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Mathematics LibreTexts

7.5: Systems of Measurement (Part 2)

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 U.S. system. But it may be easier because of the relation of the units to the powers of 10. We still must make sure to add or subtract like units.

Example 7.53:

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

Solution

We will subtract the lengths in meters. Convert 85 centimeters to meters by moving the decimal 2 places to the left; 85 cm is the same as 0.85 m.

Now that both measurements are in meters, subtract to find out how much taller Ryland is than his brother.

$$\begin{split} 1.60\; &m \\ -\; 0.85\; &m \\ \hline 0.75\; &m \end{split}$$

Ryland is 0.75 meters taller than his brother.

Exercise 7.105:

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.

Exercise 7.106:

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.

Example 7.54:

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 milliliters then convert to liters.

Translate to algebra. 3 • 150 mL
Multiply. 450 mL
Convert to liters. $$450\; mL\; \cdot \frac{0.001\; L}{1\; mL}$$
Simplify. 0.45 L

Dena needs 0.45 liter of olive oil.

Exercise 7.107:

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?

Exercise 7.108:

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?

Convert Between U.S. and Metric Systems of Measurement

Many measurements in the United States are made in metric units. A drink may come in 2-liter bottles, calcium may come in 500-mg capsules, and we may run a 5-K race. To work easily in both systems, we need to be able to convert between the two systems. Table 7.69 shows some of the most common conversions.

Table 7.69

Conversion Factors Between U.S. and Metric Systems
Length Weight Volume

1 in = 2.54 cm

1 ft = 0.305 m

1 yd = 0.914 m

1 mi = 1.61 km

1 lb = 0.45 kg

1 oz = 28 g

1 qt = 0.95 L

1 fl oz = 30 mL

1 m = 3.28 ft 1 kg = 2.2 lb 1 L = 1.06 qt

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

Example 7.55:

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

Solution
Multiply by a unit conversion factor relating mL and ounces. $$500\; mL\; \cdot \frac{1\; fl\; oz}{30\; mL} \tag{7.5.29}$$
Simplify. $$\frac{500\; fl\; oz}{30} \tag{7.5.30}$$
Divide. 16.7 fl. oz

The water bottle holds 16.7 fluid ounces.

Exercise 7.109:

How many quarts of soda are in a 2-liter bottle?

Exercise 7.110:

How many liters are in 4 quarts of milk?

The conversion factors in Table 7.69 are not exact, but the approximations they give are close enough for everyday purposes. In Example 7.55, we rounded the number of fluid ounces to the nearest tenth.

Example 7.56:

Soleil lives in Minnesota but often travels in Canada for work. While driving on a Canadian highway, she passes a sign that says the next rest stop is in 100 kilometers. How many miles until the next rest stop? Round your answer to the nearest mile.

Solution
Multiply by a unit conversion factor relating kilometers and miles. $$100\; kilometers\; \cdot \frac{1\; mile}{1.61\; kilometers} \tag{7.5.31}$$
Simplify. $$100 \cdot \frac{1\; mi}{1.61\; km} \tag{7.5.32}$$
Divide. 62 mi

It is about 62 miles to the next rest stop.

Exercise 7.111:

The height of Mount Kilimanjaro is 5,895 meters. Convert the height to feet. Round to the nearest foot.

Exercise 7.112:

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

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°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 °F. The metric system uses degrees Celsius, written °C. Figure 7.9 shows the relationship between the two systems.

Figure 7.9 - A temperature of 37°C is equivalent to 98.6°F.

If we know the temperature in one system, we can use a formula to convert it to the other system.

Definition: Temperature Conversion

To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula

$$C = \frac{5}{9} (F − 32) \tag{7.5.33}$$

To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula

$$F = \frac{9}{5} C + 32 \tag{7.5.34}$$

Example 7.57:

Convert 50°F into degrees Celsius.

Solution

We will substitute 50°F into the formula to find C.

Use the formula for converting °F to °C $$C = \frac{5}{9} (F − 32) \tag{7.5.35}$$
Substitute \(\textcolor{red}{50}\) for F. $$C = \frac{5}{9} (\textcolor{red}{50} − 32) \tag{7.5.36}$$
Simplify in parentheses. $$C = \frac{5}{9} (18) \tag{7.5.37}$$
Multiply. $$C = 10 \tag{7.5.38}$$

A temperature of 50°F is equivalent to 10°C.

Exercise 7.113:

Convert the Fahrenheit temperatures to degrees Celsius: 59°F.

Exercise 7.114:

Convert the Fahrenheit temperatures to degrees Celsius: 41°F.

Example 7.58:

The weather forecast for Paris predicts a high of 20°C. Convert the temperature into degrees Fahrenheit.

Solution

We will substitute 20°C into the formula to find F.

Use the formula for converting °F to °C $$F = \frac{9}{5} C + 32 \tag{7.5.39}$$
Substitute \(\textcolor{red}{20}\) $$F = \frac{9}{5} (\textcolor{red}{20}) + 32 \tag{7.5.40}$$
Multiply. $$F = 36 + 32 \tag{7.5.41}$$
Add. $$F = 68 \tag{7.5.42}$$

So 20°C is equivalent to 68°F.

Exercise 7.115:

Convert the Celsius temperatures to degrees Fahrenheit: The temperature in Helsinki, Finland was 15°C.

Exercise 7.116:

Convert the Celsius temperatures to degrees Fahrenheit: The temperature in Sydney, Australia was 10°C.

Practice  Makes Perfect

Make Unit Conversions in the U.S. System

In the following exercises, convert the units.

  1. A park bench is 6 feet long. Convert the length to inches.
  2. A floor tile is 2 feet wide. Convert the width to inches.
  3. A ribbon is 18 inches long. Convert the length to feet.
  4. Carson is 45 inches tall. Convert his height to feet.
  5. Jon is 6 feet 4 inches tall. Convert his height to inches.
  6. Faye is 4 feet 10 inches tall. Convert her height to inches.
  7. A football field is 160 feet wide. Convert the width to yards.
  8. On a baseball diamond, the distance from home plate to first base is 30 yards. Convert the distance to feet.
  9. Ulises lives 1.5 miles from school. Convert the distance to feet.
  10. Denver, Colorado, is 5,183 feet above sea level. Convert the height to miles.
  11. A killer whale weighs 4.6 tons. Convert the weight to pounds.
  12. Blue whales can weigh as much as 150 tons. Convert the weight to pounds.
  13. An empty bus weighs 35,000 pounds. Convert the weight to tons.
  14. At take-off, an airplane weighs 220,000 pounds. Convert the weight to tons.
  15. The voyage of the Mayflower took 2 months and 5 days. Convert the time to days.
  16. Lynn’s cruise lasted 6 days and 18 hours. Convert the time to hours.
  17. Rocco waited \(1 \frac{1}{2}\) hours for his appointment. Convert the time to seconds.
  18. Misty’s surgery lasted \(2 \frac{1}{4}\) hours. Convert the time to seconds.
  19. How many teaspoons are in a pint?
  20. How many tablespoons are in a gallon?
  21. JJ’s cat, Posy, weighs 14 pounds. Convert her weight to ounces.
  22. April’s dog, Beans, weighs 8 pounds. Convert his weight to ounces.
  23. Baby Preston weighed 7 pounds 3 ounces at birth. Convert his weight to ounces.
  24. Baby Audrey weighed 6 pounds 15 ounces at birth. Convert her weight to ounces.
  25. Crista will serve 20 cups of juice at her son’s party. Convert the volume to gallons.
  26. Lance needs 500 cups of water for the runners in a race. Convert the volume to gallons.

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

In the following exercises, solve and write your answer in mixed units.

  1. Eli caught three fish. The weights of the fish were 2 pounds 4 ounces, 1 pound 11 ounces, and 4 pounds 14 ounces. What was the total weight of the three fish?
  2. Judy bought 1 pound 6 ounces of almonds, 2 pounds 3 ounces of walnuts, and 8 ounces of cashews. What was the total weight of the nuts?
  3. One day Anya kept track of the number of minutes she spent driving. She recorded trips of 45, 10, 8, 65, 20, and 35 minutes. How much time (in hours and minutes) did Anya spend driving?
  4. Last year Eric went on 6 business trips. The number of days of each was 5, 2, 8, 12, 6, and 3. How much time (in weeks and days) did Eric spend on business trips last year?
  5. Renee attached a 6-foot-6-inch extension cord to her computer’s 3-foot-8-inch power cord. What was the total length of the cords?
  6. Fawzi’s SUV is 6 feet 4 inches tall. If he puts a 2-foot-10-inch box on top of his SUV, what is the total height of the SUV and the box?
  7. Leilani wants to make 8 placemats. For each placemat she needs 18 inches of fabric. How many yards of fabric will she need for the 8 placemats?
  8. Mireille needs to cut 24 inches of ribbon for each of the 12 girls in her dance class. How many yards of ribbon will she need altogether?

Make Unit Conversions in the Metric System

In the following exercises, convert the units.

