[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.