A Job to Be Done
Imagine that a room needs to be painted. One painter can complete the room in six hours. Another painter can complete the same room in four hours. If both painters work together, how long should the job take?
At first, it may be tempting to work directly with the numbers \(6\) and \(4\). Perhaps they should be added, averaged, or otherwise combined. But those numbers describe completion times, not the amount of work each painter performs during one hour.
The important quantity is therefore not simply how long each painter takes to finish the job. The important quantity is the painter's rate of work.
Thinking of a Job as One Whole
To study work mathematically, it is useful to represent one complete job as a single whole. The job might be painting a room, assembling a machine, filling a tank, processing a batch of data, or completing a construction project. The specific task may change, but the mathematical idea remains the same.
We represent the entire completed job by
\[ 1. \]
A person who has completed half of the job has completed
\[ \frac{1}{2}, \]
while someone who has completed three-fourths of the job has completed
\[ \frac{3}{4}. \]
This way of thinking turns work into a measurable quantity. Instead of saying only that someone has "worked for three hours," we can ask how much of the entire job was completed during those three hours.
Completion Time and Work Rate
Suppose one worker can complete an entire job in five hours. If the worker proceeds at a constant pace, then each hour accounts for one-fifth of the job.
After one hour, the worker has completed
\[ \frac{1}{5} \]
of the job. After two hours, the worker has completed
\[ \frac{2}{5}. \]
After five hours, the worker has completed
\[ \frac{5}{5}=1, \]
which represents the entire job.
This reveals an important relationship: the time required to complete one job and the rate at which the job is completed are reciprocals of one another.
Faster Completion Means a Greater Rate
Consider two workers. Worker A can complete a job in six hours, while Worker B can complete the same job in four hours.
Worker A completes
\[ \frac{1}{6} \]
of the job each hour, while Worker B completes
\[ \frac{1}{4} \]
of the job each hour.
Although \(4\) is smaller than \(6\), the rate \(\frac{1}{4}\) is greater than \(\frac{1}{6}\). This is why a shorter completion time corresponds to a faster work rate.
This inverse relationship is one of the central ideas in work-rate problems:
as completion time decreases, the rate of work increases.
Why Work Rates Add
Return to the two painters. One paints \(\frac{1}{6}\) of the room each hour, and the other paints \(\frac{1}{4}\) of the room each hour.
If they work at the same time without interfering with one another, both contribute toward the same completed job during each hour. Their contributions therefore add.
In one hour, together they complete
\[ \frac{1}{6}+\frac{1}{4}. \]
The completion times \(6\) and \(4\) are not added because they do not represent work performed during the same interval. The rates are added because each rate measures how much of the job is completed during one unit of time.
This distinction is fundamental:
when workers operate together, combine their rates rather than their individual completion times.
When One Process Opposes Another
Not every process contributes positively toward finishing a job. Imagine a pipe filling a tank while a drain allows water to escape. The pipe adds water, while the drain removes it.
The two processes still have rates, but they act in opposite directions. The filling rate contributes positively toward a full tank, while the draining rate works against that goal.
The same idea can appear in many settings: a pump and a leak, production and spoilage, deposits and withdrawals, or any situation in which one process increases a quantity while another decreases it.
The mathematical framework remains the same. What changes is whether a rate contributes to the work or opposes it.
Rate Alone Does Not Determine the Work Completed
A fast worker does not necessarily complete more work than a slower worker if the two work for different lengths of time. Both rate and time must be considered.
For example, suppose Worker A completes \(\frac{1}{4}\) of a job each hour but works for only one hour. Worker A completes
\[ \frac{1}{4} \]
of the job.
Suppose Worker B completes only \(\frac{1}{6}\) of the job each hour but works for three hours. Worker B completes
\[ 3\left(\frac{1}{6}\right)=\frac{1}{2}. \]
Worker B has the slower rate but completes more of the job because Worker B works longer.
This observation leads directly to the relationship between the three quantities at the center of every work problem: work, rate, and time.
The Three Quantities
Every work-rate situation can be analyzed by identifying three quantities:
- Work: how much of the job has been completed.
- Rate: how much of the job is completed per unit of time.
- Time: how long the worker or process operates.
These quantities are not independent. Changing one affects the relationship among the others. A faster rate allows the same amount of work to be completed in less time. More time allows a worker operating at a constant rate to complete more work.
The next part of this lesson formalizes this relationship algebraically as the Work Principle.
Continue the Lesson
The narrative ideas developed here can now be summarized by a single algebraic relationship connecting work, rate, and time.