Chapter 8 linear appliCaTionS 285

They will need 1623 hours or 16 hours 40 minutes (^23 of an hour is

(^23) of 60 minutes— 2
3 ⋅=^6040 ) to complete the run.
(b) Press 1 has produced 3(1200) = 3600 buttons alone, so there remains
45,000 − 3600 = 41,400 buttons to be produced. Let t represent the
number of hours the presses, running together, need to complete
the job.
(^) Worker Quantity Rate Time
Press 1 1200 t 1200 t
Press 2 1500 t 1500 t
Together 41,400 41,400/t t
The equation to solve is 1200t + 1500t = 41,400. (Another equation
that works is 1200 + 1500 = 41,400/t.)
1200 1500 41 400
2700 41 400
41 400
2700
151
tt
t
t
t
+=

=

,
,
,
33
The presses will need^1531 hours or 15 hours 20 minutes to com-
plete the run.

Distance Problems

Another common word problem is the distance problem, sometimes called the

uniform rate problem. The underlying formula is d = rt (distance equals rate

times time). From d = rt, we have two other equations: r = d/t and t = d/r. These

problems come in many forms: two bodies traveling in opposite directions, two

bodies traveling in the same direction, two bodies traveling away from each

other or toward each other at right angles. Sometimes the bodies leave at the

same time, sometimes one gets a head start. Usually they are traveling at differ-

ent rates, or speeds. As in all applied problems, the units of measure must be

consistent throughout the problem. For instance, if rates are given in miles per

hour and time is given in minutes, we convert minutes to hours. We could con-

vert miles per hour into miles per minute, but this can be awkward.