MATH GRADE 6 UNIT 6 RATE ANSWERS FOR EXERCISES

Transcription

1 MATH GRADE 6 UNIT 6 RATE FOR EXERCISES

2 LESSON 2: PRICE AS A RATE 1. $ $.625, or $ $ $.3125, or $ a. $2.5 b. $ a. Amount (pt) Cost non-organic ($) $.75 $1.5 $2.25 $3. $3.75 $4.5 Cost organic ($) $2.25 $4.5 $6.75 $9. $11.25 $13.5 Amount (pt) Cost non-organic ($) $5.25 $6. $6.75 $7.5 $8.25 $9. Cost organic ($) $15.75 $18. $2.25 $22.5 $24.75 $27. b. 1 pint = $ pints = 6.5 $.75 = $4.875, or $4.88 c. See the table in problem 6a. 7. a. $3. per dozen 12 eggs = $.25 per egg Elijah can buy only 1 egg, because eggs cost $.25 each. b. $.43 + $.32 = $.75 Elijah needs $.43 more, for a total of $ The car costs about $6.47 per pound. At this rate, a 2-pound wheel costs about $ Most people would not judge a car by dollars per pound. While it can be mathematically correct, the information may not influence a decision. Copyright 214 Pearson Education, Inc. 26

3 LESSON 3: FUEL EFFICIENCY AS A RATE gallons gallons miles miles 5. a. Distance (mi) Gas (gal) b. The fuel efficiency is 12.5 miles per gallon. The unit rate is given in the first column of the table. c miles 6. a. Gas (gal) Distance (mi) b. The fuel efficiency is =.8 gallon per mile The unit rate is given in the first column of the table. c gallons 7. Answers will vary. Here is one example: Most of the time people talk about miles per gallon. Miles per gallon is a number greater than. The inverse rate, gallons per mile, is a decimal number less than 1. People usually find it easier to work with numbers greater than 1 when comparing rates. 8. $.2 per mile Copyright 214 Pearson Education, Inc. 27

4 LESSON 4: POPULATION DENSITY AS A RATE 1. City Population 21 Census Area (sq mi) Population Density (to nearest tenth) Chicago 2,695, ,843.6 people/sq mi San Francisco 85, ,169.2 people/sq mi 2. Even though it has a smaller population, San Francisco is more crowded, because its population density is greater than that of Chicago. 3. The units are people per square mile. Other units would depend on the units used for area, such as people per square kilometer, and so on. 4. State Population 2 Census Area (sq mi) Population Density (to nearest tenth) California 37,593, , people/sq mi Georgia 8,186,453 59, people/sq mi Montana 92, , people/sq mi New Jersey 8,414,35 8, people/sq mi 5. New Jersey is the most crowded because it has the highest population density, and Montana is the least crowded because it has the lowest population density. 6. Population density would increase if the state s area were to shrink (assuming the population in the state remains fixed). 7. Population density would decrease if the state s population were to shrink (assuming the state s area remains fixed). 8. Answers will vary. Copyright 214 Pearson Education, Inc. 28

5 LESSON 5: WHAT IS A RATE? 1. Answers will vary. You might say that a rate is a type of quantity in which the unit is of the form A per B. A rate measures the relationship between two aspects of a situation. Situation Rate Not a Rate Reason 2. Emma earns $6.5 per hour for babysitting. This is a unit price Height (ft) Only height is measured. 1 Alex Bella Craig Desiree 4. The rate is drops per minute. 3 drops per minute 5. Denzel read 3 books this week to gather information for his science project. He could read any number of books in the following weeks. Copyright 214 Pearson Education, Inc. 29

6 LESSON 5: WHAT IS A RATE? Situation Rate Not a Rate Reason 6. The amount lists the rate of liters per quart. 1 quart (.946 liter) 7. The distance between where you live and your school. Only distance is measured. 8. Tickets to a play cost $9. for children and $12. for adults. The rates are price per ticket. 9. Answers will vary. Here are three examples: price per pound (e.g., use to find the price of 3 pounds of grapes at $3.29 per pound) miles per hour (e.g., use to find speed) heart beats per minute (e.g., use to find heart rate) Copyright 214 Pearson Education, Inc. 3

