Question

Problem 72 of the Rhind Mathematical Papyrus reads " 100 loaves of pesu 10 are exchanged for loaves of pesu 45 . How many of these loaves are there? The solution is given as, "Find the excess of 45 over \(10 .\) It is 35 . Divide this 35 by 10. You get \(3 \overline{2}\). Multiply \(3 \overline{2}\) by 100. Result: 350. Add 100 to this 350 . You get 450 . Say then that the exchange is 100 loaves of pesu 10 for 450 loaves of pesu \(45 . "^{18}\) Translate this solution into modern terminology. How does this solution demonstrate proportionality?

Short answer

Answer: In this exchange problem, 100 loaves of pesu 10 are equivalent to 450 loaves of pesu 45. The solution demonstrates proportionality as the ratio of the number of loaves in the exchange is equal to the ratio of the pesu values (\(\frac{\text{loaves of pesu 45}}{\text{loaves of pesu 10}} = \frac{45}{10}\)), indicating that as the pesu value increases, so does the quantity of loaves required for an equivalent exchange.

Step-by-step solution

Understand the given problem

We are given that 100 loaves of pesu 10 are to be exchanged for loaves of pesu 45. We need to find out how many loaves of pesu 45 are there in the exchange.

Find the excess of pesu 45 over pesu 10

Calculate the difference between pesu 45 and pesu 10. Excess of pesu 45 over pesu 10 = 45 - 10 = 35.

Divide the difference by 10

Next, we divide the difference of pesu values by 10: 35 ÷ 10 = 3.5.

Multiply the result by 100

Now we will multiply the result obtained in Step 3 by 100: 3.5 × 100 = 350.

Add 100 to the result

Finally, we will add 100 to the result of Step 4: 350 + 100 = 450.

Interpret the result

Therefore, 100 loaves of pesu 10 are equivalent to 450 loaves of pesu 45 in the exchange.

Explain proportionality

The given solution demonstrates proportionality because the ratio of the number of loaves in the exchange (\(\frac{\text{loaves of pesu 45}}{\text{loaves of pesu 10}}\)) is equal to the ratio of the pesu values (\(\frac{45}{10}\)): \(\frac{\text{loaves of pesu 45}}{\text{loaves of pesu 10}} = \frac{45}{10}\) \(\frac{450}{100} = \frac{45}{10}\) This shows that as the pesu value increases, so does the quantity of loaves required for an equivalent exchange, thus demonstrating proportionality between the pesu values and the number of loaves in the exchange.

Key concepts

Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus is one of the most significant documents from ancient Egypt, dating back to around 1650 BC. It serves as a remarkable snapshot of Egyptian mathematical practices.
This ancient manuscript provides insight into how the Egyptians understood and solved mathematical problems, showcasing their methodologies.
Written by a scribe named Ahmose, the papyrus contains a variety of mathematical problems and solutions. These problems were practical in nature, often dealing with everyday issues like measuring land, calculating the distribution of bread, or even solving geometric problems. Through these practical exercises, the Rhind Papyrus highlights the Egyptians' interest in effective problem-solving techniques and their reliance on arithmetic and proportional reasoning.

Proportionality

Proportionality is a mathematical concept that describes a relationship between two quantities where their ratio remains constant. In the context of the Rhind Mathematical Papyrus, understanding proportionality was crucial.
This concept allowed ancient Egyptians to solve various problems efficiently, by ensuring that the relationships between numbers remained balanced. An easy way to visualize proportionality is through simple ratio equations, like how the papyrus illustrates exchanges of loaves.
For instance, if the number of loaves is inversely proportional to their pesu value, doubling the pesu must halve the number of loaves for the exchange to remain fair. Therefore, when ancient scribes computed ratios, they were applying a principle very similar to modern-day mathematical techniques.

Pesu Value

In the context of the Rhind Mathematical Papyrus, the term 'pesu' specifically refers to a value indicating quality or quantity associated with goods, including bread. The pesu of a loaf is a measure of its exchange rate and, in ancient Egypt, offered a way to standardize trade between different goods.
For example, a pesu of 10 meant that it would take 10 standard loaves to equate to another value, whereas a pesu of 45 referred to another, perhaps richer or more valuable type of bread. This concept of pesu value allowed transactions, like those in Problem 72, to be calculated with consistency, ensuring that exchanges were fair and equitable.
Ultimately, pesu values provided a method to express and negotiate differences in worth, paving the way for more sophisticated financial mathematics.

Mathematical Problem-Solving

The method of solving mathematical problems in ancient Egypt, as demonstrated through the Rhind Mathematical Papyrus, relied heavily on logical reasoning and pragmatic approaches. Ancient Egyptian scribes utilized step-by-step solutions to ensure clarity and accuracy when dealing with numbers.
They usually started with an understanding of the problem, identified relationships (such as excess or deficit), and used basic arithmetic operations like addition, subtraction, multiplication, and division. Each step was clearly defined and executed systematically.
By doing so, they could break down complex arithmetic into manageable parts, making seemingly complicated calculations more approachable. This method mirrors modern problem-solving techniques by emphasizing the importance of breaking down problems into smaller, solvable steps, ensuring each part contributes to an accurate solution.