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Chapter 1: Problem 44

For about 10 years after the French revolution, the French government attempted to base measures of time on multiples of ten: One week consisted of 10 days, 1 day consisted of 10 hours, 1 hour consisted of 100 minutes, and 1 minute consisted of 100 seconds. What are the ratios of (a) the French decimal week to the standard week and (b) the French decimal second to the standard second?

Short Answer

Expert verified
(a) The ratio of the French decimal week to the standard week is \(\frac{10}{7}\). (b) The ratio of the French decimal second to the standard second is \(\frac{125}{108}\).

Step by step solution

01

- Understand the French Decimal Time System

The French decimal time system redefined the traditional time system. Here are the conversions: 1 week = 10 days, 1 day = 10 hours, 1 hour = 100 minutes, and 1 minute = 100 seconds.
02

- Understand the Standard Time System

In the standard time system: 1 week = 7 days, 1 day = 24 hours, 1 hour = 60 minutes, and 1 minute = 60 seconds.
03

- Calculate the Ratio of the French Decimal Week to the Standard Week

To find the ratio of the French decimal week to the standard week, divide the number of days in a French decimal week by the number of days in a standard week.The ratio is: \[ \text{Ratio} = \frac{10 \text{ days}}{7 \text{ days}} = \frac{10}{7} \]
04

- Calculate the Total Number of Seconds in Both Systems

Calculate the total number of seconds in one day for both systems.In the French decimal system, 1 day = 10 hours, 1 hour = 100 minutes, 1 minute = 100 seconds. Therefore,\[ \text{Total Seconds in 1 French day} = 10 \times 100 \times 100 = 100,000 \text{ seconds} \]In the standard system, 1 day = 24 hours, 1 hour = 60 minutes, 1 minute = 60 seconds. Therefore,\[ \text{Total Seconds in 1 Standard day} = 24 \times 60 \times 60 = 86,400 \text{ seconds} \]
05

- Calculate the Ratio of the French Decimal Second to the Standard Second

To find the ratio of one French decimal second to a standard second, divide the total number of seconds in one day in the French system by the total number of seconds in one day in the standard system.The ratio is: \[ \text{Ratio} = \frac{100,000 \text{ seconds}}{86,400 \text{ seconds}} = \frac{100,000}{86,400} = \frac{625}{540} = \frac{125}{108} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

time measurement
Time measurement is a fundamental concept in both daily life and scientific endeavors.
In most systems, we use intervals based on historical and astronomical observations.
For an example, the traditional system splits a day into 24 hours, an hour into 60 minutes, and a minute into 60 seconds.
The French decimal time system is different, breaking time into units of ten:
  • 1 day = 10 hours
  • 1 hour = 100 minutes
  • 1 minute = 100 seconds
Such divisions make it easier to perform mathematical calculations.
As such, understanding different ways to measure time can be very useful, particularly when delving into various historical contexts or specialized fields like physics.
ratios in physics
Ratios play a significant role in physics.
They allow us to compare quantities and understand relationships between different systems.
When we talk about the ratio of the French decimal week to the standard week, we are comparing the length of time in two different systems.
So the calculation would be:
  • French Decimal Week: 10 days
  • Standard Week: 7 days
The ratio is: \( \frac{10}{7} \).
Similarly, to compare the French decimal second to the standard second, we first calculate how many seconds are in one day under both systems:
  • Total Seconds in 1 French day: 100,000 seconds
  • Total Seconds in 1 Standard day: 86,400 seconds
    • Then, the ratio is: \ \frac{100,000}{86,400} = \frac{125}{108} \.
    This comparison helps us understand how different units relate within different frameworks.
    historical time systems
    Throughout history, many different time systems have been used.
    Each culture had its own method calibrated to societal needs and astronomical observations.
    The French decimal time system was an attempt to simplify and standardize time measurement.
    It was used for about 10 years following the French Revolution.
    The goal was to make calculations easier by using a base-10 system for all measurements:
    • 10 days per week
    • 10 hours per day
    • 100 minutes per hour
    • 100 seconds per minute
    Though it was eventually abandoned, it illustrates the creative attempts of societies to find the most effective ways to measure time.
    Seeing these systems reminds us that time measurement is a flexible concept that evolves with societal needs.
    conversion of units
    Converting units is a crucial skill when dealing with different measurement systems.
    To convert between the French decimal time and the traditional time system, we need to establish conversion factors.
    For example, when calculating the ratio of French decimal week to standard week, the steps are:
    • French Decimal Week = 10 days
    • Standard Week = 7 days
    The conversion factor becomes \[ \frac{10 \text{ days}}{7 \text{ days}} \].
    For converting seconds:
    • French day has 100,000 seconds
    • Standard day has 86,400 seconds
    The conversion is \ \frac{100,000}{86,400} = \frac{125}{108} \.
    Understanding these conversions help you move seamlessly between units in different systems, a handy skill in both academic and real-world scenarios.

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