Applications of Ratios and Proportions

Transcription

1 Applications of Ratios and Proportions Connections Have you ever... Needed to convert a measurement from inches to feet? Calculated the unit price of something sold by the case? Tried to plan the amount of food needed for a party? Ratios are a useful tool in mathematics because they apply to so many real-world situations. Ratios are all about relationships. Converting measurements, finding percents, and measuring distances on maps can all be done using ratios. Ratios are fractions that relate one type of thing to something else. For eample: 2 inches 2 inches to a foot can be shown as a ratio: foot 25 miles 25 miles per gallon can be shown as a ratio: gallon Seven pounds of pasta for 55 guests can be shown as a ratio: 7 pounds 55 guests Once you have a ratio, you can divide to find the unit rate, the amount per one unit of measurement. For eample, if you have driven 225 miles on five gallons of gas in your new hybrid car, you can divide to find a unit rate of 45 miles per gallon. You can also use ratios to solve proportions, which are equations that involve ratios and an unknown. For eample, if you know your hybrid car gets 45 miles per gallon, you can calculate the amount of gas for a 398-mile trip with a proportion: 45 miles 398 gallon miles 79

2 Learn It! Essential Math Skills Solving Problems Using Ratios A ratio is simply a fraction, but it may have different units in the numerator and denominator. You will often want to find a unit rate from a ratio. How many miles per gallon did you get on that last road trip? How many stuffed mushrooms will you have for each guest? Miles per gallon, price per pound, and inches per foot are all unit rates. When you write a unit rate as a fraction, the denominator is one. To find the unit rate if the denominator is not one, divide. If you have seven pounds of pasta 7 for 55 guests, that would be 55, or about 0.3 pounds per guest. Marie is following a pattern for building a birdhouse, but the measurements are in centimeters and her measuring tape is in inches. She knows that there are 2.54 centimeters in an inch. The roof of the birdhouse should be 25 cm by 30 cm. What are its dimensions in inches?? Write a Ratio You can write a relationship between two values as a ratio. It is a correct ratio whichever units are in the numerator and the denominator, but to avoid etra steps to solve the problem, put the units that you want in your final answer in the numerator.. Write a ratio relating centimeters to inches. When you have a relationship between two things, such as centimeters and inches, you have a ratio. You can write the ratio of 2.54 centimeters to one inch in two ways: inch 254. centimeters or 254. centimeters inch Since your final answer should be in inches, the first ratio will be easiest. The ratio of centimeters to inches is a conversion factor. Since one inch and 2.54 centimeters are equal, the value of the ratio is one. This is true of any fraction where the numerator and the denominator are equal, just as 4 4. Because the ratio is equal to one, you can multiply any number by this ratio and not change the equivalent value of that number when the units change. 80

3 Applications of Ratios and Proportions? Write and Solve an Equation Since ratios are fractions, you can cancel common units to help you solve an equation. To get rid of the old units (centimeters in this case) make sure that your conversion factor has centimeters in the denominator. Once you cancel all the units you can, the remaining units should match the units you want in your final answer. 2. Multiply the original measurement by the conversion factor to find the measurements in inches. Round your answer to the nearest tenth. Multiply both the length and width by the conversion factor. 25 cm # inch inches. 254cm cm inch # 30.. inches cm Rounded to the nearest tenth, the dimensions are 9.8 inches by.8 inches.? Check Your Answer When doing conversions, it is easy to lose track of the decimal place. Check to make sure that your answer is sensible, and review your math for common errors. 3. Is a birdhouse roof that is about 0 by 0 inches reasonable? The size of the birdhouse is reasonable. If your answer were 0.98 by.8 inches, that would be a hummingbird house! If it were 98 by 8 inches, it would be more appropriate for a family of eagles. The measurements sound reasonable. Build Your Math Skills Try solving this problem using the conversion ratio with one inch in the denominator: 254. cm inch What etra steps do you need? Why? 8

4 Practice It! Essential Math Skills Solve the following problems using ratios. quart. The conversion factor for cups to a quart is 4 cups. Manolo is making a party punch that calls for 4.5 cups of limeade, but he only has a quart jar to use to measure the limeade. How many quarts should he use? a. Write the ratio or ratios in the best way for this problem. Eplain why you wrote the ratios this way. b. Find the solution and check your answer. 2. Aikiko is building a wood shed. The blueprint she is using says that one inch on the drawing is equivalent to.5 feet. She needs to cut 2 two-by-fours to the right length. In the blueprint, they are five inches long. How long will the two-by-fours be? a. Write the ratio or ratios in the best way for this problem. Eplain why you wrote the ratios this way. b. Find the solution and check your answer. 3. Javier is in chemistry lab. He needs 250 ml of 0% sodium hydroide, w/v. This means that for every liter of water, he will add 00 grams of sodium hydroide. How many grams of sodium hydroide does he need? a. Write the ratio or ratios in the best way for this problem. Eplain why you wrote the ratios this way. b. Find the solution and check your answer. 82

