At what temperature (in degree Celsius) do the numerical values on the Celsius and Fahrenheit temperature scales become equal?

Introduction / Context:
The Celsius and Fahrenheit scales are two commonly used temperature scales. They have different zero points and degree sizes, so the same physical temperature has different numerical values on each scale. Interestingly, there is one temperature at which the Celsius and Fahrenheit readings are numerically equal. This question asks you to determine that temperature in degree Celsius.

Given Data / Assumptions:

• Celsius and Fahrenheit scales are related by a linear conversion formula.
• The general relation is C = (5 / 9) * (F minus 32), where C is in degree Celsius and F is in degree Fahrenheit.
• We want the temperature where C and F have the same numerical value.
• We assume standard definitions of both scales.

Concept / Approach:
To find the temperature where the two scales coincide numerically, we can set C equal to F in the conversion equation and solve for that common value. This reduces the problem to a simple algebraic equation. The solution will give a temperature that is negative, reflecting the fact that the intersection of the two linear scales occurs below the freezing point of water.

Step-by-Step Solution:
Step 1: Start with the conversion formula: C = (5 / 9) * (F minus 32). Step 2: At the temperature where the scales have equal numerical values, set C = F. Let this common value be T. Step 3: Substitute C = T and F = T into the formula: T = (5 / 9) * (T minus 32). Step 4: Multiply both sides by 9 to remove the denominator: 9 * T = 5 * (T minus 32). Step 5: Expand the right side: 9 * T = 5 * T minus 160. Bring terms involving T to one side: 9 * T minus 5 * T = minus 160, so 4 * T = minus 160. Divide both sides by 4 to get T = minus 40.

Verification / Alternative check:
Check the result with direct conversion. If C = minus 40, then F = (9 / 5) * C + 32 = (9 / 5) * (minus 40) + 32 = minus 72 + 32 = minus 40. This confirms that at minus 40 degree Celsius, the Fahrenheit reading is also minus 40 degree. This is the only point where the linear conversion graphs of the two scales intersect. Temperature values such as 0, 100, or 273 would give very different values on the other scale, so minus 40 is unique.

Why Other Options Are Wrong:
40: At 40 degree Celsius, Fahrenheit is (9 / 5) * 40 + 32 = 72 + 32 = 104 degree Fahrenheit, not equal. 273: 273 degree Celsius corresponds to a very high Fahrenheit value and is associated with absolute temperature in kelvin, not with equality of the two scales. -273: This is close to absolute zero in degree Celsius, but the Fahrenheit value at this point is not numerically equal to minus 273; instead it is even lower.

Common Pitfalls:
Some learners try to guess the answer using approximate values or by confusing Celsius with kelvin. Others may incorrectly rearrange the conversion formula or forget to set C equal to F. The safest method is to treat the common temperature as an unknown T and solve the simple linear equation carefully. Doing this step by step leads straightforwardly to T = minus 40 degree.

Final Answer:
The Celsius and Fahrenheit scales give equal numerical readings at -40 degree Celsius (which is also -40 degree Fahrenheit).