Question

Solve by the method of false position: A quantity and its \(2 / 3\) are added together and from the sum \(1 / 3\) of the sum is subtracted, and 10 remains. What is the quantity? (problem 28 of the Rhind Mathematical Papyrus)

Short answer

Answer: The value of the unknown quantity is 9.

Step-by-step solution

Write down the given information in an equation

According to the problem statement, we have the following equation: \(Quantity + \frac{2}{3}Quantity - \frac{1}{3}(Quantity + \frac{2}{3}Quantity) = 10\)

Simplify the equation

Now, let's simplify the equation by combining the like terms and finding a common denominator for the fractions: \(Quantity + \frac{2}{3}Quantity - \frac{1}{3}( \frac{5}{3}Quantity) = 10\) \(Quantity + \frac{2}{3}Quantity - \frac{5}{9}Quantity = 10\) Now, we combine the fractions with a common denominator of 9: \(\frac{9}{9}Quantity + \frac{6}{9}Quantity - \frac{5}{9}Quantity = 10\) \(\frac{10}{9}Quantity = 10\)

Solve for the unknown quantity using the method of false position

In the method of false position, we make an initial guess for the unknown quantity and then refine it iteratively to find its correct value. For this problem, we can make an initial guess that the quantity is 9. Step 3.1: Make the initial guess and calculate the error: Guess: Quantity = 9 \(\frac{10}{9}(9) = 10\) \(10 = 10\) Error = \(Actual\ value - Calculated\ value = 10 - 10 = 0\) Since the error is 0, the initial guess was correct, and the value of the unknown quantity is 9.

Write down the solution

The value of the unknown quantity by the method of false position is 9.

Key concepts

Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus is an ancient Egyptian document dating to around 1650 B.C., serving as a textbook of sorts in arithmetic and geometry. Discovered by Alexander Henry Rhind, it is one of the best examples we have of Egyptian mathematical practice. The papyrus contains a collection of problems and solutions that provide insight into how the ancient Egyptians approached mathematical calculations, especially in the context of daily life and practical tasks such as measuring fields and distributing food.

Among the problems is the one our exercise addresses, problem 28, which showcases a method similar to false position—a technique for solving algebraic equations. It demonstrates the level of sophistication in ancient Egyptian mathematics and their ability to solve what we would consider today to be algebraic problems using the concepts they had developed.

Ancient Mathematics Problems

Ancient mathematics problems, such as those found in the Rhind Mathematical Papyrus, are fascinating for their ingenuity and the wisdom they display in mathematical understanding, despite the absence of modern notation and concepts. These problems weren't just theoretical exercises; they were highly applicable to real-world scenarios like distribution of resources, construction, and surveying.

The basic principles employed in these ancient problems are remarkably similar to the algebraic problem-solving methods we use today. They often used intuitive techniques that resembled linear equations and proportionality, and they already had a grasp of fractions and basic operations. Exploring problems like problem 28 not only connects us to the past but also shows the timeless nature of mathematical discovery.

Algebraic Problem Solving

Algebraic problem solving involves finding the unknowns in equations using systematic techniques. The method of false position, used in the exercise above, is an early algebraic method that involves making a 'false' assumption—hence the name—to start the problem-solving process, and then correcting the assumption iteratively until the right solution is found.

This method might feel unconventional compared to modern algebraic techniques, but it highlights the historical development of mathematical concepts. In our case, the problem revealed that the false assumption was, in fact, correct, which is less common and indicates the simplicity of the particular equation. Most often, further iterations would be required to home in on the correct answer, adjusting the false position by considering the error magnitude and direction signaled by its negativity or positivity.