Systems of Measurement m81330 Systems of Measurement By the end of this section, you will be able to: Make unit conversions in the U.S. system Use mixed units of measurement in the U.S. system Make unit conversions in the metric system Use mixed units of measurement in the metric system Convert between the U.S. and the metric systems of measurement Convert between Fahrenheit and Celsius temperatures b02cd091-9e2c-47b1-bad6-680868b4ee8e Before you get started, take this readiness quiz. Multiply: 4.29(1000). If you missed this problem, review . 4,290 Simplify: 3054. If you missed this problem, review . 59 Multiply: 715·2528. If you missed this problem, review . 512 In this section we will see how to convert among different types of units, such as feet to miles or kilograms to pounds. The basic idea in all of the unit conversions will be to use a form of 1, the multiplicative identity, to change the units but not the value of a quantity.
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 United States 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, or hours.The equivalencies among the basic units of the U.S. system of measurement are listed in . The table also shows, in parentheses, the common abbreviations for each measurement. U.S. System Units Length Volume 1 foot (ft) = 12 inches (in)1 yard (yd) = 3 feet (ft)1 mile (mi) = 5280 feet (ft) 3 teaspoons (t) = 1 tablespoon (T)16 Tablespoons (T) = 1 cup (C)1 cup (C) = 8 fluid ounces (fl oz)1 pint (pt) = 2 cups (C)1 quart (qt) = 2 pints (pt)1 gallon (gal) = 4 quarts (qt) Weight Time 1 pound (lb) = 16 ounces (oz)1 ton = 2000 pounds (lb) 1 minute (min) = 60 seconds (s)1 hour (h) = 60 minutes (min)1 day = 24 hours (h)1 week (wk) = 7 days1 year (yr) = 365 days
In many real-life applications, we need to convert between units of measurement. We will use the identity property of multiplication to do these conversions. We’ll restate the Identity Property of Multiplication here for easy reference. To use the identity property of multiplication, we write 1 in a form that will help us convert the units. For example, suppose we want to convert inches to feet. We know that 1 foot is equal to 12 inches, so we can write 1 as the fraction 1 ft12 in. When we multiply by this fraction, we do not change the value but just change the units. But 12 in1 ft also equals 1. How do we decide whether to multiply by 1 ft12 in or 12 in1 ft? We choose the fraction that will make the units we want to convert from divide out. For example, suppose we wanted to convert 60 inches to feet. If we choose the fraction that has inches in the denominator, we can eliminate the inches. On the other hand, if we wanted to convert 5 feet to inches, we would choose the fraction that has feet in the denominator. We treat the unit words like factors and ‘divide out’ common units like we do common factors. Make unit conversions.Multiply the measurement to be converted by 1; write 1 as a fraction relating the units given and the units needed. Multiply. Simplify the fraction, performing the indicated operations and removing the common units. Mary Anne is 66 inches tall. What is her height in feet? Solution Convert 66 inches into feet. Multiply the measurement to be converted by 1. 66 inches ·1 Write 1 as a fraction relating the units given and the units needed. 66 inches·1 foot12 inches Multiply. 66 inches·1 foot12 inches Simplify the fraction. 66inches·1 foot12inches 66 feet12 5.5 feet
Notice that the when we simplified the fraction, we first divided out the inches. Mary Anne is 5.5 feet tall.
Lexie is 30 inches tall. Convert her height to feet. 2.5 feet Rene bought a hose that is 18 yards long. Convert the length to feet. 54 feet When we use the Identity Property of Multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator. Ndula, an elephant at the San Diego Safari Park, weighs almost 3.2 tons. Convert her weight to pounds.
(credit: Guldo Da Rozze, Flickr)
Solution We will convert 3.2 tons into pounds, using the equivalencies in . We will use the Identity Property of Multiplication, writing 1 as the fraction 2000 pounds1 ton. 3.2 tons Multiply the measurement to be converted by 1. 3.2 tons·1 Write 1 as a fraction relating tons and pounds. 3.2 tons·2000 lbs1 ton Simplify. 3.2tons·2000 lbs1ton Multiply. 6400 lbs Ndula weighs almost 6,400 pounds.
