TCM Arithmetic 2: Arithmetic of Fractions

By: Tao Steven Zheng (郑涛)

Chinese Fractions

Figure 1. Fractions visualized

In the Chinese language, fractions are expressed in words using the characters 分之 (fen zhi, literally "parts of") between the denominator (分母 fen mu) and the numerator (分子 fen zi). This encapsulates the concept of dividing a whole number into a certain number of equal parts. For example, the fraction three-quarters is written 四分之三 (si fen zhi san, literally "four parts of three"). For mixed numbers, the integer part is written first, followed by the fractional part. For example, $3 \frac{1}{7}$ is written 三、七分之一 (san, qi fen zhi yi, literally "three, seven parts of one"). Aside from the standard expressions for fractions, there exists special names for four specific fractions that are commonly found in ancient Chinese mathematical texts (Table 1).

Table 1. Four specific fractions

弱半 ruo ban (weak half)	1/4
少半 shao ban (lesser half)	1/3
中半 zhong ban (middle half)	1/2
太半 tai ban (greater half)	2/3

Computation with Fractions

The mathematicians of ancient China were expected to be proficient with calculations involving fractions. In Chapter 1 of the Jiuzhang Suanshu, there are seven major arithmetic operations to master (Table 2). In the following problem studies, we will examine in detail the procedures and relevant terminologies for three operations: 约分术 (yue fen shu, method of reducing fractions), 合分术 (he fen shu, method of adding fractions), and 经分术 (jing fen shu, method of dividing fractions).

Table 2. Arithmetic operations on fractions

(1) 约分术 yue fen shu (method of reducing fractions)

(2) 合分术 he fen shu (method of adding fractions)

(3) 减分术 jian fen shu (method of subtracting fractions)

(4) 课分术 ke fen shu (method of comparing fractions)

(5) 平分术 ping fen shu (method of averaging fractions)

(6) 经分术 jing fen shu (method of dividing fractions)

(7) 乘分术 cheng fen shu (method of multiplying fractions)

Problem Study 1: Reducing Fractions

Problem 5 and Problem 6 from Chapter 1 of the Jiuzhang Suanshu illustrates the method for obtaining the fully reduced form of a given fraction. The solution method for reducing fractions to lowest terms given in the Jiuzhang Suanshu illustrates the general method taught in elementary schools (called the Euclidean algorithm in the West). Here, the term deng shu (等数，literally "equal number") is equivalent to the greatest common divisor (GCD).

[1.05] 今有十八分之十二。问：约之得几何？ 答曰：三分之二。 [1.06] 又有九十一分之四十九。问：约之得几何？ 答曰：十三分之七。                        约分术曰：可半者半之，不可半者，副置分母子之数，以少减多，更相减损，求其等也。以等数约之。

[1.05] Suppose there are 12/18. Question: What is the reduced form? Answer: 2/3. [1.06] Suppose there are 49/91. Question: What is the reduced form? Answer: 7/13. Method for reducing fractions: If both the numerator and denominator can be halved, halve them. If they cannot be halved, set the numbers of the denominator and numerator separately, subtract the smaller from the larger, and continue subtracting until equality, to obtain the deng shu (等数, literally "equal number"). Use the deng shu to reduce the fraction.

Solution for [1.05]

The numerator 12 and the denominator 18 are even numbers, so halve them.

${\frac {12\div 2}{19\div 2}}={\frac {6}{9}}$

Now the numerator 6 and the denominator 18 are odd numbers, so they cannot be halved. Use the Euclidean algorithm to find the deng shu (equal number).

$\begin{align} 9 - 6 &= 3 \\ 6 - 3 &= 3 \\ 3 &= 3 \end{align}$

Thus, the deng shu is 3. Now divide the numerator and denominator by 3 to obtain the fully reduced fraction 2/3.

$\frac{6 \div 3}{9 \div 3} = \frac{2}{3}$

Solution for [1.06]

The numerator 49 and the denominator 91 are odd numbers, so they cannot be halved. Use the Euclidean algorithm to find the deng shu (equal number).

