3.4
Rates
1. Solve problems involving two objects traveling in
opposite directions.
2. Solve problems involving two objects traveling in the
same direction.
Rate Problems
distance = rate ∙ time
distance = distance
d=d
same direction
distance 1 + distance 2 = total distance opposite directions
d1 + d 2 = dt
Find rate and time for each vehicle.
Rate Problems
What did we find? Did we answer the question?
Two trains are traveling on parallel tracks toward each other
from a distance of 533 miles. If the freight train is traveling at
27 miles per hour and the passenger train is moving at 55 miles
per hour, how long will it take for them to meet each other?
d= d
55t + 27t = 533
82t = 533
27 mph
55 mph
d1 + d2 = dt
t = 6.5
533 miles
The trains will meet in 6.5 hours.
d=rt
Categories
Passenger
Rate
Time
Distance
55 mph
t = 6.5
55t
Freight
27 mph
t
27t
Rate Problems
What did we find? Did we answer the question?
Two planes leave Denver at the same time, one flying north and
the other flying south. If the plane flying south is traveling at
653 miles per hour and the plane flying north is traveling 560
miles per hour, after how many hours will the planes be 2426
miles apart.
d= d
653t + 560t = 2426
1213t = 2426
t=2
653 mph
560 mph
2426 miles
d1 + d2 = dt
The planes will be 2426 miles apart in 2 hours.
d=rt
Categories
South
Rate
Time
Distance
653 mph
t =2
653t
North
560 mph
t
560t
Morrison leaves Cincinnati at 4:00 pm, heading for NYC.
Barbara leaves at 4:30 pm on the same highway, heading toward
NYC. By driving 10 mph faster, Barbara overtakes Morrison at
7:30 pm. How fast is Barbara driving and how far have they
traveled when Barbara catches up with Morrison?
M
B
4:00
d1 + d 2 = dt
d= d
3.5r = 3(r + 10)
+ 10 mph
4:30
3.5r = 3r + 30
Be sure you answer
the question that is
35r = 30r + 300
What did we find?
asked!!
5r
=
300
Did we answer the question?
r = 60
How fast is Barbara driving? r + 10 = 60 + 10 = 70 mph
How far have they traveled? d = rt = (70)(3) = 210 miles
d=rt
Categories
Morrison
Rate
r = 60
Barbara
r + 10 = 70
Time
Distance
3.5
3.5 r
3
3(r + 10)
Rate Problems
What did we find? Did we answer the question?
Monica and Chandler pass each other along a straight road
traveling in opposite directions. Monica is driving at 40 mph
and Chandler at 50 mph. In how many hours will the cars be
225 miles apart?
Monica
Chandler
50 mph
40 mph
d= d
d1 + d2 = dt
40t + 50t = 225
90t = 225
t = 2.5
225 miles
They will be 225 miles apart in 2.5 hours.
d=rt
Categories
Monica
Rate
Time
Distance
40 mph
t = 2.5
40t
Chandler
50 mph
t
50t
At 6 a.m. a freight train leaves Washington D.C. traveling at 50
mph. At 9 a.m. a passenger train leaves the same station traveling
in the same direction at 75 mph. How long will it take the
passenger train to overtake the freight train? How far from the
station will they be at that time?
6:00
50 mph
d= d
50t = 75(t – 3)
50t = 75t – 225
– 25t = – 225
t=9
Freight train
9:00
75 mph
Passenger train
d1 + d 2 = dt
What did we find?
Did we answer the question?
How long will it take for the passenger train to overtake the freight train? 6 hours
How far from the station will they be at that time?
d = rt = (50)(9) = 450 miles
Categories
Freight
Rate
Time
50
t =9
Passenger
75
t–3=6
d=rt
Distance
50t
75(t – 3)