Chapter 10: Work Problems

what is the rate-time-work formula?

work = rate x time

rate =

work / time

rate is considered the

pace that work is performed

work is considered

the work completed

time is considered

the time to complete the work

if Alex can build a shed in 6 days what is his rate?

1 shed / 6 days

be careful to express the work rate as _____________ and not as time per unit of work

work per unit of time

single worker problems ask us to find

an object’s rate, work time, or work performed

how would you solve this problem: if cal can paint f figurines in m minutes, what expression represents the number of figurines he can paint in z seconds?

rate = f / m

z seconds x 1 minutes / 60 seconds = z/60 minutes

work = f/m x z/60 min = fz/60m

what is the formula when two objects work together?

work _{object 1} + work _{object 2} = work_total

how do you set up the equations for two objects that work together for the same amount of time?

work_{object 1} + work _object2=work_total

time_{object 1}= time _{object 2}

if two objects work together for the same unknown amount of time, let t=

each object’s work time

for problems where two objects begin a job together, but one object stops before completion, the object that finishes the remaining job alone will have the ______ individual time

greater

if objects A and B work together for t minutes, and then object B works alone for an extra x minutes, then objects A’s work time is _____ minutes and object B’s work time is _______ minutes

t, t+x

Ann and Bob must wash 130 dishes. Ann washes 4 dishes per minute, and Bob washes 3 dishes per minute. they work together for 10 minutes, after which Bob stops working. How many additional minutes will it take Ann to finish alone?

Ann—> rate=4 dishes/minute, time = 10+a, work = 40+4a dishes

Bob—> rate= 3 dishes/minute, time 10 minutes, work = 30 dishes

40+4a + 30 = 130

4a + 70 = 130

4a = 60

a = 15

when two objects work together, but one object has an unknown time, if t= the time it takes an object to complete the job, then _____ is the object’s rate

1/t

working alone at a constant rate, Wan can paint a fence in 5 hours. If Wan and Fran work together at their respective constant rates, they can paint the fence in 3 hours. How many hours would it take Fran to paint the fence by herself?

wan—> rate= 1/5 hours, time=3 hours, work = 3/5fence

fran—> rate=1/t hours, time = 3 hours, work =3/t hours

3/5 fence +3/t fence = 1 fence

5t (3/5 + 3/t =1)

3t +15= 5t

15= 2t

the fraction of the job done by a certain object is simply the ratio of

that object’s rate to the combined rate of all the objects

Mary can wash a batch of dishes in 4 hours and Al can wash the same batch of dishes in 6 hours. If the two dishwashers start washing the batch of dishes together for the same amount of time what fraction of the dishes will Al wash?

work_mary= 1 batch/4 hours x t hours = t/4 batch

work_al= 1 batch/6hours x t hours = t/6 batch

work total= t/4 +t/6 = 3t/12 + 2t/12 = 5t/12 batch

t/6 / 5t/12 = t/6 × 12/5t = 12/30 = 2/5

sometimes a problem will present objects that are working in opposition, in which case the total work equals

the difference between the individual work values of the two objects

in opposing worker problems what is the equation?

work_object1 - work_object2 = work_total

if object 1 is x times as fast as object 2 and we let object 2’s rate be the variable r, then object 1’s rate would be

xr

Joe catches fish as the rate that is 4 times as fast as the rate at which Mark catches fish. Together they catch 60 fish per hour. If Mark starts catching fish as fast as Joe does, how many fish will they be able to catch in two hours?

rate_joe = 4r

rate_mark = r

4r+r=60

r=12

4×12 = 48

48 + 48 = 96

96 × 2 = 192

if the rate of one worker is slower or faster than the rate of another worker, what will the rate of the slower object be?

1/t+x

if object 2 takes t minutes to complete a job, and object 1 takes x% less time, then their respective work times and times are

time_{object 1} = t(100-x/100)

time_{object 2} = t minutes

rate_{object 1}= 1/t(100-x/100_ = 100/t(100-x)

rate_{object 2}= 1/t

if object 2 takes t minutes to complete a job, and object 1 takes x% less time, then their respective work times and rates are

time_{object 1}= t(100-x/100)

time_{object 2}= t minutes

rate_{object 1}= 100/t(100-x)

rateobject 2= 1/t

how would you use the proportion method to solve this problem: five workers, each working at the same constant rate, can complete a job working together in six days. If three workers are removed from the group before they begin working, what is the combined rate of the two remaining workers?

start by defining the combined rate of the five workers, which is 1job/6 days

we have the following proportion: 5 workers/combined rate of 5 workers = 2 workers/combined rate of 2 workers

5/1/6=2/n

30n=2

n=2/30 = 1/15