7 LESSON 6: SPEED AS RATE 1. 2 meters per second or 12 meters per minute 2. a..125 mile per minute b. 7.5 miles per hour c..5 lap per minute d. 11 feet per second 3. 1 minutes 4..4 hour, or 24 minutes 5. a. t = 2n b. n =.5t 6. a. Distance (mi) Time (min) b. His speed is.25 mile per minute, or 15 miles per hour. You can find the speed in miles per hour by multiplying the miles per minute by 6. c. 18 miles 7. a. Time (min) Distance (mi) b. His speed is 4 minutes per mile, or 1 hour per mile. You can find the hours 15 per mile by dividing the minutes per mile by 6. In this case, minutes per mile probably makes more sense in reporting the speed. c. 92 minutes Copyright 214 Pearson Education, Inc. 31

8 LESSON 6: SPEED AS RATE 8. The rate is 4 minutes per mile = 1 mile per minute. In this case, minutes per mile is a 4 number greater than 1, so you would discuss minutes per mile rather than miles per minute. Minutes makes more sense in this situation. Copyright 214 Pearson Education, Inc. 32

9 LESSON 7: CONVERSION FACTORS cm cm 3. About inches 4. About inches 5. 4,68 gallons 6. 1 divided by 1.61 is about.62 mile in 1 kilometer. 7. a. 297 ft 2.4 acres = 14,544 sq ft b. 352 feet 8. a. 599 gallon containers b. 161,11 cubic inches c. 93 cubic feet 9. A rate is one quantity per another quantity. It compares two quantities. A conversion factor compares two quantities. Copyright 214 Pearson Education, Inc. 33

10 LESSON 8: RATES AND UNITS 1. a..5 block per minute b. 2 minutes 2. This calculation tells their average speed in miles per hour, since Suraj divides the number of miles by the number of hours. 3. This calculation tells their average speed in hours per mile, since Suraj s brother divides the number of hours by the number of miles miles 6.5 hours = 68 miles per hour. 68 gives their average speed in miles per hour because you are dividing miles by hours hours 442 miles.147 hour per mile..147 gives their average speed in hours per mile because you are dividing hours by miles miles per gallon 7..5 gallon per mile gallons per hour 9. Days and years both cancel out, so the units would be in dollars. Copyright 214 Pearson Education, Inc. 34

11 LESSON 13: RATES AND GRAPHS 1. Train A: 7 miles per hour (fastest) Train B: 6 miles per hour Train C: 55 miles per hour (slowest; given) 2. y 5 Distance (mi) Train A Train B Train C Time (hr) x Train A produces the steepest line. 3. Caroline: yards per second Aiko: about yards per second Copyright 214 Pearson Education, Inc. 35

12 LESSON 13: RATES AND GRAPHS 4. y 45 4 Aiko Caroline 35 Distance (yd) Time (sec) x 5. y 45 Distance (yd) Caroline Aiko Time (sec) x Copyright 214 Pearson Education, Inc. 36

13 LESSON 13: RATES AND GRAPHS 6. Distance (yd) y Aiko Caroline Time (sec) x 7. Answers will vary. If your graph has average speed on the y-axis, it will be composed of horizontal straight lines, since speed is assumed constant. If your graph has average speed on the x-axis, it will be composed of vertical straight lines. Copyright 214 Pearson Education, Inc. 37

14 LESSON 14: RATES AND FORMULAS Answers may vary for problems 1 4. Here are examples: 1. d = 3t, where d gives distance in miles and t the time in hours 2. p = 3 a, where p gives the price in dollars and a the number of artichokes 2 3. d = 7t, where d gives the distance in miles and t the time in hours 4. d = 25g, where d gives the distance in miles and g the number of gallons pages per second; seconds per page p = 125m; p = s 7. m = 1 12 p; s = p 8. Answers will vary. Copyright 214 Pearson Education, Inc. 38

15 LESSON 15: REPRESENTATIONS OF RATES 1. The graph is a straight line and goes through the origin and the point (1, 6). 2. Multiplying any amount of time in minutes by 6 gives the appropriate number of pages. 3. The line must be a straight line that passes through the origin and the point (1, 3). y 15 Number of Pages p = 3t x Time (min) 5. The line must be a straight line that passes through the origin and the point (1, 12). y 6 Number of Pages x Time (min) 6. p = 12t Copyright 214 Pearson Education, Inc. 39

16 LESSON 15: REPRESENTATIONS OF RATES 7. The three graphs all show the relationship between the number of pages and the time it takes to print them. The three graphs appear to have the same slope when you scale the axes; however, greater rates would produce steeper slopes if you drew all graphs on the same coordinate plane. Copyright 214 Pearson Education, Inc. 4