Is a temperature change in Celsius larger than a temperature change in Fahrenheit?

Question

Is a temperature change in Celsius larger than a temperature change in Fahrenheit?

The teacher offers this second way of thinking about the question.

If the temperature increases by 1 degree Celsius, does it also increase by 1 degree Fahrenheit? Or is one temperature change larger than the other?

My Attempt: $$T_F = \frac95T_C + 32 \implies (T_F + T_{\triangle F}) = \frac95(T_C+T_{\triangle C}) + 32$$ Where $T_{\triangle F}$ is some change on the Fahranheit scale and $T_{\triangle C}$ is the respective change on the Celsius scale. (I am confident this is true as long as we assume the changes in temperature are respective to each other.)

Thus, $$T_{\triangle F} = \frac95T_C + 32 - T_F + \frac95T_{\triangle C} \implies T_{\triangle F} = T_F - T_F + \frac95T_{\triangle C}$$ $$\implies T_{\triangle C} = \frac59T_{\triangle F} \checkmark$$

For some degree change on the Fahrenheit scale, $T_{\triangle F}$, the resulting change on the Celsius scale, $T_{\triangle C}$, is $\frac59$ that change. Therefore, a temperature change on the Celsius scale is larger than a temperature change on the Fahrenheit scale.

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Question: Does this show what I want? And does it answer the original question? Thanks in advance! This one is making me over think a bit I think.

Answer 1

Is a temperature change in Celsius larger than a temperature change in Fahrenheit?

is a very poorly-phrased question for which there can be two diametrically-different interpretations.

1. A temperature change of one degree Celsius is a larger temperature change than a temperature change of one degree Fahrenheit. This is the interpretation that the OP has come up with.

2. A fixed temperature difference (say the difference at sea level between the boiling point of water and the freezing point of water) measures 100 degrees on the Celsius scale but 180 degrees on the Fahrenheit scale; that is, a fixed temperature difference is more when measured on the Fahrenheit scale than when it is measured on the Celsius scale.

Answer 2

What you have so far is correct, but in my opinion, you haven't quite answered the question that was asked.

Before I get to that, though, I have just a minor comment. You write that

$$T_{\triangle F} = \frac95T_C + 32 - T_F + \frac95T_{\triangle C}.$$

This is correct, but in my opinion, you should write one more intermediate step:

$$T_{\triangle F} = \frac95 \left (T_C + T_{\triangle C} \right) + 32 - T_F = \frac95T_C + 32 - T_F + \frac95T_{\triangle C}.$$

I say this simply because when I read through your steps, it took me a minute to figure out how you got from the one expression to the next.

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Now, the original question was:

If the temperature increases by 1 degree Celsius, does it also increase by 1 degree Fahrenheit? Or is one temperature change larger than the other?

You've shown that when a temperature change is measured in each of the two scales, the number of degrees Fahrenheit is larger than the number of degrees Celsius. Equal temperature changes, different numbers of degrees.

But that's not quite what the question was asking. It's asking whether a change of 1 degree Celsius is larger, smaller, or the same as a change of 1 degree Fahrenheit. Equal numbers of degrees, different temperature changes.

So, you should take a change of 1 degree Fahrenheit, convert that to Celsius, and use the result to show that this is smaller than a change of 1 degree Celsius. Alternatively, take a change of 1 degree Celsius, convert that to Fahrenheit, and use the result to show that this is larger than a change of 1 degree Fahrenheit.

Great work so far!