Question

There is a certain army composed of dukes, earls, and soldiers. Each duke has under him twice as many earls asthere are dukes. Each earl has under him four times as many soldiers as there are dukes. The 200th part of the number of soldiers is 9 times as many as the number of dukes. How many of each are there? (This problem and the next two are from Recorde's The Whetstone of Witte.)

Short answer

Answer: There are 50 dukes, 100 earls, and 200 soldiers in the army.

Step-by-step solution

Identify the variables

Let x be the number of dukes, y be the number of earls, and z be the number of soldiers. Step 2: Set up the equations based on the relationships given

Set up the equations

We are given that each duke has under him twice as many earls as there are dukes. Therefore, 2x = y. Also, each earl has under him four times as many soldiers as there are dukes. Therefore, 4x = z. Finally, the 200th part of the number of soldiers is 9 times as many as the number of dukes, or z/200 = 9x. Step 3: Solve the system of equations

Solve the system of equations

Since we have three equations with three variables, we can solve the system: Equation 1: y = 2x Equation 2: z = 4x Equation 3: z/200 = 9x Now substitute Equation 2 in Equation 3 and solve for x: (4x)/200 = 9x x = 50 Now that we have the value of x, we can find the values of y and z by substituting the value of x in Equation 1 and Equation 2: Equation 1: y = 2(50) y = 100 Equation 2: z = 4(50) z = 200 Step 4: Report the answer

Report the answer

The army has 50 dukes, 100 earls, and 200 soldiers.

Key concepts

Algebraic Equations

An algebraic equation is a mathematical statement that presents a relationship between variables and constants. These equations typically involve operations such as addition, subtraction, multiplication, and division, and they form the foundation of algebra. In the context of the original problem, we have to identify and set up equations that represent the hierarchical structure of an army composed of dukes, earls, and soldiers.

To create these equations, information provided in the problem is translated into algebraic expressions. For instance, the statement 'Each duke has under him twice as many earls as there are dukes' can be written in algebraic form as \(y = 2x\), where \(x\) is the number of dukes and \(y\) is the number of earls. Similar translation occurs for other relationships.

In order to solve the problem, you must understand how to translate English statements into algebraic expressions, how to manipulate these expressions, and finally, how to solve the equations for the variables. By mastering algebraic equations, students can solve a wide variety of practical problems, such as the one described.

System of Equations

A system of equations refers to a collection of two or more equations with a common set of variables. In the given problem, we have a system of three equations. These equations must all be true at the same time, and the solution involves finding the values of the variables that satisfy every equation in the system.

To solve a system, there are several methods such as substitution, elimination, and graphical analysis. Here, substitution is used effectively by replacing \(y\) and \(z\) in terms of \(x\) using the initial relationships of the problem. Once a single variable is isolated, as \(x\) is found to be 50 in this situation, others can be easily calculated.

Systems of equations are a critical part of algebra and are used in various fields including science, engineering, and economics to describe complex relationships between different factors. Understanding how to set up and solve these systems is crucial for problem-solving in many areas.

Mathematical Reasoning

Mathematical reasoning involves the process of logically deducing new information from given premises and known facts. It enables students to make informed judgments and solve problems based on mathematical principles. This type of reasoning is crucial when translating a word problem into a mathematical model.

For the army problem, mathematical reasoning is applied to comprehend the hierarchy structure and to conclude the proportional relationships between dukes, earls, and soldiers. It requires attention to detail and the ability to connect various parts of the problem to set up correct equations.

Developing mathematical reasoning skills involves practice and exposure to different types of problems. It allows learners to recognize patterns, understand the logic behind the mathematical operations, and build problem-solving strategies. As such, when solving any mathematical exercise, it's essential to thoughtfully analyze information, reason out the connections between different elements, and verify the results by checking if they satisfy the given conditions.