5 Applications of Ratios and Proportions 4. Liz and Deshane work for a painting company. Liz can paint 25 square feet in eight minutes, and Deshane can paint 25 square feet in si minutes. Working together, how much can they paint in four hours? a. Write two ratios and an equation relating them. b. Answer the question and check the answer. 5. Destinee and Sarah are working together on a problem that says: Convert 743 millimeters to inches using the following information. There are 000 millimeters in a meter, 00 centimeters in a meter, and 2.54 centimeters in an inch. Sarah showed the following work. She knows it is wrong because it is so large. Inches aren t that much larger than millimeters. How could Destinee correct it for her? mm m mm m cm in 743 a 000 ka 00 ka 254. cm k 29, 25, a. Identify the ratio or ratios that are incorrectly used. b. Solve the problem. Round to the nearest tenth. c. What advice would you give Sarah to avoid this error in the future? 83

6 Learn It! Essential Math Skills Cross-Multiply and Divide You can solve any proportion with the cross-multiply and divide strategy. If you have one fraction equal to another fraction, multiplying denominators across the equals sign eliminates fractions. Then, divide to isolate the variable. Cross-Multiply a b c a bc Divide a a bc a bc a At the grocery store, peaches are priced at five pounds for $3.00. You have $4.50. Assuming there is no ta, how many pounds of peaches could you buy?? Find the Relationship Find the relationship in the question. Look for the words per or for. These words usually indicate a relationship that can be written as a ratio. Write the ratio as a fraction, including the units.. Identify the given relationship in the problem and write it as a fraction with units. The given relationship is five pounds for $3.00. You can write this as a ratio: 5 pounds 3 dollars You could also say that three dollars buys you five pounds and write it 3 pounds 5 dollars. The ratio can be written either way, since it is equivalent to one. Three dollars is equivalent to five pounds of peaches at the grocery store, so the ratio is equivalent to one. Write a Proportion Using the ratio on one side of the equation, write a proportion. There should be a fraction on the other side of the equation. The other fraction will have the same units, in the same places, and will use another given value (in this case, $4.50). You can use to represent the unknown. Once you re sure your units line up, you can remove them. 84

7 ? Applications of Ratios and Proportions 2. Write a proportion to solve the problem. Using the ratio of pounds to dollars, you can write a proportion to solve the problem: 5 3 pounds dollars dollars Review & Practice Get etra practice with the worksheet for ratios and proportions on pages ? Cross-Multiply and Divide You want to know the value of, so you want to isolate on one side of the equation. You can solve any proportion by first cross-multiplying (multiply the denominator of each fraction across the equal sign) and then dividing by the new coefficient of. 3. Solve the proportion. Step : Cross-multiply hg 45. # Step 2: Divide. You can buy or 7½ pounds of peaches ? Check Your Answer Make sure that the answer is logical and check for errors in your math. 4. If you could buy five pounds for $3.00, does it make sense that you can buy seven and a half pounds for $4.50? Why? $4.50 is one and a half times $3.00. It makes sense that you could buy one and a half times as many peaches for $4.50 as for $3.00. Since one-and-a-half times five pounds is seven and a half pounds, the answer makes sense. 85

8 Practice It! Essential Math Skills Write and solve proportions to answer the following questions.. Sophia wants to find the gas mileage of her car. When she fills the tank, the odometer reads 92,48 miles. When she fills it again, the odometer reads 92,692. It takes 9.3 gallons to completely fill the tank. What is the gas mileage of Sophia s car, to the nearest hundredth? a. Find the relationship and write a proportion. b. Solve the proportion and check your answer. c. Identify an error you might make in solving this problem. How could you avoid that error? 2. A new test for Lyme disease is 87% accurate in detecting if someone has the disease. If 2,400 people who have Lyme disease are tested, how many cases will not be detected? a. Find the relationship and write a proportion. b. Solve the proportion and check your answer. c. Identify an error you might make in solving this problem. How could you avoid that error? 86

