Algebra Demystified 2nd Ed

430 algebra De mystif ieD

“Stream” distance problems usually involve boats (traveling upstream or

downstream) and planes (traveling against a headwind or with a tailwind).

The boat or plane generally travels in one direction then turns around and

travels in the opposite direction. The distance upstream and downstream is

usually the same. If r represents the boat’s or plane’s average speed traveling

without the “stream,” then “r + stream’s speed” represents the boat’s or plane’s

average speed traveling with the stream, and “r – stream’s speed” represents

the boat’s or plane’s average speed traveling against the stream. Usually, we

are told the total trip time, that is, the time downstream and upstream. If this

is the case, then we use the model “Total time = Time downstream + Time

upstream,” where Timedownstream Distancedownstream

Averagespee

=

dds+ treams’ speed

and

Time upstream = Distanceupstream

Averagespeed–streams’ speed

. You might recognize that

these representations come from Dr=t and t D

r

=. These facts give us the

following equation to solve:

Totaltime Distancedownstream

Averagespeed

=

+streamsspeed

Distanceupstream

Averagespee

’

+

dds− treams’ speed

EXAMPLE

Miami and Pittsburgh are 1000 miles apart. A plane flew into a 50-mph

headwind from Miami to Pittsburgh. On the return flight the 50-mph wind

became a tailwind. The plane was in the air a total of 412 hours for the

round trip. What would have been the plane’s average speed without the

wind?

Let r represent the plane’s average speed (in mph) without the wind. The

plane’s average speed against the wind is r – 50 (from Miami to Pittsburgh)

and the plane’s average speed with the wind is r + 50 (from Pittsburgh to

Miami). The distance from Miami to Pittsburgh is 1000 miles. With this infor-

mation we can use t D

r

 =  to compute the time in the air in each direction.

The time in the air from Miami to Pittsburgh is^1000

r –  50

. The time in the air from

EXAMPLE

Miami and Pittsburgh are 1000 miles apart. A plane flew into a 50-mph

headwind from Miami to Pittsburgh. On the return flight the 50-mph wind

EXAMPLE

Miami and Pittsburgh are 1000 miles apart. A plane flew into a 50-mph