Why is the slope 9/5 (or 5/9) for °C↔°F conversion?

Because one hundred degrees Celsius equals one hundred eighty degrees Fahrenheit, the ratio of change is one hundred eighty divided by one hundred which reduces to nine fifths, and the reciprocal is five ninths.

Think of Celsius and Fahrenheit like two different rulers that measure the same temperature but use different sized marks; moving one hundred ticks on the Celsius ruler covers the same real temperature change as moving one hundred eighty ticks on the Fahrenheit ruler, so each single tick on Celsius equals one hundred eighty divided by one hundred or nine fifths of a Fahrenheit tick, and if you want to go the other way you multiply by five ninths so the size differences cancel out. The fixed points used to make these rulers are useful too: water freezes at zero degrees on the Celsius ruler and at thirty-two degrees on the Fahrenheit ruler, and water boils at one hundred degrees Celsius and at two hundred twelve degrees Fahrenheit; those anchor points show both the offset and the stretch between scales. The conversion formula includes adding or subtracting the thirty-two degree offset when switching between scales because the zeroes are not at the same physical temperature, and then you multiply by nine fifths or five ninths to account for the different step sizes between ticks.

A common mistake is forgetting the thirty-two degree shift when converting from Fahrenheit to Celsius, which makes your answer way off, so remember to subtract thirty-two before scaling; another is using nine fifths when you should use five ninths, which flips the conversion and gives the inverse result, so check the direction you are converting; some students also try to convert just by adding or subtracting without scaling, which ignores the different tick sizes and leads to wrong answers, so always include both the offset and the ratio.

Check out w w w dot gumball degree dot com to find more videos and resources that walk through these conversions step by step.

Why the 9/5 and 5/9 in °C↔°F Conversions

1. Celsius and Fahrenheit are different rulers
2. 100°C equals 180°F change
3. Multiply by 9/5 to convert
4. Multiply by 5/9 to reverse
5. Account for 32° offset when converting

Common °C↔°F Conversion Mistakes

1. Forgetting the 32° offset
2. Using 9/5 instead of 5/9
3. Converting by adding/subtracting only

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