Question

Solve problem 26 of chapter 6 of the Nine Chapters: There is a reservoir with five channels bringing in water. If only the first channel is open, the reservoir can be filled in \(1 / 3\) of a day. The second channel by itself will fill the reservoirin 1 day, the third channel in \(21 / 2\) days, the fourth one in 3 days, and the fifth one in 5 days. If all the channels are open together, how long will it take to fill the reservoir? (This problem is the earliest known one of this type. Similar problems appear in later Greek, Indian, and Western mathematics texts.)\\}

Short answer

Answer: The time it takes to fill the reservoir with all channels open is 105/556 days.

Step-by-step solution

Calculate the rate of each channel

Let's first define the filling rate for each channel. For this, we will use the given information and simply find the reciprocal of each individual time. For channel 1: The rate is \(\frac{1}{1/3}\), which gives us \(3\). For channel 2: The rate is \(\frac{1}{1}\), which gives us \(1\). For channel 3: The rate is \(\frac{1}{21/2}\), which gives us \(\frac{2}{21}\). For channel 4: The rate is \(\frac{1}{3}\), which gives us \(1\). For channel 5: The rate is \(\frac{1}{5}\), which gives us \(1/5\).

Find the combined rate for all channels

Now that we have the filling rate for each channel, we can find the combined filling rate by adding up the filling rates for all channels: Combined_rate = Rate_channel1 + Rate_channel2 + Rate_channel3 + Rate_channel4 + Rate_channel5 Combined_rate = \(3 + 1 + \frac{2}{21} + 1 + \frac{1}{5}\). First, let's find a common denominator in order to simplify the expression. The common denominator for 21 and 5 is 105. So, we can rewrite the expression as: Combined_rate = \(3 + 1 + \frac{10}{105} + 1 + \frac{21}{105}\). Now, we can add all the terms together: Combined_rate = \(5 + \frac{10 + 21}{105} = 5 + \frac{31}{105}\).

Calculate the time required for all channels together

We have the combined filling rate, and we need to find the time required to fill the reservoir with this rate. To do this, we will use the formula: Time = \(\frac{1}{\textrm{Combined_rate}}\) Time = \(\frac{1}{5 + \frac{31}{105}}\) Now, let's find a common denominator for the expression in the denominator. A common denominator for 5 and 105 is 105. So, we can rewrite the expression as: Time = \(\frac{1}{\frac{525+31}{105}}\) Now, we can find the reciprocal of the denominator to get the time required: Time = \(\frac{105}{556}\) So, the time it takes to fill the reservoir with all channels open is \(\frac{105}{556}\) days.

Key concepts

Nine Chapters on the Mathematical Art

The "Nine Chapters on the Mathematical Art" is a vital ancient Chinese mathematical text. Written between 100 BCE and 50 CE, it encompasses a wide array of practical problems. These problems offer insights into the daily life and economy of early China. The text aims to provide solutions to practical issues such as trade, taxation, and measurement.
It serves as an early example of how mathematics was applied to real-world problems, demonstrating concepts like fractions, ratios, and proportions. The book's structured approach to solving problems embeds it as a fundamental document in the annals of mathematics history.
Understanding these chapters provides insight into ancient Chinese society's mathematical prowess and how they influenced later civilizations. The problems discussed and solved are grounded in actual application, such as the water reservoir problem, thereby making mathematical theory accessible and practical.

Water Reservoir Problems

Water reservoir problems deal with understanding how different sources contribute to filling or emptying a reservoir. These types of problems are classical in nature and were among the earliest mathematical exercises documented in texts like the "Nine Chapters on the Mathematical Art."
These problems typically involve calculating rates where multiple flows interact, similar to a modern plumbing system. Each channel or flow in a reservoir problem has its filling capacity, and when combined, these rates can have a cumulative effect. The goal often is to determine the time it takes to fill or empty a reservoir given these combined rates.
To solve such problems, the approach involves determining the rate of each contributing channel by taking the reciprocal of the time to fill by itself. The next step is to find a combined rate by adding all individual rates. Finally, the problem is solved by determining the time taken using the situation's combined rate. Such exercises help students understand distribution, accumulation, and modern-day implications like water management.

Mathematical Problem Solving

Mathematical problem solving is a skill that involves identifying solutions to complex problems using mathematical techniques. It requires an understanding of mathematical operations, logical reasoning, and critical thinking.
To approach these problems systematically, one must:

• Identify the problem clearly.
• Understand and gather relevant data.
• Formulate a strategy based on mathematical principles.
• Execute computations precisely and diligently.
• Review and analyze solutions for accuracy.

This approach is reflected in solving the water reservoir problem, where one sets the stage by determining each flow rate, combines them, computes accurately, and interprets the results.
Problem-solving in mathematics not only enhances computational skills but also develops analytical thinking, enabling versatility in tackling diverse real-life challenges.

Historical Texts in Mathematics

Historical mathematical texts offer a glimpse into the development and evolution of mathematics across cultures and eras. They reflect the cultural and intellectual contexts of their times and articulate the progression of mathematical ideas. Such texts, like the "Nine Chapters on the Mathematical Art," constitute primary sources that convey how past civilizations approached mathematical problems.
They document the influence of mathematics on society, illustrating the practical application of mathematical concepts in areas like agriculture, construction, and astronomy. Holding historical significance, they bridge the ancient and modern worlds, outlining the origins and evolutions of mathematical thoughts and processes.
The inclusion of the water reservoir problem demonstrates how mathematical problem-solving has always been tied to real-world applications, allowing us to better understand the practical needs of historical societies and how they applied logic and reasoning to meet them.