Temperature Scales
Learning Objectives
- State the freezing and boiling points of water on the Celsius and Fahrenheit temperature scales.
- Convert from one temperature scale to the other, using conversion formulas.
Introduction
Turn on the television any morning and you will see meteorologists talking about the day’s weather forecast. In addition to telling you what the weather conditions will be like (sunny, cloudy, rainy, muggy), they also tell you the day’s forecast for high and low temperatures. A hot summer day may reach `100^@` in Philadelphia, while a cool spring day may have a low of `40^@` in Seattle.
If you have been to other countries, though, you may notice that meteorologists measure heat and cold differently outside of the United States. For example, a TV weatherman in San Diego may forecast a high of `89^@`, but a similar forecaster in Tijuana, Mexico, which is only `20` miles south, may look at the same weather pattern and say that the day’s high temperature is going to be `32^@`. What’s going on here?
The difference is that the two countries use different temperature scales. In the United States, temperatures are usually measured using the FahrenheitA measure of temperature commonly used in the United States. On the Fahrenheit scale, water freezes at `32^@` Fahrenheit and boils at `212^@` Fahrenheit. scale, while most countries that use the metric system use the CelsiusA measure of temperature commonly used in countries that use the metric system. On the Celsius scale, water freezes at `0^@` Celsius and boils at `100^@` Celsius. scale to record temperatures. Learning about the different scales, including how to convert between them, will help you figure out what the weather is going to be like, no matter which country you find yourself in.
Measuring Temperature on Two Scales
Fahrenheit and Celsius are two different scales for measuring temperature.
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A thermometer measuring a temperature of `22^@" Celsius"` is shown here.
On the Celsius scale, water freezes at `0^@` and boils at `100^@`.
If the United States were to adopt the Celsius scale, forecast temperatures would rarely go below `-30^@` or above `45^@`. (A temperature of `-18^@` may be forecast for a cold winter day in Michigan, while a temperature of `43^@`may be predicted for a hot summer day in Arizona.)
Most office buildings maintain an indoor temperature between `18^@" Celsius"` and `24^@" Celsius"` to keep employees comfortable.
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Celsius Fahrenheit
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A thermometer measuring a temperature of `72^@" Fahrenheit"` is shown here.
On the Fahrenheit scale, water freezes at `32^@` and boils at `212^@`.
In the United States, forecast temperatures measured in Fahrenheit rarely go below `-20^@` or above `120^@`. (A temperature of `0^@` may be forecast for a cold winter day in Michigan, while a temperature of `110^@`may be predicted for a hot summer day in Arizona.)
Most office buildings maintain an indoor temperature between `65^@" Fahrenheit"` and `75^@" Fahrenheit"` to keep employees comfortable.
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A cook puts a thermometer into a pot of water to see how hot it is. The thermometer reads `132^@`, but the water is not boiling yet. Which temperature scale is the thermometer measuring?
A) Celsius
B) Fahrenheit
A) Celsius
Incorrect. On the Celsius scale, water boils at `100^@`, so if the water is not boiling and the measurement is over `100^@`, then it cannot be Celsius. The correct answer is Fahrenheit.
B) Fahrenheit
Correct. Water boils at `212^@` on the Fahrenheit scale, so a measurement of `132^@` on a Fahrenheit scale is legitimate for hot (but non-boiling) water.
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Converting Between the Scales
By looking at the two thermometers shown, you can make some general comparisons between the scales. For example, many people tend to be comfortable in outdoor temperatures between `50^@` Fahrenheit and `80^@` Fahrenheit (or between `10^@` Celsius and `25^@` Celsius). If a meteorologist predicts an average temperature of `0^@` Celsius (or `32^@` Fahrenheit), then it is a safe bet that you will need a winter jacket.
Sometimes, it is necessary to convert a Celsius measurement to its exact Fahrenheit measurement or vice versa. For example, what if you want to know the temperature of your child in Fahrenheit, and the only thermometer you have measures temperature in Celsius measurement? Converting temperature between the systems is a straightforward process as long as you use the formulas provided below.
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Temperature Conversion Formulas
To convert a Fahrenheit measurement to a Celsius measurement, use this formula.
`C=5/9(F-32)`
To convert a Celsius measurement to a Fahrenheit measurement, use this formula.
`F=9/5C+32`
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How were these formulas developed? They came from comparing the two scales. Since the freezing point is `0^@` on the Celsius scale and `32^@` on the Fahrenheit scale, we subtract `32` when converting from Fahrenheit to Celsius, and add `32` when converting from Celsius to Fahrenheit.
There is also a reason for the fractions `5/9` and `9/5`. There are `100` degrees between the freezing (`0^@`) and boiling points (`100^@`) of water on the Celsius scale and `180` degrees between the similar points (`32^@` and `212^@`) on the Fahrenheit scale. Writing these two scales as a ratio, `(F^@)/(C^@)`, gives `(180^@)/(100^@)=(180^@-:20)/(100^@-:20)=9/5`. If you flip the ratio to be `(C^@)/(F^@)`, you get `(100^@)/(180^@)=(100^@-:20)/(180^@-:20)=5/9`. Notice how these fractions are used in the conversion formulas.
