From an Idea to an Equation
In the narrative introduction, work was treated as a measurable quantity. A complete job was represented by \(1\), and a work rate described how much of that job could be completed during one unit of time.
Those ideas can now be summarized by a single algebraic relationship known as the Work Principle:
\[ \boxed{W = RT} \]
where \(W\) represents the amount of work completed, \(R\) represents the rate of work, and \(T\) represents the amount of time spent working.
This relationship is the foundation of nearly every work-rate problem. The details of a problem may change, but the underlying structure remains the same.
The Work Principle
The Work Principle states that the amount of work completed equals the rate at which the work is performed multiplied by the amount of time spent working:
\[ W = RT. \]
The equation can be rearranged to solve for any of the three quantities.
To solve for rate:
\[ R = \frac{W}{T}. \]
To solve for time:
\[ T = \frac{W}{R}. \]
These three forms express the same relationship:
\[ \boxed{W = RT} \qquad \boxed{R = \frac{W}{T}} \qquad \boxed{T = \frac{W}{R}}. \]
Representing a Complete Job
In most algebraic work problems, one entire completed job is assigned the value
\[ W = 1. \]
This convention makes it possible to describe partial work using fractions. If one-half of the job has been completed, then
\[ W = \frac{1}{2}. \]
If three-fourths has been completed, then
\[ W = \frac{3}{4}. \]
Representing a complete job by \(1\) allows different workers, machines, or processes to be compared even when the physical nature of the task differs.
Completion Time and Reciprocal Work Rates
Suppose a worker can complete one entire job in \(t\) hours. Since one complete job is represented by \(1\), substitute
\[ W = 1 \]
and
\[ T = t \]
into the rate formula:
\[ R = \frac{W}{T}. \]
Then
\[ R = \frac{1}{t}. \]
Therefore, a worker who completes a job in \(t\) hours has the work rate
\[ \boxed{R = \frac{1}{t}}. \]
This reciprocal relationship is one of the most important ideas in work-rate problems. Completion time is usually expressed in units such as hours per job, while work rate is expressed in jobs per hour.
Example: Finding a Work Rate
Suppose a machine can complete one production cycle in \(8\) hours. What is its work rate?
A complete job is
\[ W = 1, \]
and the completion time is
\[ T = 8. \]
Using
\[ R = \frac{W}{T}, \]
we obtain
\[ R = \frac{1}{8}. \]
The machine therefore completes
\[ \boxed{\frac{1}{8}} \]
of the job each hour.
Finding Partial Work
Once a work rate is known, the amount of work completed during any period of time can be found using
\[ W = RT. \]
If a worker can complete a job in \(8\) hours, the rate is
\[ R = \frac{1}{8}. \]
If the worker operates for \(3\) hours, then
\[ W = \frac{1}{8}(3). \]
Therefore,
\[ \boxed{W = \frac{3}{8}}. \]
The worker completes three-eighths of the job in three hours.
More generally, if a worker requires \(a\) hours to complete a job and works for \(t\) hours, then
\[ W = \frac{1}{a}t, \]
or equivalently,
\[ \boxed{W = \frac{t}{a}}. \]
Combining Work Rates
When two or more workers or processes contribute toward the same job at the same time, their work rates are added.
If Worker A has rate \(R_A\) and Worker B has rate \(R_B\), then the combined rate is
\[ R_{\text{combined}} = R_A + R_B. \]
Suppose Worker A can complete a job in \(6\) hours and Worker B can complete the same job in \(4\) hours. Their individual rates are
\[ R_A = \frac{1}{6} \]
and
\[ R_B = \frac{1}{4}. \]
Their combined rate is therefore
\[ R_{\text{combined}} = \frac{1}{6}+\frac{1}{4}. \]
Using a common denominator,
\[ R_{\text{combined}} = \frac{2}{12}+\frac{3}{12} = \frac{5}{12}. \]
Together, the two workers complete five-twelfths of the job each hour.
Finding Combined Completion Time
If the two workers continue until the entire job is complete, then
\[ W = 1. \]
The combined rate is
\[ R = \frac{5}{12}. \]
Using
\[ T = \frac{W}{R}, \]
we obtain
\[ T = \frac{1}{\frac{5}{12}} = \frac{12}{5}. \]
Thus,
\[ \boxed{T = 2.4\text{ hours}}. \]
The two workers together complete the job in \(2.4\) hours, or \(2\) hours and \(24\) minutes.
The Standard Combined-Work Formula
Suppose Worker A can complete a job in \(a\) hours, Worker B can complete it in \(b\) hours, and together they complete the job in \(t\) hours.