9 Applications of Ratios and Proportions 3. You are driving across the country from Phoeni to Boston, a distance of 2,648 miles. You make the first 500 miles in seven hours, including stops. If you can keep up this rate, how many more hours will it take you to get to Boston? Round your answer to the nearest tenth. a. Find the relationship and write a proportion. b. Solve the proportion and check your answer. c. Identify an error you might make in solving this problem. How could you avoid that error? ẋ a. Solve the proportion. b. What is a different way that you could solve this proportion? Compare the two methods. 5. Henderson s Groceries offers three pounds of medium prawns for $8.6, and Grocery Time offers five pounds for $2.50. Which store has the best price for prawns? a. Solve the problem. b. What is a different way that you could solve this problem? Compare the two methods. 87

10 Essential Math Skills 6. John and Hasim are studying together. The problem they are working on says: The sales ta on a $400 stereo is $38. At the same ta rate, how much would the ta be on a $650 stereo? John does the following work. Hasim knows that the ta wouldn t be less on a more epensive item. a. Identify the mistake. 400 dollars 650 dollars 38 dollars 38( 400) ( 400) 650 $ b. Solve the proportion. c. What advice would you give John to avoid this error in the future? 7. Carlos and Tam both commute to work. Carlos traveled nine miles through bumper to bumper traffic at 5 miles per hour. Tam lives farther away, 2 miles. His commute as distance d in miles to time t in minutes is represented by the graph. Who arrives at work sooner, and by how much time? d t 88

11 Applications of Ratios and Proportions Check Your Skills Answer the following questions using your knowledge of ratios and proportions.. If your truck gets 5 miles to the gallon, how many miles can you go on an 8 gallon tank of gas? a..2 miles b. 270 miles c. 27 miles d. 2,700 miles 2. You are looking at buying a used car. One car costs $8,00 and gets 22 miles per gallon. Another costs $0,500 and gets 27 miles per gallon. With gas prices averaging $4 per gallon, how many miles would it take for the more epensive car to make up for the difference in cost? a. 2,400 miles b. 3,000 miles c. 600 miles d.,920 miles 3. Construction has begun on the lot net door to your house. On the blueprint, the building measures 27 cm wide and 5 cm long. The scale of the blueprint is cm 3 feet. If the lot for this building is 00 feet wide, and the building is centered on the lot, how many feet are there between the building and the side of the lot? 4. A manufacturing company has set a goal of a maimum of two defects per,000 capacitors produced. The company ships out 350,828 capacitors. Use the following figures to create an equation showing how many capacitors should be functional for the company to meet its goal., ,828 89

12 Essential Math Skills 5. The large meteorite that blasted through Russia s sky in February 203 had an average density of 3.6 g/cm 3. Dmitri found a rock that weighed 0.2 pounds and had a volume of 3.2 cm 3. The density would need to be within 0.2 g/cm 3 of the average for the rock to have been possibly part of the meteorite. How far is the density of the rock from 3.6 g/cm 3? Round your answer to the nearest tenth. 6. Grayson and Mo are heading downtown, a distance of 3.5 miles. The bicycle odometer says they are going 24 miles per hour. How many meters per minute are they traveling? Math Tip Learn common values for conversions. mile,609 meters mile 5,280 feet pound grams See page 304 for a table of common units of measure. 7. Andre s Grocery has a special on mangos, four for $3.50. Pearson s is advertising mangos for five for $7.25. How much do you save if you are buying 0 mangos from the cheaper store? a. $4.00 b. $5.75 c. $6.50 d. $ An eperienced mechanic can change a head gasket in about five hours. Her apprentice would take eight hours to complete the task. If the mechanic works for two hours and then turns the job over to the apprentice, how long does it take the apprentice to complete the job? 9. Jude is on the swim team. He swam the 50-meter breaststroke in seconds. What was his speed in miles per hour, rounded to the nearest hundredth? a mph b mph c mph d mph Remember the Concept Ratios are fractions that show a relationship. Use a ratio to convert units of measure. Cross-multiply and divide to solve proportion problems. 90

