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### Deriving the Celsius-to-Fahrenheit Conversion Formula

```
Date: 12/03/2001 at 23:33:00
From: Calixta
Subject: Temperature conversions

What is 27 degrees above freezing in Fahrenheit?

Thanks.
```

```
Date: 12/04/2001 at 01:40:05
From: Doctor Jeremiah
Subject: Re: Temperature conversions

Hi Calixta,

It depends on what temperature scale you are converting from. Since
you don't say, I am going to assume that you meant to ask what 27
degrees Celsius is when converted to Farenheit. If that's not what you
meant to ask, then let me know.

Let's pretend that there is a straight-line (linear) relation between
the two temperature scales (and it turns out that in fact there is).
If that is so, then we need two "points" to find the equation for
that line.

Freezing in Celsius is 0 degrees, and in Fahrenheit it is 32.
Boiling in Celsius is 100 degrees, and in Fahrenheit it is 212.

Farenheit  Celsius
--------------------
32         0
212       100

The equation for a line that converts Celsius to Farenheit is:

F = mC + b

where C is the Celsius temperature
m is the "slope" of the line
F is the Fahrenheit temperature
and b is the "intercept" of the line

So if we use our two "points" to fill in this equation we get:

Farenheit  Celsius      equation
--------------------   ------------
32         0       32 = 0m + b
212       100      212 = 100m + b

Now we have two equations and two unknown values (m and b) so we can
solve for these values and make an equation that converts Celsius to
Farenheit:

The two equations:

32 = 0m + b
212 = 100m + b

The first thing to do is to make both equations equal to the same
thing. The easy way to do that is to get the b terms alone on one side
of the equation:

32 = b
212 - 100m = b

Notice that we already know the value of b. And since both equations
are equal to b, they are also equal to each other:

32 = b = 212 - 100m
32 = 212 - 100m

And we can solve for m like this:

32 = 212 - 100m
32 - 212 = 212 - 100m - 212
-180 = -100m
-180 / -100 = -100m / -100
(-180/-100) = (-100/-100)m
9/5 = m

If we put the values for m and b into C = mF + b, we can come up with
an equation that converts Celsius to Fahrenheit:

F = mC + b   <==   m=9/5 , 32=b
F = (9/5)C + 32

So this is the conversion function. We want to convert 27 degrees
Celsius to Fahrenheit, so we plug 27 into this equation for C and
solve for F:

F = (9/5)C + 32   <==   C=27
F = (9/5)27 + 32
F = 80.6

So 27 degrees Celsius is 80.6 degrees Fahrenheit.

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Temperature

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