1. 3.4
Rates
1. Solve problems involving two objects traveling in opposite directions.
2. Solve problems involving two objects traveling in the same direction.

2. distance = rate ∙ time Rate Problems distance = distance d = d
same direction
distance 1 + distance 2 = total distance
d1 + d2 = dt
opposite directions
Find rate and time for each vehicle.

3. Rate Problems d = d d1 + d2 = dt The trains will meet in 6.5 hours.
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
d1 + d2 = dt
55t + 27t =
533
82t = 533
55 mph
27 mph
t = 6.5
533 miles
The trains will meet in 6.5 hours.
d = r t
Categories
Rate
Time
Distance
Passenger
Freight
55 mph t
= 6.5
55t
27 mph t
27t

4. Rate Problems d = d d1 + d2 = dt
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
d1 + d2 = dt
653t + 560t =
2426
1213t = 2426
560 mph
653 mph
t = 2
2426 miles
The planes will be 2426 miles apart in 2 hours.
d = r t
Categories
Rate
Time
Distance
South
North
653 mph t
= 2
653t
560 mph t
560t

5. Morrison leaves Cincinnati at 4:00 pm, heading for NYC
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
d = d
d1 + d2 = dt
3.5r = 3(r + 10)
4:30
+ 10 mph
3.5r = 3r + 30
Be sure you answer the question that is asked!!
35r = 30r + 300
What did we find?
5r = 300
Did we answer the question?
r = 60
How fast is Barbara driving?
r + 10 =
= 70 mph
How far have they traveled?
d = rt
= (70)(3) =
210 miles
d = r t
Categories
Rate
Time
Distance
Morrison
Barbara r
= 60
3.5
3.5 r
r + 10
= 70 3
3(r + 10)

6. Rate Problems d = d d1 + d2 = dt
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
d = d
d1 + d2 = dt
40 mph
50 mph
40t + 50t =
225
90t = 225
225 miles
t = 2.5
They will be 225 miles apart in 2.5 hours.
d = r t
Categories
Rate
Time
Distance
Monica
Chandler
40 mph t
= 2.5
40t
50 mph t
50t

7. At 6 a. m. a freight train leaves Washington D. C. traveling at 50 mph
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
d1 + d2 = dt
Freight train
50t = 75(t – 3)
9:00
75 mph
50t = 75t – 225
Passenger train
– 25t = – 225
What did we find?
t = 9
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
d = r t
Categories
Rate
Time
Distance
Freight
Passenger 50 t
= 9
50t 75
t – 3
= 6
75(t – 3)