Lesson Objectives
• Demonstrate an understanding of the six-step method to solve a word problem
• Demonstrate an understanding of how to solve a rational equation
• Learn how to solve a work rate word problem
• Learn how to solve a motion word problem with rational expressions

## Applications of Rational Expressions

We have previously learned how to solve word problems with linear equations and systems of linear equations. In this lesson, we will learn how to solve word problems that involve rational expressions. Let’s begin with our six-step method for solving a word problem:

### Six-Step Method for Applications of Rational Expressions

1. Read the problem carefully and determine what you are asked to find
2. Assign a variable to represent the unknown
3. Write out an equation which describes the given situation
4. Solve the equation
5. State the answer using a nice clear sentence
6. Check the result by reading back through the problem

### Work Rate Word Problems

A common word problem that involves rational expressions is known as a rate of work or work rate problem. These problems seem challenging at first, simply because the format is new and unfamiliar. The fact is most work rate problems are very easy to solve. The premise behind work rate problems involves finding the amount of time two or more people can accomplish a certain task, given their individual rates of work. Let's work through a few examples.
Example 1: Solve each word problem
Working alone, it takes Emily 20 hours to clean a warehouse. Ashley can clean the same warehouse in 5 hours. How long would it take Emily and Ashley to clean the warehouse if they worked together?
Step 1) Read the problem carefully and determine what you are asked to find
We are asked to find the amount of time it takes for Emily and Ashley to clean the warehouse when working together.
Step 2) Assign a variable to represent the unknown
Our unknown is the amount of time in hours it will take to complete the job (clean the warehouse)
Let x = amount of time in hours to complete the job when working together
Step 3) Write out an equation which describes the given situation
The equation setup can be a bit tricky. We want to think about what gets accomplished in one unit of time. Since we are working with hours, we think about what Emily can do in one hour and what Ashley can do in one hour.
Emily takes 20 hours to complete the job, so in one hour, she is 1/20 complete.
Ashley takes 5 hours to complete the job, so in one hour, she is 1/5 complete.
If we sum the combined efforts for the two ladies in a one hour time period, we would get: $$\frac{1}{20} + \frac{1}{5} = \frac{1}{4}$$ Working together, the two ladies will complete 1/4 of the job in one hour. At this point, you can probably see that it will take 4 hours to complete the job. Since completing 1/4 of the job in each hour for 4 hours would result in 1 completed job. Algebraically, we can find this result by multiplying 1/4 by x and setting this equal to 1: $$\frac{1}{4}x = 1$$ To think about why this equation makes sense, start with the 1/4 part. This represents the amount of work the two ladies achieve in one hour. A good way to think about this is as a fractional amount of 1 completed job. Now if we multiply this by the number of hours it takes to complete the job, which is represented with x, we will get 1 (1 whole job or 1 completed job).
Step 4) Solve the equation $$\frac{1}{4}x = 1$$ Let's multiply both sides by the reciprocal of 1/4, which is 4: $$\require{cancel} 4 \cdot \frac{1}{4}x = 4 \cdot 1$$ $$\cancel{4} \cdot \frac{1}{\cancel{4}}x = 4$$ $$x = 4$$ Step 5) State the answer using a nice clear sentence
Since x is 4, this tells us it will take 4 hours to complete the job. Let's state our answer as:
It will take Ashley and Emily 4 hours to clean the warehouse if they work together.
Step 6) Check the result by reading back through the problem
We can think about each person's fractional contribution and the number of hours they work together:
Emily: Completes 1/20 of the job in one hour, and works for four hours: $$\frac{1}{20} \cdot 4 = \frac{4}{20} = \frac{1}{5}$$ Ashley: Completes 1/5 of the job in one hour, and works for four hours: $$\frac{1}{5} \cdot 4 = \frac{4}{5}$$ Now if we sum the individual contributions for the two girls over the four hour period, we get one completed job: $$\frac{1}{5} + \frac{4}{5} = 1$$ This tells us our answer is correct.
We may also see a work rate problem where there are two competing forces. A usual example involves filling a tub or swimming pool. Let's look at an example.
Example 2: Solve each word problem
A local swimming pool contains two pipes. An inlet pipe, which fills the pool and an outlet pipe, which drains the pool. The inlet pipe can completely fill an empty swimming pool in 3 hours. The outlet pipe can drain a full pool in 12 hours. If both pipes are turned on by mistake, how long will it take to fill an empty pool?
Step 1) Read the problem carefully and determine what you are asked to find
Our goal is to find the amount of time it will take to fill the empty pool, given that both pipes (inlet and outlet) are turned on.
Step 2) Assign a variable to represent the unknown
The unknown here is the amount of time in hours it will take to fill the pool.
Let x = number of hours it will take to fill the pool
Step 3) Write out an equation which describes the given situation
Here's where we need to think for a moment. We have two competing forces. The inlet pipe is pumping water into the pool, while the outlet pool is pumping water out. We want to think about what happens in one unit of time. In this case, we are working with hours, so what happens in one hour?
The inlet pipe takes 3 hours to fill and empty the pool, if it works by itself for one hour, the pool is 1/3 full.
The outlet pipe takes 12 hours to drain a full pool, if it works by itself for one hour, the outlet pipe will drain 1/12 of a full pool.
Now, these two pipes compete since they are both on at the same time. We will subtract away the amount the outlet pipe takes away in one hour, from the amount the inlet pipe pumps in during the same one hour period: $$\frac{1}{3} - \frac{1}{12} = \frac{1}{4}$$ This tells us in one hour, the pool will be 1/4 of the way full. We can set up an equation as: $$\frac{1}{4}x = 1$$ We have the fractional amount of the pool that is full after 1 hour, multiplied by the number of hours required to fill the pool. The result is 1 completed job or a full pool.
Step 4) Solve the equation $$\frac{1}{4}x = 1$$ Let's multiply both sides by the reciprocal of 1/4, which is 4: $$4 \cdot \frac{1}{4}x = 4 \cdot 1$$ $$\cancel{4} \cdot \frac{1}{\cancel{4}}x = 4$$ $$x = 4$$ Step 5) State the answer using a nice clear sentence
Since x is 4, this tells us it will take 4 hours to fill the pool while the two pipes are left on. Let's state our answer as:
It will take 4 hours to fill the pool.
Step 6) Check the result by reading back through the problem
We can think about the two pipe's fractional contribution and the number of hours they are left on together:
Inlet pipe: fills 1/3 of the pool in one hour, works for 4 hours: $$\frac{1}{3} \cdot 4 = \frac{4}{3}$$ Outlet pipe: empties 1/12 of the pool in one hour, works for 4 hours: $$\frac{1}{12} \cdot 4 = \frac{1}{3}$$ If we subtract what the outlet pipe pumps out from what the inlet pipes pumps in over the four hour period, we get one completed job: $$\frac{4}{3} - \frac{1}{3} = 1$$

### Motion Word Problems with Rational Expressions

Another common application of rational expressions involves motion problems. These problem types were studied earlier in algebra and involve our distance formula:
d = r • t
Where d is the distance, r is the rate of speed, and t is the time. Let's look at an example.
Example 3: Solve each word problem
The Jameson river in New Gogi has a current of 3 mph. A local boat charter takes as long to go 12 miles downstream (with the current) as to go 8 miles upstream (against the current). What is the speed of the boat in still water?
Step 1) Read the problem carefully and determine what you are asked to find
We are asked to find the speed of the boat in still water (no current).
Step 2) Assign a variable to represent the unknown
Let x = the speed of the boat in still water
Step 3) Write out an equation which describes the given situation
Let's first organize our information using a table.
Trip Rate (mph) Time (hours) Distance (miles)
Downstream x + 3 12/(x + 3) 12
Upstream x - 3 8/(x - 3) 8
Since the problem tells us it takes the same amount of time for each scenario, we can set the time from our downstream scenario equal to our time from the upstream scenario. This leads to the following equation: $$\frac{12}{x + 3} = \frac{8}{x-3}$$ Step 4) Solve the equation
We can multiply both sides of the equation by the LCD (x + 3)(x - 3). This will clear all denominators: $$(x+3)(x-3) \cdot \frac{12}{(x +3)} = (x + 3)(x - 3) \cdot \frac{8}{(x-3)}$$ $$(x+3)(x-3) \cdot \frac{12}{(x +3)} =$$$$(x + 3)(x - 3) \cdot \frac{8}{(x-3)}$$ $$\cancel{(x+3)}(x-3) \cdot \frac{12}{\cancel{(x +3)}} = (x + 3)\cancel{(x - 3)} \cdot \frac{8}{\cancel{(x-3)}}$$ $$\cancel{(x+3)}(x-3) \cdot \frac{12}{\cancel{(x +3)}} =$$$$(x + 3)\cancel{(x - 3)} \cdot \frac{8}{\cancel{(x-3)}}$$ $$12(x-3) = 8(x+3)$$ $$12x - 36 = 8x + 24$$ $$4x = 60$$ $$x = 15$$ Step 5) State the answer using a nice clear sentence
Since x is 15, this tells us the speed of the boat in still water is 15 miles per hour. Let's state our answer as:
The boat has an average speed of 15 mph in still water.
Step 6) Check the result by reading back through the problem
If we think about the boat traveling downstream, the speed would be 18 mph (15 + 3). This means our time to go 12 miles would be: $$\frac{12}{18} = \frac{2}{3}$$ If the boat travels upstream, the speed would be 12 mph (15 - 3). This means our time to go 8 miles would be: $$\frac{8}{12} = \frac{2}{3}$$ In each case, the time would is the same. This is consistent with what our problem told us. We can say our answer is correct.