Problem 11 - A History of Mathematics: An Introduction

Question

A quantity, its \(1 / 3\), and its \(1 / 4\), added together, become 2 . What is the quantity? (problem 32 of the Rhind Mathematical Papyrus)

Step-by-Step Solution

Answer

Answer: \(\frac{24}{19}\)

Step 1: Create an equation

Let’s represent the unknown quantity as x, and we know that the \(1/3\) of x and \(1/4\) of x added together with x itself result in 2. So we can write this as an equation: \[ x + \frac{1}{3}x + \frac{1}{4}x = 2 \]

Step 2: Find a common denominator

To work with the fractions, we will find a common denominator. In this case, the common denominator for 3 and 4 is 12. Multiply each term by 12 to eliminate the fractions: \[12(x) + 12 (\frac{1}{3}x) + 12 (\frac{1}{4}x) = 2 \cdot 12\]

Step 3: Simplify the equation

Now, we can simplify the equation by performing the multiplications: \[ 12x + 4x + 3x = 24 \]

Step 4: Combine like terms

Combine the x terms: \[ 19x = 24 \]

Step 5: Solve for x

To find the value of x, divide both sides by 19: \[ x = \frac{24}{19} \] The unknown quantity, x, is equal to \(\frac{24}{19}\).

Key Concepts

Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus is an exceptional documentation of ancient Egyptian mathematics. Named after Alexander Henry Rhind, a Scottish antiquarian who purchased the papyrus in 1858, it is believed to have been written around 1550 BCE.

The papyrus is a compilation of math problems and solutions that provide insights into the mathematical understanding and practical applications used by the Egyptians. It contains arithmetic, algebraic, and geometric problems, making it akin to a modern-day textbook. Notably, it exhibits the Egyptians' prowess in handling fractions, particularly unit fractions, which are fractions with a numerator of 1 and a different denominator.

Understanding this historic document not only illuminates the history of mathematical development but also demonstrates the ancient origins of many mathematical practices still in use today. The exercise we explore, problem 32, reflects the type of algebraic reasoning contained within the papyrus.

Fractions in Mathematics

Fractions are a fundamental concept in mathematics, representing parts of whole numbers. They are composed of a numerator and a denominator. In ancient Egypt, particular emphasis was placed on unit fractions, which have a numerator of 1 and are used to represent other fractions in sums.

Today, fractions remain a cornerstone of mathematical education. Learning to work with them involves understanding how to perform operations such as addition, subtraction, multiplication, and division. To solve equations involving fractions, as demonstrated in the historical exercise from the Rhind Papyrus, we typically find a common denominator to simplify calculations. This practice, deeply rooted in ancient mathematics, is essential for handling complex mathematical operations in various fields of study.

Algebraic Problem Solving

Algebraic problem solving is the process of finding unknown variables within equations through a series of systematic steps. It's a fundamental aspect of algebra, a branch of mathematics that deals with symbols and the rules for manipulating these symbols.

To solve algebraic problems, one usually starts by defining a variable to represent the unknown quantity. Then, by following algebraic principles, such as combining like terms and balancing equations, the value of the variable can be isolated and calculated. As observed in the solution to the Rhind Papyrus problem, these steps still form the crux of algebraic problem-solving techniques used in contemporary mathematics education.

The clear and methodical approach to algebra helps students understand relationships between numbers and foster analytical thinking, which has applications beyond mathematics into fields such as science, engineering, and economics.