  1. Ghalib ran 5 kilometers. Convert the length to meters.
  2. Kitaka hiked 8 kilometers. Convert the length to meters.
  3. Estrella is 1.55 meters tall. Convert her height to centimeters.
  4. The width of the wading pool is 2.45 meters. Convert the width to centimeters.
  5. Mount Whitney is 3,072 meters tall. Convert the height to kilometers.
  6. The depth of the Mariana Trench is 10,911 meters. Convert the depth to kilometers.
  7. June’s multivitamin contains 1,500 milligrams of calcium. Convert this to grams.
  8. A typical ruby-throated hummingbird weights 3 grams. Convert this to milligrams.
  9. One stick of butter contains 91.6 grams of fat. Convert this to milligrams.
  10. One serving of gourmet ice cream has 25 grams of fat. Convert this to milligrams.
  11. The maximum mass of an airmail letter is 2 kilograms. Convert this to grams.
  12. Dimitri’s daughter weighed 3.8 kilograms at birth. Convert this to grams.
  13. A bottle of wine contained 750 milliliters. Convert this to liters.
  14. A bottle of medicine contained 300 milliliters. Convert this to liters.

Use Mixed Units of Measurement in the Metric System

In the following exercises, solve and write your answer in mixed units.

  1. Matthias is 1.8 meters tall. His son is 89 centimeters tall. How much taller, in centimeters, is Matthias than his son?
  2. Stavros is 1.6 meters tall. His sister is 95 centimeters tall. How much taller, in centimeters, is Stavros than his sister?
  3. A typical dove weighs 345 grams. A typical duck weighs 1.2 kilograms. What is the difference, in grams, of the weights of a duck and a dove?
  4. Concetta had a 2-kilogram bag of flour. She used 180 grams of flour to make biscotti. How many kilograms of flour are left in the bag?
  5. Harry mailed 5 packages that weighed 420 grams each. What was the total weight of the packages in kilograms?
  6. One glass of orange juice provides 560 milligrams of potassium. Linda drinks one glass of orange juice every morning. How many grams of potassium does Linda get from her orange juice in 30 days?
  7. Jonas drinks 200 milliliters of water 8 times a day. How many liters of water does Jonas drink in a day?
  8. One serving of whole grain sandwich bread provides 6 grams of protein. How many milligrams of protein are provided by 7 servings of whole grain sandwich bread?

Convert Between U.S. and Metric Systems

In the following exercises, make the unit conversions. Round to the nearest tenth.

  1. Bill is 75 inches tall. Convert his height to centimeters.
  2. Frankie is 42 inches tall. Convert his height to centimeters.
  3. Marcus passed a football 24 yards. Convert the pass length to meters.
  4. Connie bought 9 yards of fabric to make drapes. Convert the fabric length to meters.
  5. Each American throws out an average of 1,650 pounds of garbage per year. Convert this weight to kilograms.
  6. An average American will throw away 90,000 pounds of trash over his or her lifetime. Convert this weight to kilograms.
  7. A 5K run is 5 kilometers long. Convert this length to miles.
  8. Kathryn is 1.6 meters tall. Convert her height to feet.
  9. Dawn’s suitcase weighed 20 kilograms. Convert the weight to pounds.
  10. Jackson’s backpack weighs 15 kilograms. Convert the weight to pounds.
  11. Ozzie put 14 gallons of gas in his truck. Convert the volume to liters.
  12. Bernard bought 8 gallons of paint. Convert the volume to liters.

Convert between Fahrenheit and Celsius

In the following exercises, convert the Fahrenheit temperature to degrees Celsius. Round to the nearest tenth.

  1. 86°F
  2. 77°F
  3. 104°F
  4. 14°F
  5. 72°F
  6. 4°F
  7. 0°F
  8. 120°F

In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

  1. 5°C
  2. 25°C
  3. −10°C
  4. −15°C
  5. 22°C
  6. 8°C
  7. 43°C
  8. 16°C

Everyday Math

  1. Nutrition Julian drinks one can of soda every day. Each can of soda contains 40 grams of sugar. How many kilograms of sugar does Julian get from soda in 1 year?
  2. Reflectors The reflectors in each lane-marking stripe on a highway are spaced 16 yards apart. How many reflectors are needed for a one-mile-long stretch of highway?

Writing Exercises

  1. Some people think that 65° to 75° Fahrenheit is the ideal temperature range.
    1. What is your ideal temperature range? Why do you think so?
    2. Convert your ideal temperatures from Fahrenheit to Celsius.
  2. (a) Did you grow up using the U.S. customary or the metric system of measurement? (b) Describe two examples in your life when you had to convert between systems of measurement. (c) Which system do you think is easier to use? Explain.

Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

(b) Overall, after looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?

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