Arnold’s SUV weighs about 4.3 tons. Convert the weight to pounds. 8600 pounds A cruise ship weighs 51,000 tons. Convert the weight to pounds. 102,000,000 pounds Sometimes to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions. Juliet is going with her family to their summer home. She will be away for 9 weeks. Convert the time to minutes. Solution To convert weeks into minutes, we will convert weeks to days, days to hours, and then hours to minutes. To do this, we will multiply by conversion factors of 1. 9 weeks Write 1 as 7days1week,24hours1day,60minutes1hour. Cancel common units. Multiply. 9·7·24·60min1·1·1·1=90,720min Juliet will be away for 90,720 minutes.
The distance between Earth and the moon is about 250,000 miles. Convert this length to yards. 440,000,000 yards A team of astronauts spends 15 weeks in space. Convert the time to minutes. 151,200 minutes How many fluid ounces are in 1 gallon of milk?
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Solution Use conversion factors to get the right units: convert gallons to quarts, quarts to pints, pints to cups, and cups to fluid ounces. 1 gallon Multiply the measurement to be converted by 1. 1 gal1·4 qt1 gal·2 pt1 qt·2 C1 pt·8 fl oz1 C Simplify. 1gal1·4qt1gal·2pt1qt·2C1pt·8 fl oz1C Multiply. 1·4·2·2·8 fl oz1·1·1·1·1 Simplify. 128 fluid ounces There are 128 fluid ounces in a gallon.
How many cups are in 1 gallon? 16 cups How many teaspoons are in 1 cup? 48 teaspoons
Use Mixed Units of Measurement in the U.S. System Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units. Charlie 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?
(credit: Helen Penjam, Flickr)
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. Add the pounds. 2 pounds + 1 pound, 6 ounces 3 pounds, 6 ounces Charlie bought 3 pounds 6 ounces of steak.
Laura gave birth to triplets weighing 3 pounds 12 ounces, 3 pounds 3 ounces, and 2 pounds 9 ounces. What was the total birth weight of the three babies? 9 lbs. 8 oz Seymour 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? 21 ft. 6 in. Anthony bought four planks of wood that were each 6 feet 4 inches long. If the four planks are placed end-to-end, what is the total length of the wood? Solution We will multiply the length of one plank by 4 to find the total length. Multiply the inches and then the feet. Convert 16 inches to feet. 24 feet + 1 foot 4 inches Add the feet. 25 feet 4 inches Anthony bought 25 feet 4 inches of wood.
Henri wants to triple his spaghetti sauce recipe, which calls for 1 pound 8 ounces of ground turkey. How many pounds of ground turkey will he need? 4 lbs. 8 oz. Joellen wants to double a solution of 5 gallons 3 quarts. How many gallons of solution will she have in all? 11 gal. 2 qts.
Make Unit Conversions in the Metric System In the metric system, units are related by powers of 10. The root words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is 1000 meters; the prefix kilo- means thousand. One centimeter is 1100 of a meter, because the prefix centi- means one one-hundredth (just like one cent is 1100 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 Measurements Length Mass Volume/Capacity 1 kilometer (km) = 1000 m1 hectometer (hm) = 100 m1 dekameter (dam) = 10 m1 meter (m) = 1 m1 decimeter (dm) = 0.1 m1 centimeter (cm) = 0.01 m1 millimeter (mm) = 0.001 m 1 kilogram (kg) = 1000 g1 hectogram (hg) = 100 g1 dekagram (dag) = 10 g1 gram (g) = 1 g1 decigram (dg) = 0.1 g1 centigram (cg) = 0.01 g1 milligram (mg) = 0.001 g 1 kiloliter (kL) = 1000 L1 hectoliter (hL) = 100 L1 dekaliter (daL) = 10 L1 liter (L) = 1 L1 deciliter (dL) = 0.1 L1 centiliter (cL) = 0.01 L1 milliliter (mL) = 0.001 L 1 meter = 100 centimeters1 meter = 1000 millimeters 1 gram = 100 centigrams1 gram = 1000 milligrams 1 liter = 100 centiliters1 liter = 1000 milliliters
To make conversions in the metric system, we will use the same technique we did in the U.S. 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 5 k or 10 k race? The lengths 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. Nick ran a 10-kilometer race. How many meters did he run?
