DISTANCE PROBLEMS (d=rt)

(a.) How many “30 minute increments” are in 1 hour? # 1. (a.) How many “30 minute increments” are in 1 hour? (b.) If someone traveled 20 miles in 30 minutes, then how many miles did they travel in one hour? 20 • 2 = 40 miles (c.) If someone traveled 4 miles in 30 minutes, then how many miles did they travel in one hour? 4 • 2 = 8 miles

(a.) How many “20 minute increments” are in 1 hour? # 2. (a.) How many “20 minute increments” are in 1 hour? (b.) If someone traveled 15 miles in 20 minutes, then how many miles did they travel in one hour? 15 • 3 = 45 miles

Number of 12 minute increments in an hour: # 3. If someone traveled 10 miles in 12 minutes, then how many miles did they travel in one hour? Number of 12 minute increments in an hour: 10 • 5 = 50 miles

# 4. (a.) If 1,760 yards = 1 mile, then how many miles is 7040 yards? 4 miles (b.) How many miles is 5,280 yards? 3 miles

If 5,280 ft are in one mile, then how many miles is 10,560 feet? # 5. If 5,280 ft are in one mile, then how many miles is 10,560 feet? 2 miles

# 6. Fill in the chart for miles per hour. (1,760 yards = 1 mile and 5,280 feet = 1 mile) Distance Distance in miles Time Total increments in an hour Miles per Hour 17,600 yards 10 miles 20 minutes 3 10•3 = 30 miles/hr 5 miles 30 minutes 6 miles 40 minutes 3520 yards 15,840 feet 5 • 2 = 10 mph 5 miles 2 6 • (3/2) = 9 mph 6 miles 3/2 3520/1760 = 2 • 2 = 4 mph 2 2 miles 3 • 3 = 9 mph 15840/5280 = 3 3 miles

# 7. Jack walked 2 miles in 30 minutes. Jane walked 1,760 yards in 12 minutes. In miles per hour, how much faster did Jane walk than Jack? (1760 yards = 1 mile) There are 2 “30-minute” increments in an hour and 5 “12-minute increments in an hour. Jack walked 2 miles • 2 = 4 mph Jane walked 1 mile • 5 = 5 mph Jane walked 5 mph – 4 mph = 1 mph faster

# 8. Juan ran 1 mile in 6 minutes. Estelle ran 15,840 feet in 30 minutes. In miles per hour, how much faster did Juan run than Estelle? There are 10 “6-minute” increments in an hour and 2 “30-minute increments in an hour. Juan ran 1 mile • 10 = 10 mph Estelle ran 15840/5280 miles • 2 = 3 miles • 2 = 6 mph Juan ran 10 mph – 6 mph = 4 mph faster

# 9. Sebastian walked 1 mile in 12 minutes. Julie walked 5,280 yards in 20 minutes. They both walked for 2 hours. How much further had Julie walked than Sebastian? (1,760 yards = 1 mile) There are 5 “12-minute” increments in an hour and 3 “20-minute increments in an hour. Sebastian walked 1 mile • 5 = 5 mph Sebastian walked 5 • 2 = 10 miles in 2 hours. Julie walked 5280/1760 miles • 3 = 3 miles • 3 = 9 mph Julie walked 9 • 2 = 18 miles in 2 hours. Julie walked 18 – 10 = 8 miles further.

(a.) Make a table that shows how far Katie walked for each minute. # 10. John and Katie started at the same point. Katie walked 100 ft per minute. John started 3 minutes after her and walked 150 ft per minute. (a.) Make a table that shows how far Katie walked for each minute. Min. 1 Dist. 100 2 3 4 5 9 6 7 8 200 300 400 500 600 700 800 900 (b.) Make a table that shows how far John walked for each minute. (He did not go anywhere for the first 3 min.) Min. 1 2 3 4 Dist. 150 5 6 7 8 9 300 450 600 750 900

(c.) How far had Katie walked before John caught her? # 10. KATIE Min. 1 Dist. 100 2 3 4 5 9 6 7 8 200 300 400 500 600 700 800 900 JOHN Min. 1 2 3 4 Dist. 150 5 6 7 8 9 300 450 600 750 900 (c.) How far had Katie walked before John caught her? 600

He started later and had not walked as long as Katie. # 10. (d.) The two equations: d = 100 t and d = 150(t – 3) represent John and Katie’s distance. Which one represents John? Explain why. d = 150(t – 3) He started later and had not walked as long as Katie.