The example below illustrates the conversion of Celsius temperature to Fahrenheit temperature, using the boiling point of water, which is `100^@` Celsius.
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Example
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Problem
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The boiling point of water is `100^@` Celsius. What temperature does water boil at in the Fahrenheit scale?
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`F=9/5C+32`
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A Celsius temperature is given. To convert it to the Fahrenheit scale, use the formula at the left.
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`F=9/5(100)+32`
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Substitute `100` for `C` and multiply.
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`F=900/5+32`
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`F=(900-:5)/(5-:5)+32`
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Simplify `900/5` by dividing numerator and denominator by `5`.
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`F=180/1+32`
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`F=212`
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Add `180 + 32`.
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Answer
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The boiling point of water is `212^@` Fahrenheit.
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Example
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Problem
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Water freezes at `32^@` Fahrenheit. On the Celsius scale, what temperature is this?
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`C=5/9(F-32)`
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A Fahrenheit temperature is given. To convert it to the Celsius scale, use the formula at the left.
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`C=5/9(32-32)`
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Substitute `32` for `F` and subtract.
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`C=5/9(0)`
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Any number multiplied by `0` is `0`.
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`C=0`
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Answer
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The freezing point of water is `0^@` Celsius.
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The two previous problems used the conversion formulas to verify some temperature conversions that were discussed earlier: the boiling and freezing points of water. The next example shows how these formulas can be used to solve a real-world problem using different temperature scales.
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Example
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Problem
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Two scientists are doing an experiment designed to identify the boiling point of an unknown liquid. One scientist gets a result of `120^@` Celsius; the other gets a result of `250^@` Fahrenheit. Which temperature is higher and by how much?
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What is the difference between `120^@` Celsius and `250^@` Fahrenheit?
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One temperature is given in degrees Celsius, and the other is given in degrees Fahrenheit. To find the difference between them, we need to measure them on the same scale.
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`F=9/5C+32`
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Use the conversion formula to convert `120^@` Celsius to degrees Fahrenheit.
(You could convert `250^@` Fahrenheit to degrees Celsius instead; this is explained in the text after this example.)
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`F=9/5(120)+32`
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Substitute `120` for `C`.
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`F=1080/5+32`
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Multiply.
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`F=(1080-:5)/(5-:5)+32`
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Simplify `1080/5` by dividing numerator and denominator by `5`.
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`F=216/1+32`
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Add `216+32`.
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`F=248`
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You have found that `120^@text(C)=248^@text(F)`.
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`250^@ - 248^@ = 2^@" Fahrenheit"`
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To find the difference between `248^@` Fahrenheit and `250^@` Fahrenheit, subtract.
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Answer
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`250^@` Fahrenheit is the higher temperature by `2^@` Fahrenheit.
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You could have converted `250^@` Fahrenheit to degrees Celsius instead, and then found the difference in the two measurements. (Had you done it this way, you would have found that `250^@text(F)=121.1^@text(C)`, and that `121.1^@` Celsius is `1.1^@` Celsius higher than `120^@` Celsius.) Whichever way you choose, it is important to compare the temperature measurements within the same scale, and to apply the conversion formulas accurately.
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Tatiana is researching vacation destinations, and she sees that the average summer temperature in Barcelona, Spain is around `26^@` Celsius. What is the average temperature in degrees Fahrenheit?
A) `79^@` Fahrenheit
B) `-3^@` Fahrenheit
C) `45^@` Fahrenheit
D) `58^@` Fahrenheit
A) `79^@` Fahrenheit
Correct. Tatiana can find the Fahrenheit equivalent by solving the equation `F=9/5(26)+32`. The result is `78.8^@` Fahrenheit, which rounds to `79^@` Fahrenheit.
B) `-3^@` Fahrenheit
Incorrect. You used the wrong formula. To find the Fahrenheit equivalent, use the formula `F=9/5C+32`. The correct answer is `79^@` Fahrenheit.
C) `45^@` Fahrenheit
Incorrect. You misapplied the formula; try substituting `26` for `C` in the formula `F=9/5C+32`. The correct answer is `79^@` Fahrenheit.
D) `58^@` Fahrenheit
Incorrect. You misapplied the formula; try substituting `26` for `C` in the formula `F=9/5C+32`. The correct answer is `79^@` Fahrenheit.
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Summary
Temperature is often measured in one of two scales: the Celsius scale and the Fahrenheit scale. A Celsius thermometer will measure the boiling point of water at `100^@` and its freezing point at `0^@`; a Fahrenheit thermometer will measure the same events at `212^@` for the boiling point of water and `32^@` as its freezing point. You can use conversion formulas to convert a measurement made in one scale to the other scale.