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Guides d'étude > Mathematics for the Liberal Arts Corequisite

Temperature Scales

Learning Outcomes

  • Convert from one temperature scale to the other, using conversion formulas
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.

Measuring Temperature on Two Scales

Fahrenheit and Celsius are two different scales for measuring temperature.
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°C and 24°C to keep employees comfortable.

Two thermometers, one measuring a temperature of 72 degrees Fahrenheit and the other measuring a temperature of 22 degrees Celsius. The boiling point of water (100 degrees Celsius and 212 degrees Fahrenheit) is shown on both themometers, as is the freezng point of water (0 degrees Celsius and 32 degrees Fahrenheit) and comfortable room temperature (18 to 25 degrees Celsius and 65 to 75 degrees Fahrenheit).

 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°F and 75°F to keep employees comfortable.
Celsius

Fahrenheit

order of operations

In this section, you'll need to use order of operations carefully to obtain correct results. Recall that we do operations in parentheses first, then handle exponents, then multiply or divide from left to right as encountered, then add or subtract from left to right as encountered. The order is sometimes represented using the acroynim PEMDAS.

Try It

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?

Answer:  Fahrenheit scale

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°F and 80°F (or between 10°C and 25°C). If a meteorologist predicts an average temperature of 0°C (or 32°F), 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.

Temperature Conversion Formulas

To convert a Fahrenheit measurement to a Celsius measurement, use this formula.

[latex] C=\dfrac{5}{9}(F-32)[/latex]

To convert a Celsius measurement to a Fahrenheit measurement, use this formula.

[latex] F=\dfrac{9}{5}C+32[/latex]

How were these formulas developed? They came from comparing the two scales. Since the freezing point is 0° in 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 a reason for the fractions [latex] \frac{5}{9}[/latex] and [latex] \frac{9}{5}[/latex], also. 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, [latex] \frac{F{}^\circ }{C{}^\circ }[/latex], gives [latex] \frac{180{}^\circ }{100{}^\circ }=\frac{180{}^\circ \div 20}{100{}^\circ \div 20}=\frac{9}{5}[/latex]. If you flip the ratio to be [latex] \frac{\text{C}{}^\circ }{\text{F}{}^\circ }[/latex], you get [latex] \frac{100{}^\circ }{180{}^\circ }=\frac{100{}^\circ \div 20}{180{}^\circ \div 20}=\frac{5}{9}[/latex]. 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° C.

Example

The boiling point of water is 100°C. What temperature does water boil at in the Fahrenheit scale? A Celsius temperature is given. To convert it to the Fahrenheit scale, use the formula at the left.

[latex]F=\frac{9}{5}C+32[/latex]

Substitute 100 for C and multiply.

[latex]F=\frac{9}{5}(100)+32[/latex]

[latex]F=\frac{900}{5}+32[/latex]

Simplify [latex]\frac{900}{5}[/latex] by dividing numerator and denominator by 5.

[latex]F=\frac{900\div 5}{5\div 5}+32[/latex]

[latex]F=\frac{180}{1}+32[/latex]

Add [latex]180+32[/latex].

[latex]F=212[/latex]

The boiling point of water is 212°F.

Try It

[ohm_question]1011[/ohm_question]
 

Example

Water freezes at 32°F. On the Celsius scale, what temperature is this?

Answer: A Fahrenheit temperature is given. To convert it to the Celsius scale, use the formula at the left.

[latex] C=\frac{5}{9}(F-32)[/latex]

Substitute 32 for F and subtract.

[latex] C=\frac{5}{9}(32-32)[/latex]

Any number multiplied by 0 is 0.

[latex] C=\frac{5}{9}(0)[/latex]

[latex] C=0[/latex]

The freezing point of water is [latex]0^{\circ}\text{C}[/latex].

Try It

[ohm_question]1010[/ohm_question]
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.

Example

Two scientists are doing an experiment designed to identify the boiling point of an unknown liquid. One scientist gets a result of 120°C; the other gets a result of 250°F. Which temperature is higher and by how much?

Answer: One temperature is given in °C, and the other is given in °F. To find the difference between them, we need to measure them on the same scale. What is the difference between 120°C and 250°F? Use the conversion formula to convert 120°C to °F. (You could convert 250°F to °C instead; this is explained in the text after this example.)

[latex] F=\frac{9}{5}C+32[/latex]

Substitute 120 for C.

[latex] F=\frac{9}{5}(120)+32[/latex]

Multiply.

[latex] F=\frac{1080}{5}+32[/latex]

Simplify [latex] \frac{1080}{5}[/latex] by dividing numerator and denominator by 5.

[latex] F=\frac{1080\div 5}{5\div 5}+32[/latex]

Add [latex]216+32[/latex].

[latex] F=\frac{216}{1}+32[/latex]

You have found that [latex]120^{\circ}\text{C}=248^{\circ}\text{F}[/latex].

[latex] F=248[/latex]

To find the difference between 248°F and 250°F, subtract.

[latex]250^{\circ}\text{F}-248^{\circ}\text{F}=2^{\circ}\text{F}[/latex]

250°F is the higher temperature by 2°F.

You could have converted 250°F to °C instead, and then found the difference in the two measurements. (Had you done it this way, you would have found that [latex]250^{\circ}\text{F}=121.1^{\circ}\text{C}[/latex], and that 121.1°C is 1.1°C higher than 120°C.) Whichever way you choose, it is important to compare the temperature measurements within the same scale, and to apply the conversion formulas accurately.

Try It

Tatiana is researching vacation destinations, and she sees that the average summer temperature in Barcelona, Spain is around 26°C. What is the average temperature in degrees Fahrenheit?

Answer: 78.8 degrees Fahrenheit.

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.

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  • Question ID 1011, 1010. Authored by: Lippman, David. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.