The prior section covered Work Problems: Basic Concepts.
You'll learn to get exact answers in the next section, Work Problems: Guided Practice.
This current section discusses important estimating techniques for work problems.
What's the punchline?
Together, the time it will take Carol & Julia to mow the lawn together must
be between $\,\frac 12(5) = 2.5\,$ hours and $\,\frac 12(3) = 1.5\,$ hours.
Why? Keep reading!
Getting exact answers to work problems usually involves tools like adding fractions, renaming fractions, and solving equations.
However, you can often get inyourhead quickandeasy estimates!
When possible, you should definitely do this, because it can catch silly mistakes and give confidence in your answers.
As is typical for work problems, the Individual Rates Assumption must be true.
This intuitive fact is used in several of the estimates:
The phrase ‘twoperson time’ is used as a shorthand for ‘the time it takes to complete the job when two people are working together’.
Estimate based on a Single Person's Time Suppose two people will be working on a job. Suppose you know how long it takes one person to do the job when they work alone. However, you don't know if this is the faster or slower person. With a second person working, the job will get done faster. If the second person is wicked fast, then the twoperson time can be close to zero. If the second person is no help at all (takes nearinfinite time), then the twoperson time will be close to the oneperson time. Summarizing: then, working with a second person, the job will be completed between times $\,0\,$ and $\,p\,$. 
The black point gives one person's time to complete a job. The shaded interval shows the possible twoperson times. 

Estimate based on a FASTER Person's Time Let $\,T\,$ be the time it takes a FASTER person, working alone, to complete a job.
then, working with a second person, the job will be completed between times $\,\frac{1}{2}T\,$ and $\,T\,$. 
This point gives a FASTER person's time to complete a job. The shaded interval shows the possible twoperson times. 

Estimate based on a SLOWER Person's Time Let $\,t\,$ be the time it takes a SLOWER person, working alone, to complete a job.
then, working with a second person, the job will be completed between times $\,0\,$ and $\,\frac{1}{2}t\,$. 
This point gives a SLOWER person's time to complete a job. The shaded interval shows the possible twoperson times. 

Estimate based on BOTH Individual Times Suppose we know the times it takes both people, working alone, to complete a job. Call these two times $\,T\,$ and $\,t\,$. If the times are different:
Since $\,T\le t\,$, we have $\,\frac 12T\le \frac 12t\,$. This fact tells us some important things:
It can look two different ways, as shown below. The changingpoint in behavior is when the slower time ($\,t\,$) is twice the faster time ($\,T\,$):

The green interval shows the possible twoperson times based on the faster time alone. The purple interval is from half the faster time to half the slower time. The red interval shows the possible twoperson times based on the slower time alone. The best twoperson time estimate is between the dashed lines! It's a bit of a pain to have to think of the intersection of two intervals when trying to get a quickandeasy estimate. Here's a shortcut to finding the best estimate when you know the times for both workers:
Here are examples:

One final caution—don't fall into the trap of mixing up rates and times.
Here's an example.
Suppose Carol completes a job in $\,7\,$ minutes.
‘$\,7\,$ minutes’ is a time.
‘One job per $\,7\,$ minutes’ (or $\displaystyle\,\frac{1\text{ job}}{7\text{ min}}\,$ or $\displaystyle\,\frac 17\ \frac{\text{job}}{\text{min}}\,$) is the corresponding rate.
The numbers $\,7\,$ and $\,\frac 17\,$ are reciprocals.
If a number is big, its reciprocal is small.
If a number is small, its reciprocal is big.
You need to keep track of whether you're working with times or rates, because they behave very differently!
Under the Individual Rates Assumption, rates will combine.
But, times don't combine!
For example, suppose it takes Ray $\,5\,$ minutes to do the same job that Carol does in $\,7\,$ minutes.
Carol's rate is $\displaystyle\,\frac{1\text{ job}}{7\text{ min}} = \frac 17\ \frac{\text{job}}{\text{min}}\,$.
Ray's rate is $\displaystyle\,\frac{1\text{ job}}{5\text{ min}} = \frac 15\ \frac{\text{job}}{\text{min}}\,$.
The combined rate will be:
$$\,\frac{1}{7}\frac{\text{job}}{\text{min}} + \frac{1}{5}\frac{\text{job}}{\text{min}} \quad=\quad
\frac{5}{35}\frac{\text{job}}{\text{min}} + \frac{7}{35}\frac{\text{job}}{\text{min}}
\quad=\quad \frac{12}{35} \frac{\text{job}}{\text{min}}$$
However, working together, the time definitely won't be $\,5 + 7\,$ minutes!
Indeed, using the quickandeasy estimate from above,
the twoperson time must be between $\,\frac 12(5) = 2.5\,$ minutes and $\,\frac{1}{2}(7) = 3.5\,$ minutes.
The next section shows how to get the exact answer.
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.However, you can check to see if your answer is correct. 
PROBLEM TYPES:
