[distance rate time][section 19]
(1.) Sam
starts from a certain place, travelling at 4 km per
hour. Five hours later, Cliff starts from the same
place and travels in the same
direction at 6 km per hour
In how many hours will Cliff
overtake Sam?
4(t
+ 5) = 6t here is the
problem
4t + 20 = 6t multiply thru parentheses
-4t -4t subtract 4t from each side
__________________
20 = 2t subtract
___ ____
2 2
divide each side by 2
10 = t divide and cancel
result: 10 hours
(2.) A freight train leaves a station and
travels at a rate
of 30 miles per hour. Two hours later, an express train
leaves the same station travelling
in the same direction
at a rate of 50 miles per
hour. In how many hours will
the express train overtake the freight
train?
30(t
+ 2) = 50t here is the
problem
30t + 60 = 50t multiply thru parentheses
-30t -30t subtract 30t from each side
____________________
60 = 20t subtract
____ ____
20 20
divide each side by 20
3 = t divide and cancel
result: 3 hours
(3.) Sam left home on his bicycle
travelling at the rate of 6
miles per hour. One hour later, Cliff set out on the
same route travelling at the rate of
8 miles per hour. In
how many hour will Cliff overtake
Sam?
6(t + 1) = 8t here is the problem
6t + 6 = 8t multiply thru parentheses
-6t -6t
subtract 6t from each side
________________
6 = 2t subtract
____ ___
2 2
divide each side by 2
3 = t divide and cancel
result: 3 hours
(4.) A train travelling at 80 km/hr
started from West
Springfield toward East
Springfield. An hour later,
a train travelling at 64 km/hr
started from East
Springfield toward West
Springfield. West Springfield
and East Springfield are 656
kilometers apart. How
long will it take the two trains to pass
each other if
they are travelling on parallel
tracks?
80t + 64t = 656 here is the problem
144t = 656 combine like terms
_____ _____
144 144 divide each side by 144
t = 4 5/9 hours divide and cancel
result: 4 5/9 hours
(5.) Two cars start at the same time at
the same place and
travel in opposite directions. One travels at the rate
of 45 km/hr and the other at the
rate of 40 km/hr. In
how many hours will they be 255
kilometers apart?
45t + 40t = 255 here is the problem
85t = 255 combine like terms
____ ____
85 85 divide each side by 85
t = 3 divide and cancel
result: 3 hours
(6.) A plane leaves an airport at 10:00
flying north. Another
plane start from the same airport
at 12:00 flying due
south at twice the speed of the
first plane. At 2:00
they are 1920 km apart. Find the rate of speed of each
plane.
4r + 2(2r) = 1920 here is the problem
4r + 4r = 1920 multiply
8r = 1920 combine like terms
____
_____
8 8 divide each side by 8
r = 240 divide and cancel
result: Their rates of speed are 240 km/hr & 480
km/hr.
(7.) Sam left campus 4 hours after
Cliff did. They travelled
in opposite directions. Sam drove at 48 km/hr and Cliff
at 72 km/hr. How many hours after Sam left were they
1008 km apart?
48t + 72(t + 4) = 1008 here is the problem
48t + 72t + 288 = 1008 multiply thru parentheses
120t + 288 = 1008 combine like terms
-288 -288
subtract 288 from each side
_____________________
120t =
720 subtract
______ ____
120 120 divide each side by 120
t = 6 divide and cancel
result: 6 hours
(8.) A round trip in a helicopter lasted
4.5 hours. The
average speed going was 160
km/hr; the average speed
returning was 80 km/hr. What was the helicopter's
greatest distance from the
airport?
d
d
_____ + ______ = 4.5 here is the problem
160 80
d + 2d = 720 multiply thru by 160 and cancel
3d = 720 combine like terms
___ ____
3 3
divide each side by 3
d = 240 divide and cancel
result: 240 km
(9.) Sam jogs from his house to campus at a rate
of 9 km/hr.
On the return trip from campus, he
jogs at a rate of
6 km/hr. The round trip takes 1 hour and 15 minutes.
How far is the campus from Sam's
house?
(d/9) + (d/6) = 1.25 here is the problem
2d + 3d = 22.5 multiply thru by 18, cancel
5d = 22.5 combine like terms
____ ____
5 5
divide each side by 5
d = 4.5 divide and cancel
result: 4.5 km
(10.) Sam had been driving 24 km/hr for
8 hours when he
was overtaken by Cliff who left
the same starting point
5 hours after Sam. Find Cliff's rate of speed.
24(8) = 3r here is the problem
192 = 3r multiply
_____ ____
3 3
divide each side by 3
64 = r divide and cancel
result: 64 km/hr.