Question

Solve by the method of false position: A quantity and its \(1 / 7\) added together become 19. What is the quantity? (problem 24 of the Rhind Mathematical Papyrus)

Short answer

Answer: The approximate quantity of x is 11.79.

Step-by-step solution

Set up the equation

First, we need to write the problem as an equation. Let x be the quantity. Then, the given statement can be translated into the equation: $$x + \frac{1}{7}x = 19$$

Choose a false position

Let's choose a random value as a false position, which we will later utilize to adjust our value of x accordingly. Let's choose 14 as our false position value for x, assuming that the proportion to the correct value remains constant.

Plug in the false position value

Now, plug in the false position value of 14 into the equation: $$14 + \frac{1}{7}(14) = 14 + 2 = 16$$

Find the proportion

We now find what proportion the result (16) is of the actual total (19) to adjust our assumed value (14) accordingly: $$\frac{16}{19} = \frac{x}{14}$$

Solve for x

Cross-multiply and solve the equation for x: $$16 \times 14 = 19 \times x \Rightarrow x = \frac{16 \times 14}{19}$$

Calculate x

Multiply and simplify the fraction to find the value of x: $$x = \frac{16 \times 14}{19} = \frac{224}{19} = 11.79$$ The approximate quantity x is 11.79.

Key concepts

Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus is one of the earliest known sources of Egyptian mathematics, dating to around 1550 BC. It's named after the Scottish Egyptologist Alexander Henry Rhind, who purchased the papyrus in 1858. This ancient document provides a fascinating glimpse into the mathematical understanding of the Egyptians, featuring arithmetic, geometry, and algebraic problems similar to the one presented in our exercise.

The papyrus contains around 84 different mathematical problems, providing us with insight into how Egyptians approached problem-solving, notably using the method of false position to solve linear equations. Studying problems from the Rhind Mathematical Papyrus, like the one about finding a quantity with its fraction that sums up to a specific total, shows us the historical roots of algebra and the methods that lead to the mathematical principles we use today.

Mathematical Problem-Solving

Mathematical problem-solving is a fundamental skill that allows us to find solutions to various types of mathematical questions. It involves understanding the problem, devising a plan, carrying out that plan, and then checking the results. Steps such as these are integral to tackling any mathematical challenge.

Methods like the one in our textbook exercise, the method of false position, are foundational techniques in problem-solving strategies. By using a trial-and-error approach, and then adjusting proportionally based on the false attempt, students learn to navigate towards the correct solution. This technique is not only historically significant but also immensely practical, teaching a logical approach to problem-solving that can be applied beyond mathematics.

Historical Mathematics

Historical mathematics reveals the evolution of mathematical thought throughout the ages and across different cultures. From Babylonian tablets to Greek geometry, and from the mathematical advancements in ancient India to the contributions of Islamic scholars during the middle ages, each culture added to the collective understanding of math.

Studying the historical context of mathematical development, such as the use of the method of false position from the Rhind Mathematical Papyrus, underlines the continuity and progression of mathematical thinking. It highlights the ingenuity of ancient mathematicians who developed methods that remain relevant. The exchange of mathematical ideas over time and geography has been instrumental in creating the diverse field of mathematics we know today.

Algebraic Equations

Algebraic equations are the cornerstone of algebra and serve as a fundamental component of modern mathematics. They consist of mathematical statements that showcase a relationship between quantities, often involving variables and constants, where the values of the variables must be determined.

The method of false position, as shown in the textbook exercise, is an ancient technique to solve linear algebraic equations. It requires an initial guess which is then refined to reach the solution. This method represented a significant advance in algebraic thinking and contributed to the development of equations as we know them. Equations are now used to describe everything from simple relationships to complex systems in various fields such as science, finance, and engineering.