At what temperature do the Fahrenheit and Celsius temperature scales show the same numerical reading (that is, the same number on both scales)?

Introduction / Context:
This question tests basic conceptual understanding of temperature scales used in physics and everyday life. Celsius (also called centigrade) and Fahrenheit are two commonly used scales. Usually, the same physical temperature has different numerical values on these two scales. However, there is one particular temperature at which both scales show the same number. Knowing this special point helps students better understand the linear relationship between the two scales and is often directly asked in competitive exams as a simple but important fact.

Given Data / Assumptions:

• Celsius scale has fixed points: 0 degrees for freezing point of water and 100 degrees for boiling point of water at standard pressure.
• Fahrenheit scale has 32 degrees for freezing point and 212 degrees for boiling point of water.
• We are asked to find the temperature at which the numerical value on both scales is the same.
• The options include 212, 32, -40 and 0 degrees.

Concept / Approach:
The relationship between temperatures on the Celsius scale (C) and Fahrenheit scale (F) is linear and is given by the formula F = (9 / 5) * C + 32. To find the temperature at which the numbers are equal, we set F equal to C and solve the resulting simple equation. This is a straightforward substitution that shows at which point the two scales intersect. This method is often explained in basic physics and mathematics classes and is an excellent example of how linear relationships work.

Step-by-Step Solution:
Step 1: Use the conversion formula between Fahrenheit and Celsius: F = (9 / 5) * C + 32. Step 2: At the temperature where the readings are the same, set F = C. Step 3: Substitute F by C in the formula to get C = (9 / 5) * C + 32. Step 4: Rearrange the equation: C - (9 / 5) * C = 32. Step 5: Factor C: C * (1 - 9 / 5) = 32 gives C * ((5 - 9) / 5) = 32, so C * (-4 / 5) = 32. Step 6: Solve for C: C = 32 * (-5 / 4) = -40. Step 7: Therefore Celsius and Fahrenheit scales show the same numerical reading at -40 degrees.

Verification / Alternative check:
As a quick check, convert -40 degrees Celsius into Fahrenheit using the formula. Substituting C = -40 gives F = (9 / 5) * (-40) + 32 = -72 + 32 = -40. Because the calculated Fahrenheit temperature is also -40, the solution is verified. This double check confirms that -40 is indeed the temperature where both scales coincide numerically. Remembering this special value is very helpful in exams and also in practical scientific work where quick mental conversions are needed.

Why Other Options Are Wrong:
212 degrees corresponds to the boiling point of water on the Fahrenheit scale and is equal to 100 degrees on the Celsius scale, so the readings are different. 32 degrees is the freezing point of water in Fahrenheit, which equals 0 degrees Celsius, again giving different numbers. 0 degrees is the freezing point of water in Celsius, equal to 32 degrees Fahrenheit, so the two scales do not match numerically at this point.

Common Pitfalls:
Many learners try to guess from memory of freezing and boiling points and may choose 0 or 32 degrees because they recognise these as special values. Others may forget the exact conversion formula and become confused between Fahrenheit and Celsius readings. The safest approach in exams is to remember the simple derived fact that the two scales coincide at -40 degrees and, if time permits, quickly verify using the linear formula. This avoids guesswork and builds a deeper understanding of how different temperature scales relate to each other.

Final Answer:
The Fahrenheit and Celsius scales show the same numerical reading at -40 degrees.