Problem 5 - A History of Mathematics: An Introduction

Question

Solve problem 3 of chapter 3 of the Nine Chapters: Three people, who have 560,350 , and 180 coins, respectively, are required to pay a total tax of 100 coins in proportion to their wealth. How much does each pay?

Step-by-Step Solution

Answer

Answer: Person 1 pays approximately 51 coins, Person 2 pays approximately 32 coins, and Person 3 pays approximately 17 coins.

Step 1: Calculating the total wealth

First, we need to calculate the total wealth of the three people, which can be done by adding up their wealth. That is: Total wealth = 560 + 350 + 180.

Step 2: Finding the ratio of each person's wealth to the total wealth

Now that we have the total wealth, let's find the ratio of each person's wealth to the total wealth: Person 1's ratio = 560/Total wealth Person 2's ratio = 350/Total wealth Person 3's ratio = 180/Total wealth

Step 3: Calculating the tax each person should pay according to the ratio

Now that we have the ratio for each person, we can multiply these ratios by the total tax (which is 100 coins) to find the tax each person should pay: Person 1's tax = Person 1's ratio * 100 Person 2's tax = Person 2's ratio * 100 Person 3's tax = Person 3's ratio * 100 First, we need to find the total wealth: Total wealth = 560 + 350 + 180 = 1090 Now let's find the ratio of each person's wealth to the total wealth: Person 1's ratio = 560/1090 ≈ 0.51376 Person 2's ratio = 350/1090 ≈ 0.32110 Person 3's ratio = 180/1090 ≈ 0.16514 We will now multiply these ratios by the total tax of 100 coins to find each person's tax amount: Person 1's tax = 0.51376 * 100 ≈ 51.376 Person 2's tax = 0.32110 * 100 ≈ 32.110 Person 3's tax = 0.16514 * 100 ≈ 16.514 Since we cannot have a fraction of a coin, we can round these amounts to the nearest whole coin, giving: Person 1 pays approximately 51 coins Person 2 pays approximately 32 coins Person 3 pays approximately 17 coins

Key Concepts

Mathematics Problem-Solving

Solving mathematics problems often involves breaking down a complex situation into simple steps. This problem-solving methodical approach helps learners tackle a variety of math challenges effectively. Let's take the example of proportional distribution in the context of a taxation story from the Nine Chapters on the Mathematical Art.
To solve the exercise, first, compute the total wealth by summing each person's wealth, which gives you an understanding of the whole situation. Then, determine the individual proportion of wealth by dividing each person’s wealth by the total. Finally, calculate the tax contribution of each person based on these ratios. These mechanics exemplify basic operations like addition, division, and multiplication but arranged purposefully.
Mathematics problem-solving often proceeds according to logical steps:

• Identify given information and what the problem requires.
• Decompose the task into smaller, manageable steps.
• Apply arithmetic or algebraic methods to compute results.
• Interpret results in the context of the problem.

By following this sequence, students can systematically navigate through most math exercises.

Nine Chapters on the Mathematical Art

The "Nine Chapters on the Mathematical Art" is a Chinese mathematics classic, which is similar to the more familiar Greek work "Euclid's Elements." As a comprehensive ancient text, it includes a variety of problem types that are relevant even to modern contexts. Chapter 3, in particular, deals with concepts of proportional distribution, much like the exercise concerning three individuals paying taxes based on their wealth.
This ancient work covers diverse topics:

• Arithmetic operations
• Solving equations
• Proportions and ratios

By illustrating practical applications of mathematics, it stands as a foundational text for both historical and contemporary scholars. Its influence spans cultures and centuries, demonstrating consistent themes in mathematical education.
The Nine Chapters remains central to understanding the evolution of mathematical theory and its application, showcasing how ancient problems still resonate with modern math problems.

Application of Ratios

The application of ratios is fundamental for distributing quantities fairly based on given criteria. A ratio is a way to express the relative size of two or more values, and in this exercise, it helps determine how much tax each person should contribute based on wealth.
To apply ratios effectively:

• Start by finding the total or sum of all components.
• Calculate each individual's share or fraction of this total.
• Use these fractions to allocate the total quantity to be distributed (in this case, tax).

For instance, if a total tax is 100 coins and person one's wealth forms about half of the total wealth, they should pay about half of the tax. Thus, ratios allow for a fair allocation based on relative wealth.
This method is not only applicable in taxation but also in numerous real-world scenarios such as dividing profits, resources, and even time. Understanding and applying ratios can significantly increase one’s ability to analyze and solve problems logically and equitably.