答是：二十三亩七十步。 术（一种古代的数学方法）说：将两边的长度相加然后除以二，乘以高或者宽度。也可以将高或者宽度除以二，乘以两边长度之和，然后用亩的计算方法来计算。

There is a trapezoidal field (xie tian) with a height of sixty-five bu, one side length of one hundred bu, and the other side length of seventy-two bu. What is the area of the field? The answer is: twenty-three mu and seventy bu squared. The method says: Add the lengths of the two sides and half it, then multiply by the height or width. Alternatively, you can half the height or width and multiply by the sum of the two sides, then calculate according to the mu method.

现在有一块箕形田地（箕田），前宽（舌广）是二十步，后宽（踵广）是五步，高（正从）是三十步。问田地的面积是多少？
回答是：一亩一百三十五步。

There is a winnowing basket shaped field (ji tian) with a front width (she guang) of twenty bu, a back width (zong guang) of five bu, and a height (zheng cong) of thirty bu. What is the area of the field? The answer is: one mu and one hundred thirty-five bu squared.

又有一块箕形田地（箕田），前宽（舌广）是一百一十七步，后宽（踵广）是五十步，高（正从）是一百三十五步。问田地的面积是多少？
回答是：四十六亩二百三十二步半。 术（一种古代的数学方法）说：将前宽和后宽相加然后除以二，乘以高。然后用亩的计算方法来计算。

There is a winnowing basket shaped field (ji tian) with a front width (she guang) of one hundred seventeen bu, a back width (zong guang) of fifty bu, and a height (zheng cong) of one hundred thirty-five bu. What is the area of the field? The answer is: forty-six mu and two hundred thirty-two and a half bu squared. The method says: Add the front and back widths and half it, then multiply by the height. Then calculate according to the mu method.

现在有一块圆形田地（圆田），周长是三十步，直径是十步。问田地的面积是多少？
回答是：七十五步。

There is a circular field (yuan tian) with a circumference of thirty bu and a diameter of ten bu. What is the area of the field? The answer is: seventy-five bu squared.

又有一块圆形田地，周长是一百八十一步，直径是六十步又三分之一步。问这块田地的面积是多少？
回答是：十一亩九十步又十二分之一。 数学方法（术）说：半周长与半径相乘得到面积的步数。 另一个数学方法说：周长与直径相乘，然后除以四。 还有一个数学方法说：直径自乘，再乘以三，然后除以四。 最后一个数学方法说：周长自乘，然后除以十二。

There is a circular field with a circumference of one hundred and eighty-one steps, and a diameter of sixty steps plus one-third of a step. What is the area of the field? The answer is: eleven mu and ninety bu, plus one twelfth of a bu. The mathematical method (shu) says: Half the circumference multiplied by the radius gives the area in steps. Another mathematical method says: Multiply the circumference by the diameter, then divide by four. Another mathematical method says: Square the diameter, multiply by three, then divide by four. The last mathematical method says: Square the circumference, then divide by twelve.

现在有一块圆形的田地，其下周长为30步，直径为16步。问这块田地的面积是多少？
回答是：120步。

There is a circular field with a circumference of 30 steps and a diameter of 16 steps. What is the area of the field? The answer is: 120 square steps.

又有一块圆形的田地，其周长为99步，直径为51步。问这块田地的面积是多少？
回答是：五亩六十二步又四分之一步。 数学方法（术）说：用直径乘以周长，然后除以四。

There is another circular field with a circumference of ninety-nine steps and a diameter of fifty-one steps. What is the area of the field? The answer is: five mu and sixty-two bu, plus one-quarter of a bu. The mathematical method (shu) says: Multiply the diameter by the circumference, then divide by four.

现在有一块弧形的田地，其弦长为30步，矢长（即从弦到弧的垂直距离）为15步。问这块田地的面积是多少？
回答是：一亩九十七步半。

There is an arc-shaped field with a chord length of thirty steps and an arrow length (the perpendicular distance from the chord to the arc) of fifteen steps. What is the area of the field? The answer is: one mu and ninety-seven and a half bu.

又有一块弧形的田地，其弦长为78.5步（即78步又二分之一），矢长为13.77778步（即13步又九分之七）。问这块田地的面积是多少？
回答是：二亩一百五十五步又八十一分之五十六。 数学方法（术）说：用弦长乘以矢长，再加上矢长的平方，然后将结果除以二。

There is another arc-shaped field with a chord length of seventy-eight and a half steps, and an arrow length of thirteen and seventy-seven hundredths steps (which is to say, thirteen steps and seventy-seven seventy-eighths). What is the area of the field? The answer is: two mu, one hundred and fifty-five bu, plus fifty-six eighty-firsts. The method says: Multiply the chord by the arrow, add the square of the arrow, and then divide by two.

现在有一块环形的田地，内周长为92步，外周长为122步，田埂宽度为5步。问这块田地的面积是多少？
回答是：二亩五十五步。

There is a ring-shaped field with an inner circumference of ninety-two steps and an outer circumference of one hundred and twenty-two steps, and the width of the path around the field is five steps. What is the area of the field? The answer is: two mu and fifty-five bu.

又有一块环形的田地，内周长为62.75步（即62步又四分之三），外周长为113.5步（即113步又二分之一），田埂宽度为12.67步（即12步又三分之二）。问这块田地的面积是多少？
回答是：四亩一百五十六步又四分之一。 数学方法（术）说：将内外周长相加后除以二，再用得到的平均值乘以田埂宽度即为田地面积的步数。 密率术说：将内外周长的步数分别作为分子和分母写在上下两行。分母与分子相乘，得出的结果中提取整数部分，剩余的小数部分加到内周长上。同时，田埂宽度也转换为相同的分数形式，并乘以平均后的周长得出实际面积的分子。分母相乘得出法数，用实际面积的分子除以法数得到田地面积的步数及其余的小数部分。最后以亩法换算，即可得到亩数。

There is another ring-shaped field with an inner circumference of sixty-two and three-quarters steps, and an outer circumference of one hundred and thirteen and a half steps, and the width of the path around the field is twelve and two-thirds steps. What is the area of the field? The answer is: four mu and one hundred and fifty-six and a quarter bu. The method says: Add the inner and outer circumferences and divide by two, then multiply by the width of the path to get the area in steps. The detailed calculation method says: Write down the number of steps for the inner and outer circumferences with their respective denominators and numerators below each other. Multiply the denominators by the numerators, extract the whole number part from the result, and add the remaining fractional part to the inner circumference. The width of the path should also be converted into the same fractional form, and then multiplied by the averaged circumference to obtain the actual area's numerator. Multiply the denominators to get the divisor, divide this by the actual area's numerator to obtain the field's area in steps and its fractional remainder. Finally, convert using the mu system to get the number of mu.