Question 350494: The boiling and freezing temperatures of water are different in the two scales. Assuming that these scales are both linear – they really are, so that’s not an issue – is there any point at which the temperature in Fahrenheit would equal that in Celsius? What I mean is, suppose we took a piece of metal and heated – or cooled – it to a particular value on the one scale. Is there any such value for which a thermometer measuring the other scale would have the same number? What would that number be, and how did you find it?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! I originally thought no, but then I thought about it again and came up with the fact that the degrees in Celsius will be equal to the degrees in Fahrenheit when the temperature is -40 degrees.
I figured if it had to happen, it would happen when the degrees were negative.
I traced the degrees in Celsius from 0 going down by 1 degree at a time and saw that the difference between Celsius and Fahrenheit was getting smaller as I went further down into the negative territory.
At -40 degrees, they were the same.
I then went back and figured out how I could do that using a formula.
This is what I found.
f = 1.8*c + 32
This means that the degrees in Fahrenheit is equal to the degrees in Celsius * 1.8 + 32.
You might be more familiar with f = 9/5 * c + 32.
Since 9/5 = 1.8, these formulas are the same.
The reverse formula is:
c = (f-32) / 1.8
this means that the degrees in Celsius equals the degrees in Fahrenheit minus 32 and then divided by 1.8.
If they are to be equal, then f must equal c.
f = c implies:
1.8*c + 32 = (f-32) / 1.8
Multiply both sides of that equation by 1.8 to get:
1.8 * (1.8*c + 32) = f - 32
Add 32 to both sides of that equation to ge t:
1.8 * (1.8*c + 32) + 32 = f
Since we know that f = 1.8*c + 32, then we can substitute in this equation to get:
1.8 * (1.8*c + 32) + 32 = 1.8*c + 32
Subtract 32 from both sides of this equation to get:
1.8 * (1.8*c + 32) = 1.8*c
Divide both sides of this equation by 1.8 to get:
1.8*c + 32 = c
Subtract 32 from both sides of this equation and subtract c from both sides of this equation to get:
1.8*c - c = -32
Combine like terms to get:
.8*c = -32
Divide both sides of this equation by .8 to get:
c = -32 / .8
Simplify to get:
c = -40.
The formula confirms the empirical results.
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