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The conversion factor of $\frac{5}{9}$ in the formula for converting Fahrenheit to Celsius is derived from the relationship between the two temperature scales, specifically how they are defined relative to each other.

Conversion Formula

The formula to convert Fahrenheit (F) to Celsius (C) is:

Explanation of the Factors

1. Difference in Starting Points:

   • The Celsius scale is based on the freezing point (0°C) and boiling point (100°C) of water, while the Fahrenheit scale is based on different reference points—specifically, the freezing point of water at 32°F and boiling point at 212°F.

2. Scale Difference:

   • Between the freezing and boiling points of water, the Celsius scale has 100 intervals (0 to 100°C), while the Fahrenheit scale has 180 intervals (32 to 212°F).
   • To convert a temperature difference from Fahrenheit to Celsius, you need to account for this difference in the size of the degree intervals.
   • The ratio of the change in temperature in Celsius to the change in temperature in Fahrenheit is given by:

Conclusion

Thus, the conversion factor $\frac{5}{9}$ reflects how much a temperature interval in Fahrenheit corresponds to a temperature interval in Celsius. The term $(F - 32)$ adjusts for the different starting points on the scales.