Systems of Measurement

Make Unit Conversions in the U.S. System

There are two systems of measurement commonly used around the world. Most countries use the metric system. The U.S. uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.

The U.S. system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart, and gallons. Both the U.S. system and the metric system measure time in seconds, minutes, and hours.

The equivalencies of measurements are shown in . The table also shows, in parentheses, the common abbreviations for each measurement.

U.S. System of Measurement
\(\begin{array}{lllllllll}\text{Length} & & \begin{array}{lll}1\ \text{foot}\ \text{(ft.)} & = & 12\ \text{inches}\ \text{(in.)} \\ 1\ \text{yard}\ \text{(yd.)} & = & 3\ \text{feet}\ \text{(ft.)} \\ 1\ \text{mile}\ \text{(mi.)} & = & 5,280\ \text{feet}\ \text{(ft.)}\end{array}\end{array}\)	\(\begin{array}{lllllllllllllll}\text{Volume} & & \begin{array}{lll}3\ \text{teaspoons}\ \text{(t)} & = & 1\ \text{tablespoon}\ \text{(T)} \\ \text{16 tablespoons}\ \text{(T)} & = & \text{1 cup}\ \text{(C)} \\ \text{1 cup}\ \text{(C)} & = & \text{8 fluid ounces}\ \text{(fl. oz.)} \\ \text{1 pint}\ \text{(pt.)} & = & \text{2 cups}\ \text{(C)} \\ \text{1 quart}\ \text{(qt.)} & = & \text{2 pints}\ \text{(pt.)} \\ \text{1 gallon}\ \text{(gal)} & = & \text{4 quarts}\ \text{(qt.)}\end{array}\end{array}\)
\(\begin{array}{lllllll}\text{Weight} & & \begin{array}{lll}\text{1 pound}\ \text{(lb.)} & = & \text{16 ounces}\ \text{(oz.)} \\ \text{1 ton} & = & \text{2000 pounds}\ \text{(lb.)}\end{array}\end{array}\)	\(\begin{array}{lllllllllllll}\text{Time} & & \ \begin{array}{lll}\text{1 minute}\ \text{(min)} & = & \text{60 seconds}\ \text{(sec)} \\ \text{1 hour}\ \text{(hr)} & = & \text{60 minutes}\ \text{(min)} \\ \text{1 day} & = & \text{24 hours}\ \text{(hr)} \\ \text{1 week}\ \text{(wk)} & = & \text{7 days} \\ \text{1 year}\ \text{(yr)} & = & \text{365 days}\end{array}\end{array}\)

In many real-life applications, we need to convert between units of measurement, such as feet and yards, minutes and seconds, quarts and gallons, etc. We will use the identity property of multiplication to do these conversions. We’ll restate the identity property of multiplication here for easy reference.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Use Mixed Units of Measurement in the U.S. System

We often use mixed units of measurement in everyday situations. Suppose Joe is 5 feet 10 inches tall, stays at work for 7 hours and 45 minutes, and then eats a 1 pound 2 ounce steak for dinner—all these measurements have mixed units.

Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units!

Example

Try it.

Seymour bought three steaks for a barbecue. Their weights were 14 ounces, 1 pound 2 ounces and 1 pound 6 ounces. How many total pounds of steak did he buy?

Solution

We will add the weights of the steaks to find the total weight of the steaks.

Add the ounces. Then add the pounds.
Convert 22 ounces to pounds and ounces.	\(\\)2 pounds + 1 pound, 6 ounces
Add the pounds.	\(\\)3 pounds, 6 ounces
	Seymour bought 3 pounds 6 ounces of steak.

Example

Try it.

Anthony bought four planks of wood that were each 6 feet 4 inches long. What is the total length of the wood he purchased?

Solution

We will multiply the length of one plank to find the total length.

Multiply the inches and then the feet.
Convert the 16 inches to feet. Add the feet.
	Anthony bought 25 feet and 4 inches of wood.

Make Unit Conversions in the Metric System

In the metric system, units are related by powers of 10. The roots words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is 1,000 meters; the prefix kilo means thousand. One centimeter is \(\frac{1}{100}\) of a meter, just like one cent is \(\frac{1}{100}\) of one dollar.

The equivalencies of measurements in the metric system are shown in . The common abbreviations for each measurement are given in parentheses.