(e.) Set the two equations, (d = 100 t and # 10. (e.) Set the two equations, (d = 100 t and d = 150 (t – 3), equal to each other and solve. How long had they walked before they had walked the same distance? 100 t = 150 (t – 3) 100 t = 150 t – 450 - 50 t = – 450 t = 9

(f.) What is the same distance that they had walked? # 10. (f.) What is the same distance that they had walked? 100 (9) = 900 feet

# 10. (g.) Suppose that John had started 5 minutes later. What would his equation be? d = 150(t – 5)

(b.) How far had they walked before they had walked the same distance? # 11. Jackson and Tom started at the same place. Jackson walked 9 ft. per second. Tom waited three seconds later and walked 12 ft. per second. (a.) Make a dual table that shows how far each walked after each second. Time 1 2 3 4 5 6 7 8 Jackson Tom 9 18 27 36 45 54 63 72 12 24 36 48 60 (b.) How far had they walked before they had walked the same distance? 9 (12) = 108 (c.) Solve the problem algebraically. Show supporting work. 9 t = 12 (t – 3) - 3 t = - 36 9 t = 12 t – 36 t = 12

# 12. Hiroshi and Juan both rode their scooters. Hiroshi started first and rode 20 mph. Jack started 30 minutes later and rode 25 mph. How far had they traveled before they had caught up with each other? 9 t = 12 (t – .5) 9 t = 12 t – 6 - 3 t = – 6 t = 2 9 • 2 = 18 miles

# 13. d = r t d/r = t 10/15 = t = 2/3 hour = 40 minutes Distance = Speed • Time (or Rate • Time) d = r t # 13. Brady rides his bicycle 10 miles at 15 miles per hour. How long did it take him? d = r t d/r = t 10/15 = t = 2/3 hour = 40 minutes

# 14. d = r t d = r t Person Speed Time Aaron 50 mph 3 hours Bill Aaron, Bill, Colby, and Dillon all started at the same place and same time and are going to the same place. Who is the closest person to their destination? Who is the closest to their starting point? Person Speed Time Aaron 50 mph 3 hours Bill 60 mph 2.5 hours Colby 40 mph 4 hours Dillon 55 mph 2 hours and 30 min d = r t 150 mi 150 mi 80 mi 137.5 mi Aaron and Bill are closest to destination. Colby is closest to starting point.

# 15. Julie: d = 15 • 1.5 = 22.5 miles 12 • x = 22.5 miles Tom: d = r t # 15. Julie and Tom rode their bicycles to the same place. Julie rode 15 miles per hour and it took her 1.5 hours. If Tom rode 12 miles per hour, how long did it take him? Julie: d = 15 • 1.5 = 22.5 miles 12 • x = 22.5 miles Tom: x = 22.5 / 12 = 1.875 hours

d = r t # 16. Edgar, Frank, George, and Helen drove the following distances and times. Who was the fastest? Who was the slowest? Person Distance Time Edgar 100 miles 1.25 hours Frank 120 miles 2 hours George 200 miles 200 minutes Helen 150 miles 1 hour and 30 min r = d/t 80 mph 60 mph 60mph 100 mph Helen was the fastest. Frank and George were the slowest.

# 17. Jane: 50 • 3 = 150 miles 40 • 4 = 160 miles Tom: d = r t # 17. From the same starting point, Jane drove north at 50 mph for 3 hours and Tom drove east at 40 mph for 4 hours. How far are they directly apart from each other? Jane: 50 • 3 = 150 miles 40 • 4 = 160 miles Tom: They are 150 + 160 = 310 miles apart.