Their individual rates are
\[ \frac{1}{a} \]
and
\[ \frac{1}{b}. \]
The combined rate is
\[ \frac{1}{a}+\frac{1}{b}. \]
Since working together for \(t\) hours completes one whole job,
\[ t\left(\frac{1}{a}+\frac{1}{b}\right)=1. \]
Dividing by \(t\) gives
\[ \boxed{\frac{1}{a}+\frac{1}{b}=\frac{1}{t}}. \]
This formula is not a separate principle. It is a direct consequence of the Work Principle and the addition of simultaneous work rates.
Opposing Work Rates
Sometimes one process contributes toward completing a task while another process works against it. In these situations, the opposing rate is subtracted rather than added.
The net rate is
\[ R_{\text{net}} = R_{\text{positive}} - R_{\text{negative}}. \]
For example, suppose a pipe can fill a tank in \(4\) hours while a drain can empty the full tank in \(6\) hours.
The filling rate is
\[ \frac{1}{4}, \]
while the draining rate is
\[ \frac{1}{6}. \]
The net rate is therefore
\[ R_{\text{net}} = \frac{1}{4}-\frac{1}{6}. \]
Using a common denominator,
\[ R_{\text{net}} = \frac{3}{12}-\frac{2}{12} = \frac{1}{12}. \]
The tank therefore fills at a net rate of one-twelfth of the tank per hour, so it requires
\[ \boxed{12\text{ hours}} \]
to become full.
The Net-Rate Principle
Combined and opposing work problems can be summarized by one broader relationship:
\[ \boxed{R_{\text{net}} = R_1 + R_2 + R_3 + \cdots} \]
where rates that contribute toward the chosen goal are positive and rates that oppose the goal are negative.
For example,
\[ R_{\text{net}} = R_{\text{worker 1}} + R_{\text{worker 2}} - R_{\text{loss}}. \]
Once the net rate is known, the Work Principle remains
\[ W = R_{\text{net}}T. \]
Pay Attention to Units
Units provide an important way to check whether a work equation makes sense. A work rate may be expressed as
\[ \frac{\text{jobs}}{\text{hour}}. \]
Time may be expressed in hours:
\[ \text{hours}. \]
Multiplying rate by time gives
\[ \frac{\text{jobs}}{\text{hour}} \times \text{hours} = \text{jobs}. \]
The time units cancel, leaving an amount of work. This confirms the structure of
\[ W = RT. \]
Keeping track of units can prevent common mistakes, especially when a problem mixes hours, minutes, days, workers, or machines.
Common Errors
Several errors occur frequently in work-rate problems.
- Adding completion times instead of rates. If one worker takes \(6\) hours and another takes \(4\) hours, their combined rate is \(\frac{1}{6}+\frac{1}{4}\), not \(6+4\).
- Averaging completion times. The average of \(6\) and \(4\) is \(5\), but two workers working together must complete the job faster than the faster worker working alone.
- Forgetting that a complete job is represented by \(1\). Partial work must be measured relative to that whole.
- Confusing time with rate. A worker who takes \(5\) hours has a rate of \(\frac{1}{5}\) job per hour, not \(5\) jobs per hour.
- Ignoring the direction of a rate. A leak or drain should subtract from a filling or production rate when it works against the desired result.
Checking Whether an Answer Is Reasonable
Before accepting a solution, ask whether it is consistent with the situation.
If two workers cooperate continuously, their combined completion time should normally be less than the completion time of the faster worker working alone.
For example, if one worker needs \(6\) hours and another needs \(4\) hours, then the combined time must satisfy
\[ T < 4. \]
A calculated answer of \(5\) hours or \(10\) hours would immediately indicate an error.
Likewise, if a drain is open while a tank is filling, the tank should take longer to fill than it would with the filling pipe operating alone.
These qualitative checks are an important part of mathematical reasoning and often reveal mistakes before detailed calculations are reviewed.
Summary of the Work Principle
The essential relationships developed in this lesson are:
\[ \boxed{W = RT} \]
\[ \boxed{R = \frac{W}{T}} \]
\[ \boxed{T = \frac{W}{R}} \]
For one complete job performed in \(t\) units of time,
\[ \boxed{R = \frac{1}{t}}. \]
For simultaneous contributing processes,
\[ \boxed{R_{\text{net}} = R_1 + R_2 + \cdots}. \]
For opposing processes,
\[ \boxed{R_{\text{net}} = R_{\text{positive}} - R_{\text{negative}}}. \]
These relationships provide the mathematical framework needed to solve the different classes of work-rate problems.
Continue the Lesson
The Work Principle can now be applied to specific problem types, including single-rate problems, combined rates, opposing rates, staggered work, interrupted work, unknown completion times, and workforce problems.
Continue to Part 3: Work-Rate Problems