Metric System of Measurement
Length	Mass	Capacity
1 kilometer (km) = 1,000 m 1 hectometer (hm) = 100 m 1 dekameter (dam) = 10 m 1 meter (m) = 1 m 1 decimeter (dm) = 0.1 m 1 centimeter (cm) = 0.01 m 1 millimeter (mm) = 0.001 m	1 kilogram (kg) = 1,000 g 1 hectogram (hg) = 100 g 1 dekagram (dag) = 10 g 1 gram (g) = 1 g 1 decigram (dg) = 0.1 g 1 centigram (cg) = 0.01 g 1 milligram (mg) = 0.001 g	1 kiloliter (kL) = 1,000 L 1 hectoliter (hL) = 100 L 1 dekaliter (daL) = 10 L 1 liter (L) = 1 L 1 deciliter (dL) = 0.1 L 1 centiliter (cL) = 0.01 L 1 milliliter (mL) = 0.001 L
1 meter = 100 centimeters 1 meter = 1,000 millimeters	1 gram = 100 centigrams 1 gram = 1,000 milligrams	1 liter = 100 centiliters 1 liter = 1,000 milliliters

To make conversions in the metric system, we will use the same technique we did in the US system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.

Have you ever run a 5K or 10K race? The length of those races are measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.

Example

Try it.

Nick ran a 10K race. How many meters did he run?

Solution

We will convert kilometers to meters using the identity property of multiplication.

	10 kilometers
Multiply the measurement to be converted by 1.
Write 1 as a fraction relating kilometers and meters.
Simplify.
Multiply.	10,000 meters
	Nick ran 10,000 meters.

Example

Try it.

Eleanor’s newborn baby weighed 3,200 grams. How many kilograms did the baby weigh?

Solution

We will convert grams into kilograms.

Multiply the measurement to be converted by 1.
Write 1 as a function relating kilograms and grams.
Simplify.
Multiply.	\(\frac{\text{3,200 kilograms}}{1,000}\)
Divide.	3.2 kilograms The baby weighed 3.2 kilograms.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Use Mixed Units of Measurement in the Metric System

Performing arithmetic operations on measurements with mixed units of measures in the metric system requires the same care we used in the US system. But it may be easier because of the relation of the units to the powers of 10. Make sure to add or subtract like units.

Example

Try it.

Ryland is 1.6 meters tall. His younger brother is 85 centimeters tall. How much taller is Ryland than his younger brother?

Solution

We can convert both measurements to either centimeters or meters. Since meters is the larger unit, we will subtract the lengths in meters. We convert 85 centimeters to meters by moving the decimal 2 places to the left.

Write the 85 centimeters as meters.

\(\begin{array}{l}\ 1.60\ \text{m} \\ \underset{\text{_______}}{-0.85\ \text{m}} \\ 0.75\ \text{m}\end{array}\)

Ryland is \(\text{0.75 m}\) taller than his brother.

Example

Try it.

Dena’s recipe for lentil soup calls for 150 milliliters of olive oil. Dena wants to triple the recipe. How many liters of olive oil will she need?

Solution

We will find the amount of olive oil in millileters then convert to liters.

	Triple \(\text{150 mL}\)
Translate to algebra.	\(3\cdot 150\ \text{mL}\)
Multiply.	\(\text{450 mL}\)
Convert to liters.	\(450\cdot \frac{0.001\ \text{L}}{1\ \text{mL}}\)
Simplify.	\(\text{0.45 L}\)
	Dena needs 0.45 liters of olive oil.

Convert Between the U.S. and the Metric Systems of Measurement

Many measurements in the United States are made in metric units. Our soda may come in 2-liter bottles, our calcium may come in 500-mg capsules, and we may run a 5K race. To work easily in both systems, we need to be able to convert between the two systems.

shows some of the most common conversions.

Conversion Factors Between U.S. and Metric Systems
Length	Mass	Capacity
\(\begin{array}{lllllllllll}\begin{array}{lll}\text{1 in.} & = & \text{2.54 cm} \\ \text{1 ft.} & = & \text{0.305 m} \\ \text{1 yd.} & = & \text{0.914 m} \\ \text{1 mi.} & = & \text{1.61 km} \\ \text{1 m} & = & \text{3.28 ft.}\end{array}\end{array}\)	\(\begin{array}{lllllll}\begin{array}{lll}\text{1 lb.} & = & \text{0.45 kg} \\ \text{1 oz.} & = & \text{28 g} \\ \text{1 kg} & = & \text{2.2 lb.}\end{array}\end{array}\)	\(\begin{array}{lllllll}\begin{array}{lll}\text{1 qt.} & = & \text{0.95 L} \\ \text{1 fl. oz.} & = & \text{30 mL} \\ \text{1 L} & = & \text{1.06 qt.}\end{array}\end{array}\)

shows how inches and centimeters are related on a ruler.

shows the ounce and milliliter markings on a measuring cup.

shows how pounds and kilograms marked on a bathroom scale.

We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors.

Example

Try it.

Lee’s water bottle holds 500 mL of water. How many ounces are in the bottle? Round to the nearest tenth of an ounce.

Solution

	\(\text{500 mL}\)
Multiply by a unit conversion factor relating mL and ounces.	\(500\ \text{milliliters}\cdot \frac{1\ \text{ounce}}{30\ \text{milliliters}}\)
Simplify.	\(\frac{\text{500 ounce}}{30}\)
Divide.	\(\text{16.7 ounces.}\)
	The water bottle has 16.7 ounces.

Example

Try it.

Soleil was on a road trip and saw a sign that said the next rest stop was in 100 kilometers. How many miles until the next rest stop?

Solution

	\(\text{100 kilometers}\)
Multiply by a unit conversion factor relating km and mi.	\(100\ \text{kilometers}\cdot \frac{1\ \text{mile}}{1.61\ \text{kilometer}}\)
Simplify.	\(\frac{\text{100 miles}}{1.61}\)
Divide.	\(\text{62 miles.}\)
	Soleil will travel 62 miles.

Convert between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for \(22\text{^{\circ}}\text{C},\) what does that mean?

The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written \(\text{^{\circ}}\text{F}\text{.}\) The metric system uses degrees Celsius, written \(\text{^{\circ}}\text{C}\text{.}\) shows the relationship between the two systems.

Example

Try it.

Convert \(50\text{^{\circ}}\) Fahrenheit into degrees Celsius.

Solution

We will substitute \(50\text{^{\circ}}\text{F}\) into the formula to find C.

Simplify in parentheses.
Multiply.
	So we found that 50°F is equivalent to 10°C.

Example

Try it.

While visiting Paris, Woody saw the temperature was \(20\text{^{\circ}}\) Celsius. Convert the temperature into degrees Fahrenheit.

Solution

We will substitute \(20\text{^{\circ}}\text{C}\) into the formula to find F.

Multiply.
Add.
	So we found that 20°C is equivalent to 68°F.

Key Concepts

• Metric System of Measurement
  • Length
    \(\begin{array}{lll}\text{1 kilometer (km)} & = & \text{1,000 m} \\ \text{1 hectometer (hm)} & = & \text{100 m} \\ \text{1 dekameter (dam)} & = & \text{10 m} \\ \text{1 meter (m)} & = & \text{1 m} \\ \text{1 decimeter (dm)} & = & \text{0.1 m} \\ \text{1 centimeter (cm)} & = & \text{0.01 m} \\ \text{1 millimeter (mm)} & = & \text{0.001 m} \\ \text{1 meter} & = & \text{100 centimeters} \\ \text{1 meter} & = & \text{1,000 millimeters}\end{array}\)
  • Mass
    \(\begin{array}{lll}\text{1 kilogram (kg)} & = & \text{1,000 g} \\ \text{1 hectogram (hg)} & = & \text{100 g} \\ \text{1 dekagram (dag)} & = & \text{10 g} \\ \text{1 gram (g)} & = & \text{1 g} \\ \text{1 decigram (dg)} & = & \text{0.1 g} \\ \text{1 centigram (cg)} & = & \text{0.01 g} \\ \text{1 milligram (mg)} & = & \text{0.001 g} \\ \text{1 gram} & = & \text{100 centigrams} \\ \text{1 gram} & = & \text{1,000 milligrams}\end{array}\)
  • Capacity
    \(\begin{array}{lll}\text{1 kiloliter (kL)} & = & \text{1,000 L} \\ \text{1 hectoliter (hL)} & = & \text{100 L} \\ \text{1 dekaliter (daL)} & = & \text{10 L} \\ \text{1 liter (L)} & = & \text{1 L} \\ \text{1 deciliter (dL)} & = & \text{0.1 L} \\ \text{1 centiliter (cL)} & = & \text{0.01 L} \\ \text{1 milliliter (mL)} & = & \text{0.001 L} \\ \text{1 liter} & = & \text{100 centiliters} \\ \text{1 liter} & = & \text{1,000 milliliters}\end{array}\)
• Temperature Conversion
  • To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula \(\text{C}=\frac{5}{9}(\text{F}-32)\)
  • To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula \(\text{F}=\frac{9}{5}\text{C}+32\)

Practice (40)

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

1. MaryAnne is 66 inches tall. Convert her height into feet.

2. Lexie is 30 inches tall. Convert her height to feet.

   Avslöja svaret

   2.5 feet

3. Rene bought a hose that is 18 yards long. Convert the length to feet.

   Avslöja svaret

   54 feet

4. Ndula, an elephant at the San Diego Safari Park, weighs almost 3.2 tons. Convert her weight to pounds.

   Avslöja svaret

   We will convert 3.2 tons into pounds. We will use the identity property of multiplication, writing 1 as the fraction \(\frac{2000\ \text{pounds}}{1\ \text{ton}}.\)
   

   	\(\text{3.2 tons}\)
   Multiply the measurement to be converted, by 1.	\(\text{3.2 tons}⋅1\)
   Write 1 as a fraction relating tons and pounds.	\(\text{3.2 tons}⋅\frac{\text{2,000 pounds}}{\text{1 ton}}\)
   Simplify.
   Multiply.	\(\text{6,400 pounds}\)
   	Ndula weighs almost 6,400 pounds.

5. Arnold’s SUV weighs about 4.3 tons. Convert the weight to pounds.

   Avslöja svaret

   8,600 pounds

6. The Carnival Destiny cruise ship weighs 51,000 tons. Convert the weight to pounds.

   Avslöja svaret

   102,000,000 pounds

7. Juliet is going with her family to their summer home. She will be away from her boyfriend for 9 weeks. Convert the time to minutes.

   Avslöja svaret

   To convert weeks into minutes we will convert weeks into days, days into hours, and then hours into minutes. To do this we will multiply by conversion factors of 1.
   

   	9 weeks
   Write \(1\) as \(\frac{\text{7 days}}{\text{1 week}}\), \(\frac{\text{24 hours}}{\text{1 day}}\), and \(\frac{\text{60 minutes}}{\text{1 hour}}\).	\(\frac{\text{9 wk}}{1}⋅\frac{\text{7 days}}{\text{1 wk}}⋅\frac{\text{24 hr}}{\text{1 day}}⋅\frac{\text{60 min}}{\text{1 hr}}\)
   Divide out the common units.
   Multiply.	\(\frac{9⋅7⋅24⋅\text{60 min}}{1⋅1⋅1⋅1}\)
   Multiply.	\(\text{90,720 min}\)

   Juliet and her boyfriend will be apart for 90,720 minutes (although it may seem like an eternity!).

8. The distance between the earth and the moon is about 250,000 miles. Convert this length to yards.

   Avslöja svaret

   440,000,000 yards

9. The astronauts of Expedition 28 on the International Space Station spend 15 weeks in space. Convert the time to minutes.

   Avslöja svaret

   151,200 minutes

10. How many ounces are in 1 gallon?

    Avslöja svaret

    We will convert gallons to ounces by multiplying by several conversion factors. Refer to .
    

    	1 gallon
    Multiply the measurement to be converted by 1.	\(\frac{\text{1 gallon}}{1}⋅\frac{\text{4 quarts}}{\text{1 gallon}}⋅\frac{\text{2 pints}}{\text{1 quart}}⋅\frac{\text{2 cups}}{\text{1 pint}}⋅\frac{\text{8 ounces}}{\text{1 cup}}\)
    Use conversion factors to get to the right unit. Simplify.
    Multiply.	\(\frac{1⋅4⋅2⋅2⋅\text{8 ounces}}{1⋅1⋅1⋅1⋅1}\)
    Simplify.	\(\text{128 ounces}\) There are 128 ounces in a gallon.

11. How many cups are in 1 gallon?

    Avslöja svaret

    16 cups

12. How many teaspoons are in 1 cup?

    Avslöja svaret

    48 teaspoons

13. Seymour bought three steaks for a barbecue. Their weights were 14 ounces, 1 pound 2 ounces and 1 pound 6 ounces. How many total pounds of steak did he buy?

    Avslöja svaret

    We will add the weights of the steaks to find the total weight of the steaks.
    

    Add the ounces. Then add the pounds.
    Convert 22 ounces to pounds and ounces.	\(\\)2 pounds + 1 pound, 6 ounces
    Add the pounds.	\(\\)3 pounds, 6 ounces
    	Seymour bought 3 pounds 6 ounces of steak.

14. Laura gave birth to triplets weighing 3 pounds 3 ounces, 3 pounds 3 ounces, and 2 pounds 9 ounces. What was the total birth weight of the three babies?

    Avslöja svaret

    \(\text{8 lbs}.\ \text{15 oz}\)

15. Stan cut two pieces of crown molding for his family room that were 8 feet 7 inches and 12 feet 11 inches. What was the total length of the molding?

    Avslöja svaret

    21 ft. 6 in.

16. Anthony bought four planks of wood that were each 6 feet 4 inches long. What is the total length of the wood he purchased?

    Avslöja svaret

    We will multiply the length of one plank to find the total length.
    

    Multiply the inches and then the feet.
    Convert the 16 inches to feet. Add the feet.
    	Anthony bought 25 feet and 4 inches of wood.

17. Henri wants to triple his spaghetti sauce recipe that uses 1 pound 8 ounces of ground turkey. How many pounds of ground turkey will he need?

    Avslöja svaret

    4 lbs. 8 oz.

18. Joellen wants to double a solution of 5 gallons 3 quarts. How many gallons of solution will she have in all?

    Avslöja svaret

    11 gal. 2 qt.

19. Nick ran a 10K race. How many meters did he run?

    Avslöja svaret

    We will convert kilometers to meters using the identity property of multiplication.
    

    	10 kilometers
    Multiply the measurement to be converted by 1.
    Write 1 as a fraction relating kilometers and meters.
    Simplify.
    Multiply.	10,000 meters
    	Nick ran 10,000 meters.

20. Sandy completed her first 5K race! How many meters did she run?

    Avslöja svaret

    5,000 meters

21. Herman bought a rug 2.5 meters in length. How many centimeters is the length?

    Avslöja svaret

    250 centimeters

22. Eleanor’s newborn baby weighed 3,200 grams. How many kilograms did the baby weigh?

    Avslöja svaret

    We will convert grams into kilograms.
    

    Multiply the measurement to be converted by 1.
    Write 1 as a function relating kilograms and grams.
    Simplify.
    Multiply.	\(\frac{\text{3,200 kilograms}}{1,000}\)
    Divide.	3.2 kilograms The baby weighed 3.2 kilograms.

23. Kari’s newborn baby weighed 2,800 grams. How many kilograms did the baby weigh?

    Avslöja svaret

    2.8 kilograms

24. Anderson received a package that was marked 4,500 grams. How many kilograms did this package weigh?

    Avslöja svaret

    4.5 kilograms

25. Convert ⓐ 350 L to kiloliters ⓑ 4.1 L to milliliters.

    Avslöja svaret

    1. ⓐ We will convert liters to kiloliters. In , we see that \(1\ \text{kiloliter}=\text{1,000 liters.}\)
       

       	\(\text{350 L}\)
       Multiply by 1, writing 1 as a fraction relating liters to kiloliters.	\(\text{350 L}⋅\frac{\text{1 kL}}{\text{1,000 L}}\)
       Simplify.	\(350\ \text{L}⋅\frac{\text{1 kL}}{1,000\ \text{L}}\)
       	\(\text{0.35 kL}\)

       
       
       
    2. ⓑ We will convert liters to milliliters. From we see that \(\text{1 liter}\ =\ \text{1,000 milliliters.}\)
       

       Multiply by 1, writing 1 as a fraction relating liters to milliliters.
       Simplify.
       Move the decimal 3 units to the right.

26. Convert: ⓐ 725 L to kiloliters ⓑ 6.3 L to milliliters

    Avslöja svaret

    ⓐ 0.725 kiloliters ⓑ 6,300 milliliters

27. Convert: ⓐ 350 hL to liters ⓑ 4.1 L to centiliters

    Avslöja svaret

    ⓐ 35,000 liters ⓑ 410 centiliters

28. Ryland is 1.6 meters tall. His younger brother is 85 centimeters tall. How much taller is Ryland than his younger brother?

