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Measuring Temperature

Convert between Fahrenheit and Celsius.

Learning Objectives

After completing this section, you should be able to:

  1. Convert between Fahrenheit and Celsius.
  2. Identify reasonable values for temperature applications.
  3. Solve application problems involving temperature.

Converting Between Fahrenheit and Celsius Temperatures

Understanding how to convert between Fahrenheit and Celsius temperatures is an essential skill in understanding metric temperatures. You likely know that below 32 °F means freezing temperatures and perhaps that the same holds true for 0 °C. While it may be difficult to recall that water boils at 212 °F, knowing that it boils at 100 °C is a fairly easy thing to remember.

But what about all the temperatures in between? What is the temperature in degrees Celsisus on a scorching summer day? What about a cool autumn afternoon? If a recipe instructs you to preheat the oven to 350 °F, what Celsius temperature do you set the oven at?

lists common temperatures on both scales, because we don’t use Celsius temperatures daily it’s difficult to remember them. Fortunately, we don’t have to. Instead, we can convert temperatures from Fahrenheit to Celsius and from Celsius to Fahrenheit using a simple algebraic expression.

Converting Temperatures from Fahrenheit to Celsius

Try it.

A recipe calls for the oven to be set to 392 °F. What is the temperature in Celsius?

Solution

Use the formula in to convert from Fahrenheit to Celsius.

\[\begin{array}{lll}C & = & \frac{5}{9}(F-32) \\ C & = & \frac{5}{9}(392-32) \\ C & = & \frac{5}{9}(360) \\ C & = & 200\end{array}\]

So, 392 °F is equivalent to 200 °C.

Converting Temperatures from Celsius to Fahrenheit

Try it.

On a sunny afternoon in May, the temperature in London was 20 °C. What was the temperature in degrees Fahrenheit?

Solution

Use the formula in to convert from Celsius to Fahrenheit.

\[\begin{array}{lll}F & = & \frac{9}{5}C+32 \\ F & = & \frac{9}{5}(20)+32 \\ F & = & 36+32 \\ F & = & 68\end{array}\]

The temperature was 68 °F.

Comparing Temperatures in Celsius and Fahrenheit

Try it.

A manufacturer requires a vaccine to be stored in a refrigerator at temperatures between 36 °F and 46 °F. The refrigerator in the local pharmacy cools to 3 °C. Can the vaccine be stored safely in the pharmacy’s refrigerator?

Solution

Use the formula in to convert from Celsius to Fahrenheit.

\[\begin{array}{lll}F & = & \frac{9}{5}C+32 \\ F & = & \frac{9}{5}(3)+32 \\ F & = & 5.4+32 \\ F & = & 37.4\end{array}\]

Then, compare the temperatures.

\[36F^{\circ}<37.4F^{\circ}<46F^{\circ}\]

Yes. 37.4 °F falls within the acceptable range to store the vaccine, so it can be stored safely in the pharmacy’s refrigerator.

Reasonable Values for Temperature

While knowing the exact temperature is important in most cases, sometimes an approximation will do. When trying to assess the reasonableness of values for temperature, there is a quicker way to convert temperatures for an approximation using mental math. These simpler formulas are listed in .

Using Benchmark Temperatures to Determine Reasonable Values for Temperatures

Try it.

Which is the more reasonable value for the temperature of a freezer?

  • 5 °C or
  • –5 °C?
Solution

We know that water freezes at 0 °C. So, the more reasonable value for the temperature of a freezer is −5 °C, which is below 0 °C. At temperature of 5 °C is above freezing.

Using Estimation to Determine Reasonable Values for Temperatures

Try it.

The average body temperature is generally accepted as 98.6 °F. What is a reasonable value for the average body temperature in degrees Celsius:

  • 98.6 °C,
  • 64.3 °C, or
  • 34.3 °C?
Solution

To estimate the average body temperature in degrees Celsius, subtract 30 from the temperature in degrees Fahrenheit, and divide the result by 2.

\[\frac{(98.6-30)}{2}=\frac{68.6}{2}=34.3\]

A reasonable value for average body temperature is 34.3 °C.

Using Conversion to Determine Reasonable Values for Temperatures

Try it.

Which is a reasonable temperature for storing chocolate:

  • 28 °C,
  • 18 °C, or
  • 2 °C?
Solution

Use the formula in to determine the temperature in degrees Fahrenheit.

\[\begin{array}{lll}F & = & \frac{9}{5}C+32 \\ F & = & \frac{9}{5}(28)+32=82.4 \\ F & = & \frac{9}{5}(18)+32=64.4 \\ F & = & \frac{9}{5}(2)+32=35.6\end{array}\]

A temperature of 82.4 °F would be too hot, causing the chocolate to melt. A temperature of 35.6 °F is very close to freezing, which would affect the look and feel of the chocolate. So, a reasonable temperature for storing chocolate is 18 °C, or 64.4 °F.

