Problem 7 - A History of Mathematics: An Introduction

Question

Solve problem 28 of chapter 6 of the Nine Chapters: A man is carrying rice on a journey. He passes through three customs stations. At the first, he gives up \(1 / 3\) of his rice, at the second \(1 / 5\) of what was left, and at the third, \(1 / 7\) of what remains. After passing through all three customs stations, he has left 5 pounds of rice. How much did he have when he started? (Versions of this problem occur in later sources in various civilizations.)

Step-by-Step Solution

Answer

Answer: The man had approximately 11 pounds of rice initially.

Step 1: Find the fraction of rice left after passing through all customs stations

At the first customs station, he gives up 1/3 of his rice. So, he has (1 - 1/3) = 2/3 left. At the second customs station, he gives up 1/5 of his remaining rice. So, he has (1 - 1/5) = 4/5 of what was left before. At the third customs station, he gives up 1/7 of his remaining rice. So, he has (1 - 1/7) = 6/7 of what was left before. Now, let's multiply the fractions to find the total fraction of rice left after passing through all three customs stations: (2/3) * (4/5) * (6/7) = (2*4*6) / (3*5*7) = 48 / 105.

Step 2: Relate the fraction of rice remaining with the initial amount

Let's denote the initial amount of rice the man had as 'x'. Then, after passing through all three customs stations, he's left with (48/105) * x. And according to the exercise, this amount equals 5 pounds. So, we have: (48/105) * x = 5

Step 3: Solve for the initial amount of rice (x)

Now we just need to solve this equation for 'x'. To do this, we can simply divide both sides of the equation by (48/105): x = 5 / (48/105) x = 5 * (105/48)

Step 4: Calculate the initial amount of rice

Now let's perform the calculation to find the initial amount of rice: x = 5 * (105/48) = 525/48 To get the result as a whole number, we can find the nearest whole number: x ≈ 11 (rounded up to the nearest whole number) So, the man had approximately 11 pounds of rice when he started his journey.

Key Concepts

Mathematical Problem Solving

When it comes to mathematical problem solving, understanding the context and systematically breaking down a problem is key. The rice problem from the Nine Chapters is a perfect example of how to approach problem solving step-by-step.
Here are some general steps to tackle similar problems:

• **Understand the Problem:** Start by comprehending what the problem is asking. Identify all key information and what needs to be found.
• **Create a Plan:** Decide on the mathematical operations and processes that will help solve the problem. Consider using algebraic expressions to represent unknowns.
• **Execute the Plan:** Write the mathematical equations and calculate step by step, ensuring each step logically follows the previous one.
• **Review and Verify:** Once you have an answer, review each step. Check if the solution matches the conditions given in the problem.

This structured approach ensures clarity and reduces chances of errors, making it easier to arrive at the correct solution.

Fractions in Mathematics

Fractions are a fundamental concept in mathematics, essential for representing parts of a whole. In the rice problem, fractions simplify the complex task of determining how much rice remains after each deduction.
Here’s a quick guide to understanding and working with fractions in math:

• **Definition:** A fraction consists of a numerator (top number) and a denominator (bottom number) and represents a division of the numerator by the denominator.
• **Operations:**
  • Addition/Subtraction: To add or subtract fractions with different denominators, find a common denominator and adjust the numerators accordingly.
  • Multiplication: Multiply the numerators together and denominators together.
  • Division: Flip the second fraction and multiply (multiply by the reciprocal).
• **Simplifying Fractions:** Reduce fractions by dividing both the numerator and denominator by their greatest common divisor (GCD).

This understanding of fractions not only aids in solving specific problems but also builds a solid foundation for tackling more complex mathematical concepts.

Ancient Chinese Mathematics

Ancient Chinese mathematics is renowned for its sophistication and practical application. The Nine Chapters on the Mathematical Art, one of its most famous works, contains a variety of problem types that explore arithmetic, algebra, geometry, and more.
Within this ancient text, problems involving proportions, like the rice problem, reveal the Chinese mathematicians’ adept use of arithmetic principles. They pursued practical solutions for daily problems, with:

• **Real-Life Applications:** Many problems reflect the commercial and agricultural practices of the time. This made their math directly applicable to everyday needs.
• **Innovative Methods:** The text demonstrated early concepts of using fractions and algebraic thinking to reach solutions that were advanced for the era.
• **Influence on Future Mathematics:** The work influenced mathematical studies not only in Asia but also in other parts of the world as mathematical ideas spread.

Studying these methods highlights the rich history and evolution of mathematics, showing us how ancient problem-solving techniques can still be relevant today.