Jiuzhang Suanshu Junshu 26

Problem

This classic problem comes from Chapter 6 of the Jiuzhang Suanshu (九章算術).

今有池，五渠注之。其一渠開之，少半日一滿；次，一日一滿；次，二日半一滿；次，三日一滿；次，五日一滿。今皆決之，問：幾何日滿池？

Suppose there there is a cistern, and 5 canals fill into it. Open the first canal and the cistern fills in ¹/₃ day; with the second, it fills in 1 day; with the third, in 2 ¹/₂ days; with the fourth, in 3 days, and with the fifth in 5 days. Now, suppose all the canals are opened. Question: How how many days will cistern be full?

Solution

Let $T$ be the time (in days) it takes to fill the cistern if all canals are open.

Let $x$ be the volume of water.

$$\frac{x}{1/3} + \frac{x}{1} + \frac{x}{5/2} + \frac{x}{3} + \frac{x}{5} = \frac{x}{T}$$

$$x\left(\frac{1}{1/3} + \frac{1}{1} + \frac{1}{5/2} + \frac{1}{3} + \frac{1}{5}\right) = \frac{x}{T}$$

Divide $x$ on both sides of the equation.

$$3 + 1 + \frac{2}{5} + \frac{1}{3} + \frac{1}{5} = \frac{1}{T}$$

$$\frac{74}{15} = \frac{1}{T}$$

$$T = \frac{15}{74}$$

If all canals are opened, the cistern will be filled in $\frac{15}{74}$ days.