$\begin{align} 91 - 49 &= 42 \\ 49 - 42 &= 7 \\ 42 - 7 &= 35 \\ 35 - 7 &= 28 \\ 28 - 7 &= 21 \\ 21 - 7 &= 14 \\ 14 - 7 &= 7 \\ 7 &= 7 \end{align}$

Thus, the deng shu is 7. Now divide the numerator and denominator by 7 to obtain the fully reduced fraction 7/13.

$\frac{49 \div 7}{91 \div 7} = \frac{7}{13}$

Problem Study 2: Addition of Fractions

Problem 7 to Problem 9 from Chapter 1 of the Jiuzhang Suanshu is concerned with the addition of two or more fractions.

[1.07] 今有三分之一，五分之二。问：合之得几何？ 答曰：十五分之十一。 [1.08] 又有三分之二，七分之四，九分之五。问：合之得几何？ 答曰：得一、六十三分之五十。 [1.09] 又有二分之一，三分之二，四分之三，五分之四。问：合之得几何？ 答曰：得二、六十分之四十三。                        合分术曰：母互乘子，并以为实。母相乘为法。实如法而一。不满法者，以法命之。其母同者，直相从之。

[1.07] Suppose there are $\frac{1}{3}$ and $\frac{2}{5}$ . Question: What is the sum? Answer: $\frac{11}{15}$. [1.08] Again, suppose there are $\frac{2}{3}$, $\frac{4}{7}$, $\frac{5}{9}$. Question: What is the sum? Answer: The result is $1\frac{50}{63}$. [1.09] Again, suppose there are $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$,$\frac{4}{5}$. Question: What is the sum? Answer: The result is $2\frac{43}{60}$ . Method for adding fractions: Cross-multiply (互乘 hu cheng) the numerators and denominators, and combine them as the dividend (实 shi). Mutually multiply all of the denominators together as the divisor. Divide the dividend by the divisor; if it does not divide completely, then use the divisor to name (命 ming) the fraction [the results]. If their denominators are the same, then directly add them together.

The method prescribed in the Jiuzhang Suanshu describes a general algorithm for computing the sums. Given an arbitrary number of fractions

$\{{\frac {a_{1}}{b_{1}}},{\frac {a_{2}}{b_{2}}},{\frac {a_{3}}{b_{3}}},...,{\frac {a_{n}}{b_{n}}}\}$

arrange the numerators and denominators as follows:

Numerators (子 zi)	$a_1$	$a_2$	$a_3$	...	$a_n$
Denominators (母 mu)	$b_1$	$b_2$	$b_3$	...	$b_n$

Calculate the numerator by a cross-multiplication procedure called hu cheng (互乘). This process is calculated as follows:

$A_1$	$a_1 \times b_2 \times b_3 \times ... \times b_n$
$A_2$	$a_2 \times b_1 \times b_3 \times ... \times b_n$
$A_3$	$a_3 \times b_1 \times b_2 \times ... \times b_n$
$\vdots$	$\vdots$
$A_n$	$a_n \times b_1 \times b_2 \times ... \times {b}_{n-1}$

Then add all the hu cheng (互乘) products together to obtain the dividend $A$.

$A = A_1 + A_2 + A_3 + ... + A_n$

Calculate the divisor $B$ by multiplying all the given denominators:

$B = b_1 \times b_2 \times b_3 \times ... \times b_n$

Finally, divide the dividend (实 shi) by the divisor (法 fa):

$\frac{A}{B}$

Reduce the fraction if needed.

$\frac{A}{B} \rightarrow \frac{a}{b}$

If $a > b$, express the fraction as a mixed number $\frac{a}{b} = q + \frac{r}{b}$, where $a=bq+r$ and $0 < r < b$.

Solution for [1.07]

Numerators (子 zi)	$1$	$2$
Denominators (母 mu)	$3$	$5$

Dividend (实 shi):

$\begin{align} A &= 1 \times 5 + 2 \times 3 \\ &= 5 + 6 \\ &= 11 \end{align}$

Divisor (法 fa):

$B = 3 \times 5$

Divide the dividend (实 shi) by the divisor (法 fa):

$\frac{A}{B} = \frac{11}{15}$

Solution for [1.08]

Numerators (子 zi)	$2$	$4$	$5$
Denominators (母 mu)	$3$	$7$	$9$

Dividend (实 shi):

$\begin{align} A &= 2 \times 7 \times 9 + 4 \times 3 \times 9 + 5 \times 3 \times 7 \\ &= 126 + 108 + 105 \\ &= 339 \end{align}$

Divisor (法 fa):

$B = 3 \times 7 \times 9 = 189$

Divide the dividend (实 shi) by the divisor (法 fa):

$\frac{A}{B} = \frac{339}{189} = \frac{113}{63} = 1\frac{50}{63}$

Solution for [1.09]

Numerators (子 zi)	$1$	$2$	$3$	$4$
Denominators (母 mu)	$2$	$3$	$4$	$5$

Dividend (实 shi):

$\begin{align} A &= 1 \times 3 \times 4 \times 5 + 2 \times 2 \times 4 \times 5 + 3 \times 2 \times 3 \times 5 + 4 \times 2 \times 3 \times 4 \\ &= 60 + 80 + 90 + 96 \\ &= 326 \end{align}$

Divisor (法 fa):

$B = 2 \times 3 \times 4 \times 5 = 120$

Divide the dividend (实 shi) by the divisor (法 fa):

$\frac{A}{B} = \frac{326}{120} = \frac{163}{60} = 2\frac{43}{60}$

Problem Study 3: Division of Fractions

Problem 17 and Problem 18 from Chapter 1 of the Jiuzhang Suanshu focuses on the division of fractions. Each problem is about splitting a given quantity of money in units of qian (钱, literally "cash" or "coin") among a given number of people. In Problem 18, the money is divided among $3\frac{1}{3}$ people; hence, the amount of money and number of people should be treated as abstract quantities.

[1.17] 今有七人，分八钱三分钱之一。问：人得几何？ 答曰：人得一钱、二十一分钱之四。 [1.18] 又有三人，三分人之一，分六钱三分钱之一，四分钱之三。问：人得几何？ 答曰：人得二钱、八分钱之一。 经分术曰：以人数为法，钱数为实，实如法而一。有分者通之，重有分者同而通之。

[1.17] Suppose there are 7 people dividing $8\frac{1}{3}$ qian. Question: How much should each person receive? Answer: Each person receives $1 \frac{4}{21}$ qian. [1.18] Again, suppose there are $3\frac{1}{3}$ people dividing $6\frac{1}{3}$ qian and  $\frac{3}{4}$ qian. Question: How much should each person receive? Answer: Each person receives $2 \frac{1}{8}$ qian. Method for dividing fractions: Use the number of people as the divisor, and the amount of qian as the dividend; divide the dividend by the divisor. If there are mixed numbers, interconnect (通 tong) them. If both of the rates are mixed numbers, then interconnect and equalize them.

In ancient China, it is known that dividing two fracions $\frac{a}{b}$ and $\frac{c}{d}$ is equivalent to taking the first fraction and multiplying it with the reciprocal of the second fraction.

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

The term tong (通, interconnect) refers to the process of converting a mixed number into an improper fraction. The procedure is s follows:

$a \frac{b}{c} \rightarrow \frac{ac+b}{c}$

Solution for [1.17]

$\begin{align} 8 \frac{1}{3} \div 7 &= \frac{25}{3} \times \frac{1}{7} \\ &= \frac{25}{21} \\ &= 1 \frac{4}{21} \end{align}$

Solution for [1.18]

$\begin{align} \left(6 \frac{1}{3} + \frac{3}{4} \right) \div 3 \frac{1}{3} &= 7 \frac{1}{12} \div 3 \frac{1}{3} \\ &= \frac{85}{12} \div \frac{10}{3} \\ &= \frac{85}{12} \times \frac{3}{10} \\ &= \frac{17}{4} \times \frac{1}{2} \\ &= \frac{17}{8} \\ &= 2 \frac{1}{8} \end{align}$