Applications of Rational Equations Solutions

• The rate for a work problem is thought of as: $\frac{\mathbf{number}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{tasks}\mathbf{ }\mathbf{or}\mathbf{ }\mathbf{jobs}}{\mathbf{time}\mathbf{ }\mathbf{it}\mathbf{ }\mathbf{takes}\mathbf{ }\mathbf{to}\mathbf{ }\mathbf{complete}}$

• A rate of work problem is a transformation of the equation    d = rt   .

  • d is    distance    and is the job being completed

  • r is    rate   

  • t is    time     

• When the rate of work for one person is    combined    with the rate of work for another person, the task that needs to be completed can be finished    faster    because those rates of work are combined using d = rt.

• Using a table to set up problems can help determine the information you have and the information you need to solve for:
  

  	Person 1		Person 2		Working Together
  time:	x		y		t
  rate of work:	$\frac{1}{x}$	+	$\frac{1}{y}$	=	$\frac{1}{t}$

Example 6

Solve.

Natalee and Grant are shoveling snow from a very long driveway after a snow storm. It takes Natalee 6 hours to completely shovel the driveway alone. Grant is able to shovel the driveway alone in 10 hours. How long would it take if Natalee and Grant work together?

Plan
Identify key information in the problem

Organize time and rate in a chart

Write and solve an equation

Implement

	Natalee		Grant		Together
time:	6		10		t
rate:	$\frac{1}{6}$	+	$\frac{1}{10}$	=	$\frac{1}{t}$

$$\begin{gathered} \frac{1}{6}+\frac{1}{10}=\frac{1}{t}\begin{array}{cc} & \mathrm{LCD}\left(6, 10, t\right)=30t\end{array} \\ 30t\left(\frac{1}{6}+\frac{1}{10}=\frac{1}{t}\right) \mathrm{or} \frac{1}{6}\left(\frac{5t}{5t}\right)+\frac{1}{10}\left(\frac{3t}{3t}\right)=\frac{1}{t}\left(\frac{30}{30}\right) \end{gathered}$$

$\begin{array}{rcl}\frac{5t}{30t}+\frac{3t}{30t}& =& \frac{30}{30t}\\ 5t+3t& =& 30\\ 8t& =& 30\\ t& =& \frac{30}{8}=\frac{15}{4}\\ t& =& 3\frac{3}{4}\end{array}$

Explain
It will take Natalee and Grant $3\frac{3}{4}$ hours to shovel the driveway together.

Note

You do not have to use a chart to solve work problems; however, it can be helpful to organize the information and determine what value you need to solve for.