Applying Rational Equations
Learning Objective(s)
· Solve real world problems using rational functions.
Rational expressions and rational equations can be useful tools for representing real life situations and for finding answers to real problems. In particular, they are quite good for describing distancespeedtime questions, and modeling multiperson work problems.
Solving Work Problems
Work problems often ask us to calculate how long it will take different people working at different speeds to finish a task. The algebraic models of such situations often involve rational equations derived from the work formula, W = rt. The amount of work done (W) is the product of the rate of work (r) and the time spent working (t). The work formula has 3 versions:
_{}
_{}
_{}
Some work problems have multiple machines or people working on a project together for the same amount of time but at different rates. In that case, we can add their individual work rates together to get a total work rate. Let’s look at an example:
Example  
Problem  Myra takes 2 hours to plant 500 flower bulbs. Francis takes 3 hours to plant 450 flower bulbs. Working together, how long should it take them to plant 1500 bulbs? 

 
 Myra: 500 bulbs/2 hours = 250 bulbs/hour
Francis: 450 bulbs/3 hours = 150 bulbs/hour 
 Think about how many bulbs each person can plant in one hour. This is their planting rate.
 
 Myra & Francis together: 250 + 150 bulbs/hour = 400 bulbs/hour

 Combine their hourly rates to determine the rate they work together.  
 _{}

 Use one of the work formulas to write a rational equation, for example _{}. We know r, the combined work rate, and we know W, the amount of work that must be done. What we don't know is how much time it will take to do the required work at the designated rate.
 
 _{}
_{}
_{}
t = 3 ¾ hours

 Solve the equation by multiplying both sides by the common denominator, then isolating t.  
Answer  It takes 3 hours 45 minutes for Myra and Francis to plant 1500 bulbs together.  
Other work problems go the other way. We'll calculate how long it will take one person to do a job alone when we know how long it takes a group to get it done:
Example  
Problem  Jamie, Pria and Paul can paint a room together in 2 hours. If Pria does the job alone she can paint the room in 5 hours. If Paul works alone, he can paint the room in 6 hours. If Jamie works alone, how long would it take her to paint the room? 
 
Pria + Paul + Jamie = _{} room/ hour
Pria =_{} room/hour
Paul = _{} room/hour
Jamie = _{} room/hour 
 Determine the hourly rates for each person and for the whole group using the formula _{}.
Work is painting 1 room, so W = 1.
We don’t know how long Jamie will take, so we need to keep the variable t.  
_{} 
 Write an equation to show that the sum of their individual rates equals the group rate.
(Think of it this way: Pria works for one hour and paints _{} of the room. Paul works for an hour and paints _{} of the room. Jamie works for an hour and paints _{} of the room. Together they have painted half the room in one hour.  
_{}
_{}
_{}

 Solve the rational equation.
First find the least common denominator of the individual rates. It is 5 • 6 • t = 30t.
Then multiply each term on the left by a fractional form of 1 so that all rates have the same denominator and can be added. (Note: we could also have found the common denominator of the entire equation, which is also 30t, and multiplied both sides by it.)  
_{}
11t + 30 = _{}
11t + 30 =15t
11t  11t + 30 = 15t – 11t
30 = 4t
7.5 = t 
 Now multiply both sides of the equation by the common denominator, then simplify.  
Answer 
It would take Jamie 7.5 hours to paint the room by herself.  
Tanya and Cam can each wash a car and vacuum its interior in 2 hours. Pat needs 3 hours to do this same job by himself. If Pat, Cam and Tanya work together, how long will it take them to clean a car?
A) 40 minutes B) 45 minutes C) 1.2 hours D) 1 hour

Sometimes work problems describe rates in a relative way: someone works 3 times as fast as someone else or a machine takes 2 fewer hours to finish a job than another model of machine. In these instances, we express one rate using information about another rate. Let’s look at an example:
Example  
Problem  One pipe can fill a pool 1.5 times faster than a second pipe. If both pipes are open, the pool can be filled in 6 hours. If only the slower pipe is open, how long would it take to fill the pool? 
 
 _{}
fast pipe = _{}
slow pipe = _{}
fast pipe + slow pipe = _{} 
 Find the rates of each pipe alone and the two working together.
Work = 1 pool filled
Hours needed for fast pipe to fill pool: p Hours needed for slow pipe to fill pool alone: 1.5 p Hours needed for both pipes together: 6
 

_{} 
 Write an equation that shows that the amount of work completed by both pipes in one hour is equal to the sum of the work of each pipe.  

_{}
_{}
6 + 4 = p
p =10 hours 
 Solve for p. One way to do this is to rewrite the rational expressions using a common denominator.
Common denominator of p, 1.5p and 6 is 6p.
 
 10 • 1.5 = 15 hours 
 Slow pipe takes 1.5p hours to do the work alone.  
Answer 
The slower pipe will take 15 hours to fill the pool alone. 

 
Summary
Rational equations can be used to solve a variety of problems that involve rates, times and work. Using rational expressions and equations can help us answer questions about how to combine workers or machines to complete a job on schedule.