(credit: William Warby, Flickr)
Solution We will convert kilometers to meters using the Identity Property of Multiplication and the equivalencies in . 10 kilometers Multiply the measurement to be converted by 1. Write 1 as a fraction relating kilometers and meters. Simplify. Multiply. 10,000 m Nick ran 10,000 meters.
Sandy completed her first 5-km race. How many meters did she run? 5000 m Herman bought a rug 2.5 meters in length. How many centimeters is the length? 250 cm Eleanor’s newborn baby weighed 3200 grams. How many kilograms did the baby weigh? Solution We will convert grams to kilograms. Multiply the measurement to be converted by 1. Write 1 as a fraction relating kilograms and grams. Simplify. Multiply. Divide. 3.2 kilograms The baby weighed 3.2 kilograms.
Kari’s newborn baby weighed 2800 grams. How many kilograms did the baby weigh? 2.8 kilograms Anderson received a package that was marked 4500 grams. How many kilograms did this package weigh? 4.5 kilograms Since the metric system is based on multiples of ten, conversions involve multiplying by multiples of ten. In Decimal Operations, we learned how to simplify these calculations by just moving the decimal.To multiply by 10,100,or1000, we move the decimal to the right 1,2,or3 places, respectively. To multiply by 0.1,0.01,or0.001 we move the decimal to the left 1,2,or3 places respectively. We can apply this pattern when we make measurement conversions in the metric system. In , we changed 3200 grams to kilograms by multiplying by 11000(or0.001). This is the same as moving the decimal 3 places to the left. Convert: ⓐ 350 liters to kiloliters ⓑ 4.1 liters to milliliters. Solution ⓐ We will convert liters to kiloliters. In , we see that 1 kiloliter=1000 liters. 350 L Multiply by 1, writing 1 as a fraction relating liters to kiloliters. Simplify. Move the decimal 3 units to the left. 0.35 kL
ⓑ We will convert liters to milliliters. In , we see that 1 liter=1000milliliters. 4.1 L Multiply by 1, writing 1 as a fraction relating milliliters to liters. Simplify. Move the decimal 3 units to the left. 4100 mL
Convert: ⓐ 7.25 L to kL ⓑ 6.3 L to mL. ⓐ 0.00725 kLⓑ 6300 mL Convert: ⓐ 350 hL to L ⓑ 4.1 L to cL. ⓐ 35,000 Lⓑ 410 cL
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. 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. Ryland is 0.75 meters taller than his brother. 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. 83 cm 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. 1.04 m 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. Triple 150 mL Translate to algebra. 3·150mL Multiply. 450mL Convert to liters. 450mL·0.001L1mL Simplify. 0.45L Dena needs 0.45 liter of olive oil.
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? 2 L 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? 2.4 kg
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. shows some of the most common conversions. Conversion Factors Between U.S. and Metric Systems Length Weight Volume 1 in = 2.54 cm1 ft = 0.305 m1 yd = 0.914 m1 mi = 1.61 km1 m = 3.28 ft 1 lb = 0.45 kg1 oz = 28 g1 kg = 2.2 lb 1 qt = 0.95 L1 fl oz = 30 mL1 L = 1.06 qt
We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors. 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 500 mL Multiply by a unit conversion factor relating mL and ounces. 500mL·1fl oz30mL Simplify. 500fl oz30 Divide. 16.7fl. oz. The water bottle holds 16.7 fluid ounces.
How many quarts of soda are in a 2-liter bottle? 2.12 quarts How many liters are in 4 quarts of milk? 3.8 liters The conversion factors in are not exact, but the approximations they give are close enough for everyday purposes. In , we rounded the number of fluid ounces to the nearest tenth. 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 100 kilometers Multiply by a unit conversion factor relating kilometers and miles. 100kilometers·1mile1.61kilometers 100·1mi1.61km Simplify. 100mi1.61 Divide. 62 mi It is about 62 miles to the next rest stop.
The height of Mount Kilimanjaro is 5,895 meters. Convert the height to feet. Round to the nearest foot. 19,336 ft The flight distance from New York City to London is 5,586 kilometers. Convert the distance to miles. Round to the nearest mile. 3,470 mi
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. shows the relationship between the two systems.
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. Temperature Conversion To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula 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=59(F−32) Simplify in parentheses. C=59(18) Multiply. C=10 A temperature of 50°F is equivalent to 10°C.
Convert the Fahrenheit temperatures to degrees Celsius: 59°F. 15°C Convert the Fahrenheit temperatures to degrees Celsius: 41°F. 5°C 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=95C+32 Multiply. F=36+32 Add. F=68 So 20°C is equivalent to 68°F.
Convert the Celsius temperatures to degrees Fahrenheit: The temperature in Helsinki, Finland was 15°C. 59°F Convert the Celsius temperatures to degrees Fahrenheit: The temperature in Sydney, Australia was 10°C. 50°F ACCESS ADDITIONAL ONLINE RESOURCES American Unit Conversion Time Conversions Metric Unit Conversions American and Metric Conversions Convert from Celsius to Fahrenheit Convert from Fahrenheit to Celsius
Section Exercises
Practice Makes PerfectMake Unit Conversions in the U.S. System In the following exercises, convert the units. A park bench is 6 feet long. Convert the length to inches. A floor tile is 2 feet wide. Convert the width to inches. 24 inches A ribbon is 18 inches long. Convert the length to feet. Carson is 45 inches tall. Convert his height to feet. 3.75 feet Jon is 6 feet 4 inches tall. Convert his height to inches. Faye is 4 feet 10 inches tall. Convert her height to inches. 58 inches A football field is 160 feet wide. Convert the width to yards. On a baseball diamond, the distance from home plate to first base is 30 yards. Convert the distance to feet. 90 feet Ulises lives 1.5 miles from school. Convert the distance to feet. Denver, Colorado, is 5,183 feet above sea level. Convert the height to miles. 0.98 miles A killer whale weighs 4.6 tons. Convert the weight to pounds. Blue whales can weigh as much as 150 tons. Convert the weight to pounds. 300,000 pounds An empty bus weighs 35,000 pounds. Convert the weight to tons. At take-off, an airplane weighs 220,000 pounds. Convert the weight to tons. 110 tons The voyage of the Mayflower took 2 months and 5 days. Convert the time to days (30 days = 1 month). Lynn’s cruise lasted 6 days and 18 hours. Convert the time to hours. 162 hours Rocco waited 112 hours for his appointment. Convert the time to seconds. Misty’s surgery lasted 214 hours. Convert the time to seconds. 8100 seconds How many teaspoons are in a pint? How many tablespoons are in a gallon? 256 tablespoons JJ’s cat, Posy, weighs 14 pounds. Convert her weight to ounces. April’s dog, Beans, weighs 8 pounds. Convert his weight to ounces. 128 ounces Baby Preston weighed 7 pounds 3 ounces at birth. Convert his weight to ounces. Baby Audrey weighed 6 pounds 15 ounces at birth. Convert her weight to ounces. 111 ounces Crista will serve 20 cups of juice at her son’s party. Convert the volume to gallons. Lance needs 500 cups of water for the runners in a race. Convert the volume to gallons. 31.25 gallons Use Mixed Units of Measurement in the U.S. System In the following exercises, solve and write your answer in mixed units. 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? 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? 4 lbs. 1 oz. 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? 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 weeks and 1 day 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? 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? 9 ft 2 in 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? 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? 8 yards Make Unit Conversions in the Metric System In the following exercises, convert the units. Ghalib ran 5 kilometers. Convert the length to meters. Kitaka hiked 8 kilometers. Convert the length to meters. 8000 meters Estrella is 1.55 meters tall. Convert her height to centimeters. The width of the wading pool is 2.45 meters. Convert the width to centimeters. 