[chatgpt]

It seems that there is a subtle difference between a ratio and a rate. Can you explain the difference?

Yes! The difference between a ratio and a rate is subtle but important:

• A ratio is a comparison between two quantities of the same kind, expressed as a fraction or with a colon (e.g., 3:2 or 3/2). It is a pure number and does not have units. For example, if a recipe calls for 2 cups of flour and 3 cups of sugar, the ratio of flour to sugar is 2:3.
  
• A rate, on the other hand, is a special type of ratio where the two compared quantities have different units. It often describes how one quantity changes in relation to another. For example, speed is a rate because it compares distance (miles, kilometers) to time (hours, seconds), such as 60 miles per hour (mph).
  

1. Wikipedia puts it simply:
   
   1. A ratio shows how many times one number contains another.
      
   2. So, the names we give to numbers are irrelevant.
      
   3. So, the denominator is the unit.
      
2. But in metrology: ratio is the dimensionless quotient between two physical quantities measured with the same units.
   
3. A quotient of two quantities that are measured with different units may be called a rate.
   
4. So, this thing is related to units. Unit is nothing more than a standard scaling factor.
   
5. So, speed \(s=d/t\) is a rate.
   
6. But here “time” is just a name we give to the unit we scale the distance.
   
7. Time is also a distance or a magnitude. All magnitudes are distances. So, we compare two lengths.
   
8. I can measure the distance traveled with meters and I can scale it with centimeters. So instead of 1 second as my scaling unit I can have 1 cm as my scaling unit. Nothing will change. Because all I’m doing is the count how many times my unit is in the antecedant.
   
9. There is no difference between saying “1 meter per second” and “1 meter per 1 cm”.
   
10. If the speed is doubled, the object will be traveling “2 meters per 1 cm”.
    
11. So speed has nothing to do with the philosophical Time with a capital T. To measure speed we just compare two lengths, two magnitudes.
    
12. Consequent term is the unit. A unit is nothing more than the scaling factor.