E. Work
The
amount of work, W, accomplished in time, T,
depends on the rate, R, at which the work is being
accomplished. Work problems are quite similar to the problems
of uniform motion. The equation we use is
W = RT
Try
to solve by determining the rate per time period (usually per
hour or per minute). The rate, R, is most often
expressed as the job to do divided by the time, where W
= 1 job. For example, a tractor plows 1/10 of a field each hour;
the job is one field, so the rate is 1/10 of a field per hour.
If it takes x tractors to do one job in 1 hour, then each tractor
works at a rate of 1/x of the job per hour. If it takes x tractors
4 hours to do one job, then each tractor works at one quarter
of the previous rate, or at the rate of 1/4x of the job per hour.
In general, if it takes x tractors y hours to do one job, the
rate that each tractor works is 1/(x × y) of the job per
hour.
Example 18
It takes 3 men 8 hours to paint a house. How long will it take
5 men to paint the same house?
Solution
The per hour rate at which each man works is
R = 1/(3 x 8) = 1/24 houses per hour
The rate for 5 men is (5R). The
work is 1 house. Our equation gives us
1 = 5/24T
Therefore, T = 24/5 = 4.8 hours
or 4 hours and 48 minutes.
NOTE: 0.8 hours is 0.8 × 60 = 48 minutes.
Example 19
Michelle can input a day's invoices into the computer system
in 40 minutes, and John can input the same invoices in 60 minutes.
How long will it take both of them, working simultaneously, to
input the invoices?
Solution
Michelle's rate for doing the job is 1/40 of the job per minute.
John's rate is 1/60 of the job per minute. Let the time they
work be T. Then the sum of the work that Michelle does and the
work that John does must equal one job:
1=(1/40)T + (1/60)T
This is most easily solved by
multiplying by 40(60):
40(60) = [40(60)]/40 ×
T + [40(60)]/60 × T
2400 = 60T + 40T
T = 24 minutes
Example 20
Kelly and Shelley can type the manuscript in 8 hours. Kelly can
type the manuscript alone in 20 hours. How long would it take
Shelley to type the manuscript?
Solution
The rate that Kelly works is 1/20 of the job per hour. Let the
rate that Shelley works be R. To do one job in 8 hours we have
1 = 1/20(8) + R(8)
To solve for R, multiply by 20:
20 = 8 + 20R(8)
12 = 8(20)R
Therefore, R = 12/[8(20)] = 3/40
of the job per hour.
To type the entire manuscript
alone, Shelley takes