245 centimeters Mount Whitney is 3,072 meters tall. Convert the height to kilometers. The depth of the Mariana Trench is 10,911 meters. Convert the depth to kilometers. 10.911 kilometers June’s multivitamin contains 1,500 milligrams of calcium. Convert this to grams. A typical ruby-throated hummingbird weights 3 grams. Convert this to milligrams. 3000 milligrams One stick of butter contains 91.6 grams of fat. Convert this to milligrams. One serving of gourmet ice cream has 25 grams of fat. Convert this to milligrams. 25,000 milligrams The maximum mass of an airmail letter is 2 kilograms. Convert this to grams. Dimitri’s daughter weighed 3.8 kilograms at birth. Convert this to grams. 3800 grams A bottle of wine contained 750 milliliters. Convert this to liters. A bottle of medicine contained 300 milliliters. Convert this to liters. 0.3 liters Use Mixed Units of Measurement in the Metric System In the following exercises, solve and write your answer in mixed units. Matthias is 1.8 meters tall. His son is 89 centimeters tall. How much taller, in centimeters, is Matthias than his son? Stavros is 1.6 meters tall. His sister is 95 centimeters tall. How much taller, in centimeters, is Stavros than his sister? 65 centimeters 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? 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? 1.82 kilograms Harry mailed 5 packages that weighed 420 grams each. What was the total weight of the packages in kilograms? 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? 16.8 grams Jonas drinks 200 milliliters of water 8 times a day. How many liters of water does Jonas drink in a day? 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? 42,000 milligrams Convert Between U.S. and Metric Systems In the following exercises, make the unit conversions. Round to the nearest tenth. Bill is 75 inches tall. Convert his height to centimeters. Frankie is 42 inches tall. Convert his height to centimeters. 106.7 centimeters Marcus passed a football 24 yards. Convert the pass length to meters. Connie bought 9 yards of fabric to make drapes. Convert the fabric length to meters. 8.2 meters Each American throws out an average of 1,650 pounds of garbage per year. Convert this weight to kilograms (2.20 pounds = 1 kilogram). An average American will throw away 90,000 pounds of trash over his or her lifetime. Convert this weight to kilograms (2.20 pounds = 1 kilogram). 40,900 kilograms A 5K run is 5 kilometers long. Convert this length to miles. Kathryn is 1.6 meters tall. Convert her height to feet. 5.2 feet Dawn’s suitcase weighed 20 kilograms. Convert the weight to pounds. Jackson’s backpack weighs 15 kilograms. Convert the weight to pounds. 33 pounds Ozzie put 14 gallons of gas in his truck. Convert the volume to liters. Bernard bought 8 gallons of paint. Convert the volume to liters. 30.2 liters Convert between Fahrenheit and Celsius In the following exercises, convert the Fahrenheit temperature to degrees Celsius. Round to the nearest tenth. 86°F 77°F 25°C 104°F 14°F −10°C 72°F 4°F −15.6°C 0°F 120°F 48.9°C In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth. 5°C 25°C 77°F −10°C −15°C 5°F 22°C 8°C 46.4°F 43°C 16°C 60.8°F
Everyday Math 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? 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? 110 reflectors
Writing Exercises Some people think that 65° to 75° Fahrenheit is the ideal temperature range. ⓐ What is your ideal temperature range? Why do you think so? ⓑ Convert your ideal temperatures from Fahrenheit to Celsius. ⓐ Did you grow up using the U.S. customary or the metric system of measurement? ⓑ Describe two examples in your life when you had to convert between systems of measurement. ⓒ Which system do you think is easier to use? Explain.
Self Checkⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?