    Avslöja svaret

    We can convert both measurements to either centimeters or meters. Since meters is the larger unit, we will subtract the lengths in meters. We convert 85 centimeters to meters by moving the decimal 2 places to the left.

    Write the 85 centimeters as meters.

    \(\begin{array}{l}\ 1.60\ \text{m} \\ \underset{\text{_______}}{-0.85\ \text{m}} \\ 0.75\ \text{m}\end{array}\)

    Ryland is \(\text{0.75 m}\) taller than his brother.

29. Mariella is 1.58 meters tall. Her daughter is 75 centimeters tall. How much taller is Mariella than her daughter? Write the answer in centimeters.

    Avslöja svaret

    83 centimeters

30. The fence around Hank’s yard is 2 meters high. Hank is 96 centimeters tall. How much shorter than the fence is Hank? Write the answer in meters.

    Avslöja svaret

    1.04 meters

31. Dena’s recipe for lentil soup calls for 150 milliliters of olive oil. Dena wants to triple the recipe. How many liters of olive oil will she need?

    Avslöja svaret

    We will find the amount of olive oil in millileters then convert to liters.

    	Triple \(\text{150 mL}\)
    Translate to algebra.	\(3\cdot 150\ \text{mL}\)
    Multiply.	\(\text{450 mL}\)
    Convert to liters.	\(450\cdot \frac{0.001\ \text{L}}{1\ \text{mL}}\)
    Simplify.	\(\text{0.45 L}\)
    	Dena needs 0.45 liters of olive oil.

32. A recipe for Alfredo sauce calls for 250 milliliters of milk. Renata is making pasta with Alfredo sauce for a big party and needs to multiply the recipe amounts by 8. How many liters of milk will she need?

    Avslöja svaret

    2 liters

33. To make one pan of baklava, Dorothea needs 400 grams of filo pastry. If Dorothea plans to make 6 pans of baklava, how many kilograms of filo pastry will she need?

    Avslöja svaret

    2.4 kilograms

34. Lee’s water bottle holds 500 mL of water. How many ounces are in the bottle? Round to the nearest tenth of an ounce.

    Avslöja svaret

    	\(\text{500 mL}\)
    Multiply by a unit conversion factor relating mL and ounces.	\(500\ \text{milliliters}\cdot \frac{1\ \text{ounce}}{30\ \text{milliliters}}\)
    Simplify.	\(\frac{\text{500 ounce}}{30}\)
    Divide.	\(\text{16.7 ounces.}\)
    	The water bottle has 16.7 ounces.

35. How many quarts of soda are in a 2-L bottle?

    Avslöja svaret

    2.12 quarts

36. How many liters are in 4 quarts of milk?

    Avslöja svaret

    3.8 liters

37. Soleil was on a road trip and saw a sign that said the next rest stop was in 100 kilometers. How many miles until the next rest stop?

    Avslöja svaret

    	\(\text{100 kilometers}\)
    Multiply by a unit conversion factor relating km and mi.	\(100\ \text{kilometers}\cdot \frac{1\ \text{mile}}{1.61\ \text{kilometer}}\)
    Simplify.	\(\frac{\text{100 miles}}{1.61}\)
    Divide.	\(\text{62 miles.}\)
    	Soleil will travel 62 miles.

38. The height of Mount Kilimanjaro is 5,895 meters. Convert the height to feet.

    Avslöja svaret

    19,336 feet

39. The flight distance from New York City to London is 5,586 kilometers. Convert the distance to miles.

    Avslöja svaret

    3469.57 miles

40. Convert \(50\text{^{\circ}}\) Fahrenheit into degrees Celsius.

    Avslöja svaret

    We will substitute \(50\text{^{\circ}}\text{F}\) into the formula to find C.
    

    Simplify in parentheses.
    Multiply.
    	So we found that 50°F is equivalent to 10°C.

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.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

i

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Systems of Measurement

1. Make unit conversions in the US system
2. Use mixed units of measurement in the US system
3. Make unit conversions in the metric system
4. Use mixed units of measurement in the metric system
5. Convert between the US and the metric systems of measurement
6. Convert between Fahrenheit and Celsius temperatures
7. Multiply the measurement to be converted by 1; write 1 as a fraction relating the units given and the units needed.
8. Multiply.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Prova själv

⌨ Type ✎ Draw

Dina matteproblem

Lös

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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