Problem 26 - A History of Mathematics: An Introduction

Question

If an Imperial solidus is sold for \(31 \frac{1}{2}\) Pisan denarii, and a Genoese solidus is worth \(19 \frac{3}{4}\) Pisan denarii, then how many Genoese pounds will one have for 17 Imperial pounds, 11 solidi, and 5 denarii? (One pound equals 20 solidi. Note, that the exchange rate between Pisan and Genoese money is different in this exercise from that stated in the previous exercise.)

Step-by-Step Solution

Answer

Answer: 28 Genoese pounds.

Step 1: Convert Imperial pounds to Imperial solidi

As one imperial pound is equal to 20 solidi, we'll first convert 17 Imperial pounds to solidi: 17 Imperial pounds × 20 solidi/1 pound = 340 Imperial solidi

Step 2: Add the 11 solidi to the total

Next, add the 11 solidi to the 340 Imperial solidi we found in step 1: 340 Imperial solidi + 11 solidi = 351 Imperial solidi

Step 3: Convert the Imperial solidi to Pisan denarii

We are given that 1 Imperial solidus is sold for 31 and 1/2 Pisan denarii. To convert 351 Imperial solidi to Pisan denarii, we can simply multiply by the conversion rate: 351 Imperial solidi × (31 \frac{1}{2} \frac{Pisan \: denarii}{Imperial \: solidus}) = 351 Imperial solidi × \frac{63}{2} \frac{Pisan \: denarii}{Imperial \: solidus}

Step 4: Calculate the total Pisan denarii

Now, calculate the product of step 3, and add the 5 Pisan denarii to the total: 351 Imperial solidi × \frac{63}{2} \frac{Pisan \: denarii}{Imperial \: solidus} + 5 Pisan denarii = 11025.5 Pisan denarii

Step 5: Convert Pisan denarii to Genoese solidi

Next, use the conversion rate given for 1 Genoese solidus to Pisan denarii (19 and 3/4 Pisan denarii) to convert the total Pisan denarii to Genoese solidi: 11025.5 Pisan denarii × \frac{1 Genoese \: solidus}{19 \frac{3}{4} Pisan \: denarii} = 11025.5 Pisan denarii × \frac{1 Genoese \: solidus}{\frac{79}{4} Pisan \: denarii}

Step 6: Calculate the total Genoese solidi

Now, calculate the product of step 5: 11025.5 Pisan denarii × \frac{1 Genoese \: solidus}{\frac{79}{4} Pisan \: denarii} = 560 Genoese solidi

Step 7: Convert Genoese solidi to Genoese pounds

Finally, as one pound equals 20 solidi, we can convert 560 Genoese solidi to Genoese pounds: 560 Genoese solidi × \frac{1 Genoese \: pound}{20 Genoese \: solidi} = 28 Genoese pounds Hence, for 17 Imperial pounds, 11 solidi, and 5 denarii, one will have 28 Genoese pounds.

Key Concepts

Imperial pounds

Imperial pounds were used in England and other parts of the broader British Empire historically. They played a crucial role in the complex system of old English currency. Essentially, 1 Imperial pound was divided into 20 solidi. This structure means whenever you want to convert pounds into solidi, you simply multiply the number of pounds by 20.
For example, if you have 17 Imperial pounds, the conversion would be:

• Multiply 17 by 20
• Resulting in 340 Imperial solidi

In this exercise, understanding the conversion from pounds to solidi is the first step in moving forward to perform more complex currency conversions.

Genoese solidi

Genoese solidi were integral in the financial activities of Genoa, a significant maritime republic. Unlike the Imperial monetary system, where 20 solidi equals one pound, understanding the value of Genoese solidi requires connecting it with Pisan currency for conversion.
The exercise involves determining the worth of Genoese solidi in terms of Pisan denarii, for which it states that one Genoese solidus is equivalent to 19 and 3/4 Pisan denarii. Such conversion helps in computing the total value accurately when dealing with historical currencies that had regional variances.

Pisan denarii

Pisan denarii were used in Pisa, a city that had interaction with various currencies, showcasing the blend of commerce in medieval Italy. In this problem, Pisan denarii serve as the intermediary currency.
We find that 1 Imperial solidus is equivalent to 31 and 1/2 Pisan denarii. This conversion factor is crucial as it allows you to transform the Imperial solidi into Pisan denarii, forming a bridge to convert into Genoese currency eventually.
With 351 Imperial solidi, the formula used is:

• Multiply 351 by 31.5 to determine Pisan denarii
• Add any remaining smaller currency units present in the exercise

exchange rates

Exchange rates in this context refer to the conversion values between different medieval currencies. They are essential because each city-state often had its unique currency valued differently.
The exercise demonstrates various exchange rates:

• Imperial solidus to Pisan denarii: 1 to 31.5
• Genoese solidus to Pisan denarii: 1 to 19.75

Exchange rates are not static and could vary due to the economic and political backdrop, making understanding them a significant historical mathematical challenge.

mathematical conversion

Mathematical conversion within the context of historical currency requires a systematic approach:

• First, understand the equivalence between units
• Apply multiplication or division to convert amounts

This exercise involved: - Converting Imperial solidi to Pisan denarii using a multiplication conversion factor - Changing Pisan denarii to Genoese solidi by division using the given rate
Each step requires attention to details like unit fractions and ensures values reflect correctly on the right scale, ultimately leading to the final answer of 28 Genoese pounds.