Chapter Review Exercises
<link document="m81315">Rational and Irrational Numbers</link>In the following exercises, write as the ratio of two integers. 6 −5 −51 2.9 1.8 1810 In the following exercises, determine which of the numbers is rational. 0.42,0.3–,2.56813… 0.75319…,0.16—,1.95 0.16—,1.95 In the following exercises, identify whether each given number is rational or irrational. ⓐ 49 ⓑ 55 ⓐ 72 ⓑ 64 ⓐ irrationalⓑ rational In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers. −9,0,0.361....,89,16,9 −5,−214,−4,0.25—,135,4 ⓐ 4ⓑ −5,−4,4ⓒ −5,−214,−4,0.25—,135,4ⓓ noneⓔ −5,−214,−4,0.25—,135,4
<link document="m81316">Commutative and Associative Properties</link>In the following exercises, use the commutative property to rewrite the given expression. 6+4=____ −14·5=____ −14·5 = 5(−14) 3n=____ a+8=____ a + 8 = 8 + a In the following exercises, use the associative property to rewrite the given expression. (13·5)·2=_____ (22+7)+3=_____ (22 + 7) + 3 = 22 + (7 + 3) (4+9x)+x=_____ 12(22y)=_____ 12(22y)=(12·22)y In the following exercises, evaluate each expression for the given value. If y=1112, evaluate: ⓐ y+0.7+(−y) ⓑ y+(−y)+0.7 If z=−53, evaluate: ⓐ z+5.39+(−z) ⓑ z+(−z)+5.39 ⓐ 5.39ⓑ 5.39 If k=65, evaluate: ⓐ 49(94k) ⓑ (49·94)k If m=−13, evaluate: ⓐ −25(52m) ⓑ (−25·52)m ⓐ 13ⓑ 13 In the following exercises, simplify using the commutative and associative properties. 6y+37+(−6y) 14+1115+(−14) 1115 1411·359·1114 −18·15·29 −60 (712+45)+15 (3.98d+0.75d)+1.25d 5.98 d −12(4m) 30(56q) 25 q 11x+8y+16x+15y 52m+(−20n)+(−18m)+(−5n) 34 m + (−25 n)
<link document="m81317">Distributive Property</link>In the following exercises, simplify using the distributive property. 7(x+9) 9(u−4) 9y − 36 −3(6m−1) −8(−7a−12) 56a + 96 13(15n−6) (y+10)·p yp + 10p (a−4)−(6a+9) 4(x+3)−8(x−7) −4x + 68 In the following exercises, evaluate using the distributive property. If u=2, evaluate ⓐ 3(8u+9)and ⓑ 3·8u+3·9 to show that 3(8u+9)=3·8u+3·9 If n=78, evaluate ⓐ 8(n+14) and ⓑ 8·n+8·14 to show that 8(n+14)=8·n+8·14 ⓐ 9ⓑ 9 If d=14, evaluate ⓐ −100(0.1d+0.35) and ⓑ −100·(0.1d)+(−100)(0.35) to show that −100(0.1d+0.35)=−100·(0.1d)+(−100)(0.35) If y=−18, evaluate ⓐ −(y−18) and ⓑ −y+18 to show that −(y−18)=−y+18 ⓐ 36ⓑ 36
<link document="m81318">Properties of Identities, Inverses, and Zero</link>In the following exercises, identify whether each example is using the identity property of addition or multiplication. −35(1)=−35 29+0=29 identity property of addition (6x+0)+4x=6x+4x 9·1+(−3)=9+(−3) identity property of multiplication In the following exercises, find the additive inverse. −32 19.4 −19.4 35 −715 715 In the following exercises, find the multiplicative inverse. 92 −5 −15 110 −49 −94 In the following exercises, simplify. 83·0 09 0 50 0÷23 0 43+39+(−43) (n+6.75)+0.25 n + 7 513·57·135 16·17·12 34 23·28·37 9(6x−11)+15 54x − 84
<link document="m81330">Systems of Measurement</link>In the following exercises, convert between U.S. units. Round to the nearest tenth. A floral arbor is 7 feet tall. Convert the height to inches. A picture frame is 42 inches wide. Convert the width to feet. 3.5 feet Kelly is 5 feet 4 inches tall. Convert her height to inches. A playground is 45 feet wide. Convert the width to yards. 15 yards The height of Mount Shasta is 14,179 feet. Convert the height to miles. Shamu weighs 4.5 tons. Convert the weight to pounds. 9000 pounds The play lasted 134 hours. Convert the time to minutes. How many tablespoons are in a quart? 64 tablespoons Naomi’s baby weighed 5 pounds 14 ounces at birth. Convert the weight to ounces. Trinh needs 30 cups of paint for her class art project. Convert the volume to gallons. 1.9 gallons In the following exercises, solve, and state your answer in mixed units. John caught 4 lobsters. The weights of the lobsters were 1 pound 9 ounces, 1 pound 12 ounces, 4 pounds 2 ounces, and 2 pounds 15 ounces. What was the total weight of the lobsters? Every day last week, Pedro recorded the amount of time he spent reading. He read for 50,25,83,45,32,60,and135 minutes. How much time, in hours and minutes, did Pedro spend reading? 