Sunzi Suanjing Lower Scroll 33

Problem

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？ 答曰：九萬匝。 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。

Suppose the distance between Chang'an and Luoyang is 900 li. One rotation of a cart's wheel is 1 zhang 8 chi. If one wishes to get from Luoyang to Chang'an, how many rotation will the wheel make? Answer: 90,000 revolutions. Method: Set 900 li, multiply by 300 bu to obtain 270,000 bu. Then multiply by 6 chi to obtain 1,620,000 chi. Let [the circumference of ] the wheel, 1 zhang 8 chi, be the divisor. Divide to obtain the answer.

Note: 1 li = 300 bu, 1 bu = 6 chi, 1 zhang = 10 chi

Solution

This problem is an exercise of unit conversions. The first step is to convert the distance between Luoyang and Chang'an from the unit li to chi.

900 li converts to 270,000 bu.

$$900 \; li \times \frac{300 \; bu}{ 1 \; li} = 270,000 \; bu$$

270,000 bu converts to 1,620,000 chi.

$$270,000 \; bu \times \frac{6 \; chi}{1 \; bu} = 1,620,000 \; bu$$

Since one rotation of the wheel is 18 chi, divide the distance between Luoyang and Chang'an in chi by 18 chi.

$$\frac{1,620,000 \; chi}{18 \; chi} = 90,000$$

Thus, the wheel makes 90,000 rotations from Luoyang to Chang'an.