13 Answers and Eplanations Applications of Ratios and Proportions page 79 Solving Problems Using Ratios Practice It! pages qt a c # 4 c With the measurement that you have (cups) in the denominator of the conversion factor, you can multiply the quantity you have by the conversion factor. Your answer should be smaller than the original number, since quarts are larger than cups. b. 35/8 quarts 4 # ft 2a. 5 in # in With the measurement that you have (inches) in the denominator of the conversion factor, you can multiply the quantity you have by the conversion factor. Your answer should be larger than the original number, since one inch in the drawing equals more than one foot. 2b. 7.5 feet When there is a one in the denominator, you can remove it. L 3a. 250 ml # 000 ml 025. L With the measurement that you have (ml) in the denominator of the conversion factor, you can multiply the quantity you have by the conversion factor. Your answer should be smaller than the original number, since liters are larger than milliliters L 00 g sodium hydroide/ L The conversion factor tells you how much sodium hydroide to add for every liter of solution. You want liters in the denominator of the conversion factor so that you are left with grams of sodium hydroide. 3b. 25 g 250 # a. Liz: 25 ft 2 per 8 min Deshane: 25 ft 2 per 6 min 4 hr 240 min ft 25 ft ( # 8 min 240 min) + ( # 6 min 240 min) 4b.,750 square feet ` # 8 240j+ ` # 6 240j a. The ratio of millimeters to meters should have the millimeters in the denominator. The equation should change millimeters to meters by multiplying with millimeters in the denominator. 5b inches m 00 cm in 743 mmd nd na k 000 mm m 254. cm in 743` 000 j^00h` j ` 0 j` j 254. in c. You might advise Sarah to make sure that her units cancel to avoid this error in the future. Cross-Multiply and Divide Practice It! pages , , 48 miles a. 93. gallons gallon b mpg 92, , c. An error you might make is not realizing you need to subtract the first odometer reading from the second to find the miles driven in 9.3 gallons. To avoid this error, think about the end result you want: miles driven per gallon. In this problem, you know the amount of gallons used (9.3). Ask, how many miles were driven with 9.3 gallons? 2a undetected cases 3 00 tested 2, 400 tested i

14 Essential Math Skills 2b.,62 undetected cases , , ,200 00,62 2c. An error you might make is not realizing that a percent can be written as a ratio. When a percent is written as a fraction, it also is a ratio: out of miles 2648 miles 3a. 7 hours 3b. 30. hours c. One mistake you might make is forgetting to subtract the seven hours already driven from the total hours for the trip. You could subtract 500 miles from the total mileage before writing your proportion to avoid this problem. Or, you could double-check what the question is asking for before finalizing your answer. 25 4a ẋ 2.5 9(4.9) b. You could also solve the proportion by changing 2. 5/9 to a decimal. This method may be more difficult for many people, since 2.5 divided by 9 creates a long, difficult decimal. 5a. Henderson s Groceries has the better price. Compare: 3 lb 5 lb 86 and Approimately lb per dollar and 0.4 lb per dollar The first price is lower. 5b. You might create an inequality to solve this problem and see if it is true > 2. 50? 0.35 $ 0.4 Since the answer is no, Henderson s Groceries has the better price. These methods are very similar, since you are performing the same math. This is a trial-anderror method, since you start by guessing whether to use a greater than or less than sign. 6a. The proportion is set up incorrectly. The first ratio must have a similar relationship to the second ratio. You can set it up in different ways: 400 dollars 650 dollars 38 ta 400 dollars 38 ta 650 dollars 6b. $ (38) c. You might tell John to eamine his proportion. Do the numerator and denominator on one side represent a similar relationship to the numerator and denominator on the other side? In the first ratio of John s equation, the numerator is the first price and the denominator is the second price. In the second ratio, the numerator is the ta for the second price and the denominator is the ta for the first price. The relationships are different. 7. Tam arrives si minutes earlier. hour Carlos: 9 miles # 5 miles 9 5 hours 0.6 hours 60 min 06. hours ( ) 36 min hour Tam: On the graph, you can see that Tam drives 2 miles in 30 minutes. ii

15 Answers and Eplanations Check Your Skills pages b. 270 miles 5 miles gallon b. 3,000 miles gallons $ 2400 $ 4 per gallon 600 gallons 27 miles per gallon - 22 miles per gallon 5 mpg 5 5 miles gallon feet cm 27 cm 3 ft gallons The building is 8 feet wide feet left in the lot feet on each side of the building g/cm3 defects capacitors 350, lb g 3 c m 3. 2 cm lb g/cm meters per minute 24 mi 609 m c hr mi 7. b. $ $8.75 c m 350, 828 capacitors 60 hr min g/ cm. 4. g/ cm m m/ min y 5y 72.5 y $ hours 2 hours 5 hours 2 5 3/5 or 0.6 left to do 8 hours job complete 06. job hours 9. c mph 50 m 3600 sec mile 80, 000 c m sec hr c m 609 m 52, iii