7 hours 10 minutes Fouad is 6 feet 2 inches tall. If he stands on a rung of a ladder 8 feet 10 inches high, how high off the ground is the top of Fouad’s head? Dalila wants to make pillow covers. Each cover takes 30 inches of fabric. How many yards and inches of fabric does she need for 4 pillow covers? 3 yards, 12 inches In the following exercises, convert between metric units. Donna is 1.7 meters tall. Convert her height to centimeters. Mount Everest is 8,850 meters tall. Convert the height to kilometers. 8.85 kilometers One cup of yogurt contains 488 milligrams of calcium. Convert this to grams. One cup of yogurt contains 13 grams of protein. Convert this to milligrams. 13,000 milligrams Sergio weighed 2.9 kilograms at birth. Convert this to grams. A bottle of water contained 650 milliliters. Convert this to liters. 0.65 liters In the following exercises, solve. Minh is 2 meters tall. His daughter is 88 centimeters tall. How much taller, in meters, is Minh than his daughter? Selma had a 1-liter bottle of water. If she drank 145 milliliters, how much water, in milliliters, was left in the bottle? 855 milliliters One serving of cranberry juice contains 30 grams of sugar. How many kilograms of sugar are in 30 servings of cranberry juice? One ounce of tofu provides 2 grams of protein. How many milligrams of protein are provided by 5 ounces of tofu? 10,000 milligrams In the following exercises, convert between U.S. and metric units. Round to the nearest tenth. Majid is 69 inches tall. Convert his height to centimeters. A college basketball court is 84 feet long. Convert this length to meters. 25.6 meters Caroline walked 2.5 kilometers. Convert this length to miles. Lucas weighs 78 kilograms. Convert his weight to pounds. 171.6 pounds Steve’s car holds 55 liters of gas. Convert this to gallons. A box of books weighs 25 pounds. Convert this weight to kilograms. 11.4 kilograms In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth. 95°F 23°F −5°C 20°F 64°F 17.8°C In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth. 30°C −5°C 23°F −12°C 24°C 75.2°F
Chapter Practice Test For the numbers 0.18349…,0.2–,1.67, list the ⓐ rational numbers and ⓑ irrational numbers. Is 144 rational or irrational? 144=12therefore rational. From the numbers −4,−112,0,58,2,7, which are ⓐ integers ⓑ rational ⓒ irrational ⓓ real numbers? Rewrite using the commutative property: x·14=_________ x·14 = 14·x Rewrite the expression using the associative property: (y+6)+3=_______________ Rewrite the expression using the associative property: (8·2)·5=___________ (8·2)·5 = 8·(2·5) Evaluate 316(163n) when n=42. For the number 25 find the ⓐ additive inverse ⓑ multiplicative inverse. ⓐ −25ⓑ 52 In the following exercises, simplify the given expression. 34(−29)(43) −3+15y+3 15y (1.27q+0.25q)+0.75q (815+29)+79 2315 −18(32n) 14y+(−6z)+16y+2z 30y − 4z 9(q+9) 6(5x−4) 30x − 24 −10(0.4n+0.7) 14(8a+12) 2a + 3 m(n+2) 8(6p−1)+2(9p+3) 66p − 2 (12a+4)−(9a+6) 08 0 4.50 0÷(23) 0 In the following exercises, solve using the appropriate unit conversions. Azize walked 412 miles. Convert this distance to feet. (1 mile=5,280 feet). One cup of milk contains 276 milligrams of calcium. Convert this to grams. (1 milligram=0.001 gram) .276 grams Larry had 5 phone customer phone calls yesterday. The calls lasted 28,44,9,75,and55 minutes. How much time, in hours and minutes, did Larry spend on the phone? (1 hour=60 minutes) Janice ran 15 kilometers. Convert this distance to miles. Round to the nearest hundredth of a mile. (1 mile=1.61 kilometers) 9.317 miles Yolie is 63 inches tall. Convert her height to centimeters. Round to the nearest centimeter. (1 inch=2.54 centimeters) Use the formula F=95C+32 to convert 35°C to degrees F 95°F