1. THE “REALLY TOUGH STUFF”
STORY PROBLEMS II
THE “REALLY TOUGH STUFF”

2. General Directions for d = rt problems
Read the problem carefully.
Ask yourself, “What am I trying to find.”
Determine what kind of a problem it is.
Write your equation and solve
Ask yourself, “Did I answer the question asked?”

3. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
What am I trying to find?
How long does it take the MOTORBOAT to catch the canoe.
In other words, how long is the motorboat on the water?

4. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
This problem contains speed (rate) and time.

5. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
Use d = rt
Make a table of the information.

6. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
Rate
Time
Distance
Canoe
Motorboat

7. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
Rate
Time
Distance
Canoe 12
Motorboat

8. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
Rate
Time
Distance
Canoe 12
Motorboat 21

9. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
Rate
Time
Distance
Canoe 12 t
Motorboat 21

10. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
Rate
Time
Distance
Canoe 12 t
Motorboat 21
t - 2

11. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
Rate
Time
Distance
Canoe 12 t
12t
Motorboat 21
t - 2

12. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
Rate
Time
Distance
Canoe 12 t
12t
Motorboat 21
t - 2
21(t - 2)

13. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
The motorboat has caught the canoe (same direction)
The distances traveled by both must be the same.
Set the two distances equal to each other and solve.

14. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
12𝑡=21(𝑡−2)

15. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
12𝑡=21(𝑡−2)
12𝑡=21𝑡−42

16. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
12𝑡=21(𝑡−2)
12𝑡=21𝑡−42
−9𝑡=−42

17. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
12𝑡=21(𝑡−2)
12𝑡=21𝑡−42
−9𝑡=−42
𝑡= −42 −9

18. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
12𝑡=21(𝑡−2)
12𝑡=21𝑡−42
−9𝑡=−42
𝑡= −42 −9 =4. 6

19. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
12𝑡=21(𝑡−2)
You can convert your answer to hours and minutes if you like.
12𝑡=21𝑡−42
−9𝑡=−42
𝑡= −42 −9 =4. 6

20. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
12𝑡=21(𝑡−2
You can convert your answer to hours and minutes if you like.
12𝑡=21𝑡−42
The time you have found is the time for the canoe. You were not asked to find that. Find the motorboat’s time.
−9𝑡=−42
𝑡= −42 −9 =4. 6

21. A canoe leaves a campsite and travels at an average speed of 12 mph
A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe?
12𝑡=21(𝑡−2
You can convert your answer to hours and minutes if you like.
12𝑡=21𝑡−42
The time you have found is the time for the canoe. You were not asked to find that. Find the motorboat’s time.
−9𝑡=−42
𝑡= −42 −9 =4. 6
It takes the motorboat 2 hours and 40 minutes to catch the canoe. Or you can just leave it hours.

22. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
This problem contains rate and time (but it is given as “clock” time.)
If you go to a location, then come back, you have a “round trip” situation
In a round trip, the distances are equal.

23. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
Make a table.
Add a column for the “clock” time.

24. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
Rate
Time
Distance
Clock Up
Down

25. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
Rate
Time
Distance
Clock Up 4
Down

26. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
Rate
Time
Distance
Clock Up 4
Down 6

27. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
Rate
Time
Distance
Clock Up 4 t
Down 6

28. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
Rate
Time
Distance
Clock Up 4 t
Down 6
3 - t

29. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
Rate
Time
Distance
Clock Up 4 t 4t
Down 6
3 - t

30. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
Rate
Time
Distance
Clock Up 4 t 4t
Down 6
3 - t
6(3 – t)

31. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
Rate
Time
Distance
Clock Up 4 t 4t
9:00 am
Down 6
3 - t
6(3 – t)

32. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
Rate
Time
Distance
Clock Up 4 t 4t
9:00 am
Down 6
3 - t
6(3 – t)
????

33. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
4𝑡=6(3−𝑡)

34. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
4𝑡=6(3−𝑡)
4𝑡=18−6𝑡

35. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
4𝑡=6(3−𝑡)
4𝑡=18−6𝑡
10𝑡=18

36. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
4𝑡=6(3−𝑡)
4𝑡=18−6𝑡
10𝑡=18
𝑡= 18 10

37. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
4𝑡=6(3−𝑡)
4𝑡=18−6𝑡
10𝑡=18
𝑡= =1.8 hours

38. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
4𝑡=6(3−𝑡)
This does NOT answer the question asked. This is how long it takes to get to the top of the hill, not AT WHAT TIME?
4𝑡=18−6𝑡
10𝑡=18
𝑡= =1.8 hours

39. Suppose you begin to hike up a hill at 9:00 a. m
Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill?
4𝑡=6(3−𝑡)
This does NOT answer the question asked. This is how long it takes to get to the top of the hill, not AT WHAT TIME?
4𝑡=18−6𝑡
10𝑡=18
𝑡= =1.4 hours
If you left at 9:00 am, and it took 1.8 hours (or 1 hour 48 minutes), then you arrived at the top of the hill at 10:48 am.

40. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
The jets are flying in “opposite directions.”
To solve an opposite direction problem, ADD the distances traveled to find the distance apart.

41. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Make a table.

42. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
Eastbound

43. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25
Eastbound

44. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25
Eastbound r

45. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25 2
Eastbound r

46. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25 2
Eastbound r

47. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25 2
2(r+25)
Eastbound r

48. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25 2
2(r+25)
Eastbound r 2r

49. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25 2
2(r+25)
Eastbound r 2r
2𝑟+2 𝑟+25 =1250

50. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25 2
2(r+25)
Eastbound r 2r
2𝑟+2 𝑟+25 =1250
2𝑟+2𝑟+50=1250

51. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25 2
2(r+25)
Eastbound r 2r
2𝑟+2 𝑟+25 =1250
2𝑟+2𝑟+50=1250
4𝑟+50=1250

52. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25 2
2(r+25)
Eastbound r 2r
2𝑟+2 𝑟+25 =1250
4𝑟=1200
2𝑟+2𝑟+50=1250
4𝑟+50=1250

53. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25 2
2(r + 25)
Eastbound r 2r
2𝑟+2 𝑟+25 =1250
4𝑟=1200
2𝑟+2𝑟+50=1250
𝑟=300
4𝑟+50=1250

54. Two jets leave Dallas at the same time and fly in opposite directions
Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet.
Rate
Time
Distance
Westbound
r + 25 2
2(r +25)
Eastbound r 2r
2𝑟+2 𝑟+25 =1250
4𝑟=1200
2𝑟+2𝑟+50=1250
𝑟=300
4𝑟+50=1250
The rate (r) is the rate of the eastbound jet. Therefore, the westbound jet is flying at 325 mph.

55. Wrap-up There are three type of 𝑑=𝑟𝑡 problems. Same direction
Round-trip
Opposite direction
In same direction and round-trip problems, you set the distances equal to each other.
In opposite direction, you add the distances together.

56. Assignment
2.5B: