Question

Solve problem 1 of chapter 7 of the Nine Chapters using the method of surplus and deficiency: Several people purchasein common one item. If each person paid 8 coins, the surplus is 3 ; if each paid 7, the deficiency is 4 . How many people were there and what is the price of the item?

Short answer

Answer: 7 people purchase the item together, and the price of the item is 53 coins.

Step-by-step solution

Solve for y in the first equation

Start by solving the first equation, 8x = y + 3, for y: y = 8x - 3

Substitute the expression for y in the second equation

Substitute the expression for y from step 1 into the second equation, 7x = y - 4: 7x = (8x - 3) - 4 7x = 8x - 7

Solve for x

Now solve the equation from step 2 to find the value of x: 7x = 8x - 7 7 - 7x = 8x - 8x x = 7

Use the value of x to find y

With the value of x, substitute it back into the expression for y from step 1: y = 8(7) - 3 y = 56 - 3 y = 53

Interpret the results

Since x represents the number of people and y represents the price of the item, the result shows that there were 7 people who purchased the item for a total price of 53 coins.

Key concepts

Nine Chapters on the Mathematical Art

The ancient Chinese text 'Nine Chapters on the Mathematical Art' is a cornerstone of mathematical history. Written during the Han Dynasty, it encompasses a variety of mathematical concepts and problem-solving techniques. One of the significant contributions of this comprehensive work is its empirical approach to solving everyday problems ranging from surveying land to understanding proportions in taxation.

This classic piece is divided into nine sections, each dealing with different areas of mathematics such as geometry, algebra, and even contains problems similar to the one outlined in our exercise. The text marks a valuable documentation of the mathematical knowledge during that era, and it is not surprising that its influence has seeped into the current educational curriculum, providing context and historical precedent to the mathematical exercises students engage in today.

Method of Surplus and Deficiency

The method of surplus and deficiency is a mathematical approach that can be considered a precursor to algebraic reasoning. It is based on observing the impact of changing one variable on the total outcome and was traditionally used in ancient economics and commerce.

For example, the problem presented from the 'Nine Chapters' relies on this method. It requires manipulating the given conditions—how surplus or deficiency in coins relates to the number of people—to arrive at a set of equations that can be solved systematically. The elegance of this method lies in its simplicity and its power in dealing with problems that involve balancing two different scenarios to find the unknowns.

Algebraic Reasoning

Algebraic reasoning is the process of manipulating symbols and letters to represent numbers and arithmetic operations, with the goal of solving for unknown values. It involves understanding and applying the rules of algebra to construct and solve equations.

In our exercise, algebraic reasoning is demonstrated through the development and solution of mathematical equations representing the payment scenarios with surplus and deficiency. By algebraically manipulating these equations, we isolate the unknown variable corresponding to the number of people, exemplifying a direct application of algebraic reasoning in problem-solving.

Mathematical Equations

Mathematical equations are the bread and butter of problem-solving in algebra. They form a language that expresses mathematical ideas succinctly and allows for the systematic exploration of relationships between quantities. Equations consist of two expressions separated by an equals sign indicating that both expressions are of the same value.

In the context of the solved problem, the equations (8x - 3) for surplus and (7x + 4) for deficiency succinctly capture the situation. The solution process involves equalizing both expressions, manipulating them, and eventually solving for x and y, the unknowns representing the number of people and the price, respectively. Such equations are not just mathematical statements but powerful tools that enable us to decode and solve real-world scenarios.