Some work problems require part of the work being performed by one worker before the other worker joins in, or both start the job and one finishes the job. In these cases, the together quantity and one of the individual quantities will not be ‘‘1.’’ Take the time the one worker works alone divided by the time that worker requires to do the job alone, then subtract from 1. This is the proportion left over for both to work together.
Examples
Jerry needs
40 minutes to mow the lawn. Lou can mow the same lawn in 30 minutes. If Jerry
works alone for 10 minutes then Lou joins in, how long will it take for them to
finish the job?
Because Jerry worked for 10 minutes, he did
10/40 = ¼ of the job alone. So, there is 1 – ¼ = ¾ of the job remaining when Lou started working. Let t
represent the number of minutes they worked together—after Lou joins in. Even
though Lou does not work the entire job, his rate is still 1/30.
The equation
to solve is 1/40 + 1/30 = 3/4t. The LCD is 120t.
Together,
they will work 90/7 =12 6/7 minutes.
A pipe can
fill a reservoir in 6 hours. Another pipe can fill the same reservoir in 4
hours. If the second pipe is used alone for 2 ½ hours, then the first pipe
joins the second to finish the job, how long will the first pipe be used?
The amount of time Pipe I is used is the
same as the amount of time both pipes work together. Let t represent the number
of hours both pipes are used. Alone, the second pipe performed 2 ½ parts of a
4-part job.
The equation
to solve is 1/6 + 1/4 = 3/8t. The LCD is 24t.
Both pipes
together will be used for 9/10 hours or 9/10 · 60 = 54 minutes. Hence, Pipe I
will be used for 54 minutes.
Press A can
print 100 fliers per minute. Press B can print 150 fliers per minute. The
presses will be used to print 150,000 fliers.
(a) How
long will it take for both presses to complete the run if they work together?
(b) If
Press A works alone for 24 minutes then Press B joins in, how long will it take
both presses to complete the job?
These
problems are different from the previous work problems because the rates are
given, not the times.
Before, we used Q = rt implies r = Q/t.
Here, we will use Q = rt to fill in the Quantity boxes.
(a) Press
A’s rate is 100, and Press B’s rate is 150. The together quantity is 150,000.
Let t represent the number of minutes both presses work together; this is also
how much time each individual press will run.
Press A’s quantity is 100t, and Press B’s
quantity is 150t. The together rate is r = Q/t = 150,000/t.
In this problem, the quantity
produced by Press A plus the quantity produced by Press B will equal the
quantity produced together. This gives the equation 100t + 150t = 150,000. (Another
equation that works is 100 + 150 = 150,000/t.)
The presses will run for 600 minutes or 10 hours.
(b) Because
Press A works alone for 24 minutes, it has run 24 ×
100 = 2400 fliers. When Press B begins its run, there are 150,000 – 2400 =
147,600 fliers left to run. Let t represent the number of minutes both presses
are running. This is also how much time Press B spends on the run. The boxes
will represent work done together.
The equation to solve is 100t + 150t
= 147,600. (Another equation that works is 100 + 150 = 147,600/t.)
The
presses will work together for 590.4 minutes or 9 hours 50 minutes 24 seconds.
(This is 590 minutes and 0.4(60) = 24 seconds.)
Practice
1. Neil
can paint a wall in 45 minutes; Scott, in 30 minutes. If Neil begins painting
the wall and Scott joins in after 15 minutes, how long will it take both to
finish the job?
2. Two
hoses are used to fill a water trough. The first hose can fill it in 20 minutes
while the second hose needs only 16 minutes. If the second hose is used for the
first 4 minutes and then the first hose is also used, how long will the first
hose be used?
3. Jeremy
can mow a lawn in one hour. Sarah can mow the same lawn in one and a half
hours. If Jeremy works alone for 20 minutes then Sarah starts to help, how long
will it take for them to finish the lawn?
4. A
mold press can produce 1200 buttons an hour. Another mold press can produce
1500 buttons an hour. They need to produce 45,000 buttons.
(a) How
long will be needed if both presses are used to run the job?
(b) If
the first press runs for 3 hours then the second press joins in, how long will
it take for them to finish the run?
Solutions
1. Neil
worked alone for 15/45 = 1/3 of the job, so 1 – 1/3 = 2/3 of the job remains.
Let t represent the number of minutes both will work together.
The
equation to solve is 1/45 + 1/30 = 2/3t. The LCD is 90t.
It
will take Scott and Neil 12 minutes to finish painting the wall.
2. Hose
2 is used alone for 4/16 = 1/4 of the job, so 1 – ¼ = 3/4 of the job remains.
Let t represent the number of minutes both hoses will be used.
The
equation to solve is 1/20 + 1/16 = 3/4t. The LCD is 80t.
Both hoses will be used for 6 2/3 minutes or 6
minutes 40 seconds. Therefore, Hose 1 will be used for 6 2/3
minutes.
3. Some
of the information given in this problem is given in hours and other
information in minutes. We must use only one unit of measure. Using minutes as
the unit of measure will make the computations a little less messy. Let t represent
the number of minutes both Sarah and Jeremy work together. Alone, Jeremy completed 
of the job, so 
of the job remains to be done.
The
equation to solve is 1/60 + 1/90 = 2/3t. The LCD is 180t.
They
will need 24 minutes to finish the lawn.
4.
(a) Let
t represent the number of hours the presses need, working together, to complete
the job.
The equation to solve is 1200t + 1500t = 45,000. (Another equation that
works is 1200 + 1500 = 45,000/t.)
They will need 16 2/3 hours or 16 hours 40 minutes (2/3 of an hour is 2/3
of 60 minutes—2/3 – 660 = 40) 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.
The equation to solve is 1200t + 1500t = 41,400. (Another equation that
works is 1200 + 1500 = 41,400/t.)
The presses will need 15 1/3 hours or 15 hours 20 minutes to complete the run.
“Sumber Informasi”
Thanks for reading Work Problems - 2. Please share...!
