Work Problems: Guided Practice

The concepts needed to solve work problems were thoroughly discussed in Work Problems: Basic Concepts.

Work Problems: Quick-and-Easy Estimates discussed estimating techniques which are used on this page.

This lesson gives a detailed solution for the two-person work problem shown below. There is also step-by-step guided practice, so you can achieve mastery.

A Typical Work Problem

Carol mows a lawn in 5 hours.

Julia mows the same lawn in only 3 hours.

How long will it take if they mow the lawn together?

Detailed Solution: Both Individual Rates Known; Find the Time it Takes when Working Together

• Check the Individual Rates Assumption:

  Two people can mow a lawn together without changing their individual rates, providing (for example):

  • they each have their own mower
  • the lawn is big enough that they can be working on different sections at the same time
  • they don't slow down due to chatting with the other person

  In order to proceed, we'll assume that these conditions hold.

• Get an Estimate:

  From the prior section, the two-person time must be between $\,\frac{5}{2} = 2.5\ $ hours and $\,\frac{3}{2} = 1.5\ $ hours.

• Identify and Rename the Individual Rates:

  Carol's work rate is one job per $\,5\,$ hours, where the ‘job’ is ‘mowing a (particular) lawn’.

  Rename this rate so the denominator is ‘$1$ hour’ instead of ‘$5$ hours’. (Remember that dividing by $\,5\,$ is the same as multiplying by $\frac15$.)

  Carol's rate:

  $$ \begin{align} \frac{1\text{ job}}{5\text{ hr}}\ \ &=\ \ \overset{\rm divide\ top\ and\ bottom\ by\ 5} { \overbrace { \frac{1\text{ job}}{5\text{ hr}}\cdot \frac{\frac1{5}}{\frac1{5}} } }\cr\cr &= \ \ \frac{\frac1{5}\text{ job}}{1\text{ hr}} \ \ = \ \ \frac{1}{5}\ \ \frac{\text{job}}{\text{hr}} \end{align} $$

  So, Carol does $\frac15$ of the job in $1$ hour.

  Now that you've seen how it's done—you definitely don't need to show all this work! You can just ‘pull the numbers to the front’.

  Rename Julia's rate in one easy step:

  Julia's rate:

  $$ \frac{1\text{ job}}{3\text{ hr}}\ \ = \ \ \frac{1}{3}\ \ \frac{\text{job}}{\text{hr}} $$

  So, Julia does $\frac13$ of the job in $1$ hour.

• Compute the Combined Rate:

  Under the Individual Rates Assumption, the combined rate is the sum of the individual rates:

  $$ \begin{align} &\text{combined rate}\cr &\quad=\ \ \text{sum of individual rates}\cr &\quad=\ \ \overset{\rm Carol's\ rate}{\overbrace{\frac15\ \frac{\text{job}}{\text{hr}}}} + \overset{\rm Julia's\ rate}{\overbrace{\frac13\ \frac{\text{job}}{\text{hr}}}}\cr\cr &\quad=\ \ \frac{3}{15}\frac{\text{job}}{\text{hr}} + \frac{5}{15} \frac{\text{job}}{\text{hr}}\cr\cr &\quad=\ \ \frac{8}{15}\ \frac{\text{job}}{\text{hr}}\ \ \text{(add fractions)} \end{align} $$

  Working together, they do $\,\frac{8}{15}\,$ of the job in $\,1\,$ hour.

• Rename the Combined Rate:

  We want to know how long it takes to do $\,1\,$ job, working together. To find this, rename the combined rate so the numerator is $\,1\,.$

  Remember: any number, multiplied by its reciprocal, gives $\,1\,.$

  Combined rate:

  $$ \begin{align} \frac{\frac{8}{15}\text{ job}}{1\text{ hr}}\ \ &=\ \ \overset{\rm multiply\ top\ and\ bottom} { \overset{\rm by\ the\ reciprocal\ of\ 8/15} { \overbrace { \frac{\frac8{15}\text{ job}}{1\text{ hr}}\cdot \frac{\frac{15}8}{\frac{15}8} } } }\cr\cr &=\ \ \frac{1\text{ job}}{\frac{15}8\text{ hr}} \end{align} $$

  Alternatively:

  $$ \begin{align} \frac{8 \text{ job}}{15 \text{ hr}}\ \ &= \ \ \frac{8 \text{ job}}{15 \text{ hr}}\cdot\frac{\frac 18}{\frac 18}\cr &= \ \ \frac{1\text{ job}}{\frac{15}8\text{ hr}} \end{align} $$

  Together, they can do $\,1\,$ job in $\frac{15}8$ hours.

• State the Conclusion in a Desired Way:

  Note that $\,\frac{15}8\,$ hours is the exact solution. No decimal approximation has been done up to this point. Whenever possible, get an exact answer—do any needed approximation at the last step only.

  Here, we'll choose to round the answer to one decimal place. Using your calculator, $\,15/8\,$ is approximately $\,1.9\,.$

  Together, Carol and Julia can mow the lawn in about $\,1.9\,$ hours. (Note that this agrees with our estimate.)

Step-by-Step Guided Practice

• Click ‘New Problem’ to get started.
• Want a hint for each step? Click the ‘Hint?’ button.
• When you're ready to check your answer, click the ‘Show’ button.

Back to the top to try another problem!