Solving Application Problems Involving Temperature

Whether traveling abroad or working in a clinical laboratory, knowing how to solve problems involving temperature is an important skill to have. Many food labels express sizes in both ounces and grams. Most rulers and tape measures are two-sided with one side marked in inches and feet and the other in centimeters and meters. And while many thermometers have both Fahrenheit and Celsius scales, it really isn’t practical to pull out a thermometer when cooking a recipe that uses metric units. Let’s review at few instances where knowing how to fluently use the Celsius scale helps solve problems.

Using Subtraction to Solve Temperature Problems

Try it.

The temperature in the refrigerator is 4 °C. The temperature in the freezer is 21 °C lower. What is the temperature in the freezer?

Solution

Use subtraction to find the difference.

\[4-21=-17\]

So, the temperature in the freezer is −17 °C.

Using Addition to Solve Temperature Problems

Try it.

A scientist was using a liquid that was 35 °C. They needed to heat the liquid to raise the temperature by 6 °C. What was the temperature after the scientist heated it?

Solution

Use addition to find the new temperature.

\[35+6=41\]

The temperature of the liquid was 41 °C after the scientist heated it.

Solving Complex Temperature Problems

Try it.

The optimum temperature for a chemical compound to develop its unique properties is 392 °F. When the heating process begins, the temperature of the compound is 20 °C. For safety purposes the compound can only be heated 9 °C every 15 minutes. How long until the compound reaches its optimum temperature?

Solution

Step 1: Determine the optimum temperature in degrees Celsius using the formula in .

\[\begin{array}{lll}C & = & \frac{5}{9}(F-32) \\ C & = & \frac{5}{9}(392-32) \\ C & = & \frac{5}{9}(360) \\ C & = & 200\end{array}\]

Step 2: Subtract the starting temperature.

\[200C^{\circ}-20C^{\circ}=180C^{\circ}\]

Step 3: Determine the number of 15-minute cycles needed to heat the compound to its optimum temperature.

\[180\div 9=20\]

Step 4: Multiply the number of cycles needed by 15 minutes and convert the product to hours and minutes.

\[\begin{array}{l}15\text{minutes}\times 9=135\text{minutes} \\ 135\text{minutes}=2\text{hours}15\text{minutes}\end{array}\]

So, it will take 2 hours and 15 minutes for the compound to reach its optimum temperature.

Key Concepts

  • Temperature is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a stove top, or an ice cube. The faster those atoms and molecules move, the higher the temperature.
  • In the metric system, temperature is measured using the Celsius (°C) scale.
  • The Celsius scale was created with 100 degrees separating the point at which water freezes, 0 °C, and the point at which water boils, 100 °C. Scientifically, these are the points at which water molecules change from one state of matter to another—from solid (ice) to liquid (water) to gas (water vapor).

Formulas

To convert temperature from Fahrenheit to Celsius:

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

To convert temperature from Celsius to Fahrenheit:

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

To estimate temperature from Fahrenheit to Celsius: \[C=\frac{F-30}{2}\]

To estimate temperature from Celsius to Fahrenheit: \[F=2C+30\]

Projects

  1. Take a favorite recipe that uses customary measures and convert the measures and cooking temperature to the metric system.
  2. Find a recipe that uses metric measures and convert the measures and cooking temperature to the U.S. Customary System of Measurement, using cups, tablespoons, or teaspoons as required.
  3. What did you observe? Was it easier to convert from one system to another? Which system allows for more precise measurements? What kitchen tools would you need in your kitchen if you used the metric system?
  1. Compare the average gas price in California to the average gas price in Puerto Rico.
  2. What conversions did you need to make to do the comparison?
  3. Do you think that the price of the gasoline is affected by the units in which it is sold?
  1. What system of measurement is used for track and field events? Why do you think this system is used?
  2. What system of measurement is used for football? Why do you think this system is used?
  3. Research various sports records. Which units of measurement is used? What do you think influenced the unit of measure used?

Practice (9)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. A recipe calls for the oven to be set to 392 °F. What is the temperature in Celsius?

    Giải đáp

    Use the formula in to convert from Fahrenheit to Celsius.

    \[\begin{array}{lll}C & = & \frac{5}{9}(F-32) \\ C & = & \frac{5}{9}(392-32) \\ C & = & \frac{5}{9}(360) \\ C & = & 200\end{array}\]

    So, 392 °F is equivalent to 200 °C.

  2. On a sunny afternoon in May, the temperature in London was 20 °C. What was the temperature in degrees Fahrenheit?

    Giải đáp

    Use the formula in to convert from Celsius to Fahrenheit.

    \[\begin{array}{lll}F & = & \frac{9}{5}C+32 \\ F & = & \frac{9}{5}(20)+32 \\ F & = & 36+32 \\ F & = & 68\end{array}\]

    The temperature was 68 °F.

  3. A manufacturer requires a vaccine to be stored in a refrigerator at temperatures between 36 °F and 46 °F. The refrigerator in the local pharmacy cools to 3 °C. Can the vaccine be stored safely in the pharmacy’s refrigerator?

    Giải đáp

    Use the formula in to convert from Celsius to Fahrenheit.

    \[\begin{array}{lll}F & = & \frac{9}{5}C+32 \\ F & = & \frac{9}{5}(3)+32 \\ F & = & 5.4+32 \\ F & = & 37.4\end{array}\]

    Then, compare the temperatures.

    \[36F^{\circ}<37.4F^{\circ}<46F^{\circ}\]

    Yes. 37.4 °F falls within the acceptable range to store the vaccine, so it can be stored safely in the pharmacy’s refrigerator.

  4. Which is the more reasonable value for the temperature of a freezer?

    • 5 °C or
    • –5 °C?
    Giải đáp

    We know that water freezes at 0 °C. So, the more reasonable value for the temperature of a freezer is −5 °C, which is below 0 °C. At temperature of 5 °C is above freezing.

  5. The average body temperature is generally accepted as 98.6 °F. What is a reasonable value for the average body temperature in degrees Celsius:

    • 98.6 °C,
    • 64.3 °C, or
    • 34.3 °C?
    Giải đáp

    To estimate the average body temperature in degrees Celsius, subtract 30 from the temperature in degrees Fahrenheit, and divide the result by 2.

    \[\frac{(98.6-30)}{2}=\frac{68.6}{2}=34.3\]

    A reasonable value for average body temperature is 34.3 °C.

  6. Which is a reasonable temperature for storing chocolate:

    • 28 °C,
    • 18 °C, or
    • 2 °C?
    Giải đáp

    Use the formula in to determine the temperature in degrees Fahrenheit.

    \[\begin{array}{lll}F & = & \frac{9}{5}C+32 \\ F & = & \frac{9}{5}(28)+32=82.4 \\ F & = & \frac{9}{5}(18)+32=64.4 \\ F & = & \frac{9}{5}(2)+32=35.6\end{array}\]

    A temperature of 82.4 °F would be too hot, causing the chocolate to melt. A temperature of 35.6 °F is very close to freezing, which would affect the look and feel of the chocolate. So, a reasonable temperature for storing chocolate is 18 °C, or 64.4 °F.

  7. The temperature in the refrigerator is 4 °C. The temperature in the freezer is 21 °C lower. What is the temperature in the freezer?

    Giải đáp

    Use subtraction to find the difference.

    \[4-21=-17\]

    So, the temperature in the freezer is −17 °C.

  8. A scientist was using a liquid that was 35 °C. They needed to heat the liquid to raise the temperature by 6 °C. What was the temperature after the scientist heated it?

    Giải đáp

    Use addition to find the new temperature.

    \[35+6=41\]

    The temperature of the liquid was 41 °C after the scientist heated it.

  9. The optimum temperature for a chemical compound to develop its unique properties is 392 °F. When the heating process begins, the temperature of the compound is 20 °C. For safety purposes the compound can only be heated 9 °C every 15 minutes. How long until the compound reaches its optimum temperature?

    Giải đáp

    Step 1: Determine the optimum temperature in degrees Celsius using the formula in .

    \[\begin{array}{lll}C & = & \frac{5}{9}(F-32) \\ C & = & \frac{5}{9}(392-32) \\ C & = & \frac{5}{9}(360) \\ C & = & 200\end{array}\]

    Step 2: Subtract the starting temperature.

    \[200C^{\circ}-20C^{\circ}=180C^{\circ}\]

    Step 3: Determine the number of 15-minute cycles needed to heat the compound to its optimum temperature.

    \[180\div 9=20\]

    Step 4: Multiply the number of cycles needed by 15 minutes and convert the product to hours and minutes.

    \[\begin{array}{l}15\text{minutes}\times 9=135\text{minutes} \\ 135\text{minutes}=2\text{hours}15\text{minutes}\end{array}\]

    So, it will take 2 hours and 15 minutes for the compound to reach its optimum temperature.

Symbols used here

^\circ
degrees
1/360 of a full turn. 180° = π radians.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\approx
approximately equal
Equal to the precision shown, not exactly.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
a \bmod n
remainder
What is left after dividing a by n.
\%
per cent
Per hundred: 15% = 15/100.
a : b,\ \frac{a}{b}
ratio, fraction
a for every b; a divided by b.

How to: Measuring Temperature

  1. Convert between Fahrenheit and Celsius.
  2. Identify reasonable values for temperature applications.
  3. Solve application problems involving temperature.
  4. 5 °C or
  5. –5 °C?
  6. 98.6 °C,
  7. 64.3 °C, or
  8. 34.3 °C?

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Thử đi.

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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