Lesson Objectives

- Demonstrate an understanding of how to solve a word problem
- Demonstrate an understanding of how to solve an equation with rational expressions
- Learn how to solve word problems that involve rational equations

## How to Solve Word Problems with Rational Expressions

At this point in the course, we should be very comfortable with solving word problems. We will usually see some application problems that involve
rational equations. Let's begin by reviewing our six-step method for solving word problems.

Example 1: Solve each word problem

Jason's car uses 20 gallons of gas to travel 400 miles. If Jason currently has 7 gallons of gas in his car, how much gas is needed to travel 200 miles?

Step 1) Read the problem carefully and determine what you are asked to find

We are asked to find how many additional gallons of gasoline are needed to travel 200 miles.

Step 2) Assign a variable to represent the unknown

Let x = the number of additional gallons of gasoline needed to travel 200 miles.

Step 3) Write out an equation which describes the given situation

We will set up a proportion: $$\frac{400 \hspace{.2em} miles}{20 \hspace{.2em} gallons} = \frac{200 \hspace{.2em} miles}{(x + 7) \hspace{.2em} gallons}$$ Step 4) Solve the equation

We can multiply both sides by the LCD, which is 20(x + 7), or we can just cross multiply. $$400(x + 7) = 200 \cdot 20$$ $$400x + 2800 = 4000$$ $$400x = 1200$$ $$x = 3$$ Step 5) State the answer using a nice clear sentence

Since x is 3, this tells us that Jason needs an additional 3 gallons of gas. We will state our answer as:

Jason needs an additional 3 gallons of gas, to travel 200 miles.

Step 6) Check the result by reading back through the problem

We know the car gets 20 miles to the gallon. We get this from dividing 400 by 20. If Jason wants to travel 200 miles, he will need 10 gallons of gas. We get this by dividing 200 by 20. Since he started out with 7 gallons of gas, he needs 3 additional gallons to obtain a total of 10.

d = rt

Let's look at an example.

Example 2: Solve each word problem

Mary likes to travel to the next town to see the state fair. When she leaves her house, she can take one of two routes, the highway, or the interstate. Mary averages 30 miles per hour when she drives on the highway to get to the state fair. She averages 50 miles per hour when she takes the interstate. If both routes are exactly the same length, and she saves two hours by driving on the interstate, how far away is the state fair?

Step 1) Read the problem carefully and determine what you are asked to find

We are asked to find the distance in miles from her house to the state fair.

Step 2) Assign a variable to represent the unknown

Let x = The distance in miles from Mary's house to the fair.

Step 3) Write out an equation which describes the given situation

Let's organize our information in a table.

In our table, we get the rate from the problem. The distance is represented by x in each case. To get the time, we solve the distance formula for time:

d = rt

t = d/r

Time traveled is equal to the distance divided by the rate of speed. In each case, we took the distance of x and divided by the rate of speed.

To set up an equation, we think about what information is given in the problem. We are told that it takes two hours longer on the highway than on the interstate. This means the time it takes to drive on the interstate plus 2 hours will be equal to or the same as the time it takes to drive on the highway. $$\frac{x}{50} + 2 = \frac{x}{30}$$ Step 4) Solve the equation

$$\frac{x}{50} + 2 = \frac{x}{30}$$ We will begin by multiplying each side by the LCD. The LCD is 150. $$150 \cdot \frac{x}{50} + 150 \cdot 2 = 150 \cdot \frac{x}{30}$$ $$\require{cancel}3\cancel{150} \cdot \frac{x}{\cancel{50}} + 300 = 5\cancel{150} \cdot \frac{x}{\cancel{30}}$$ $$3x + 300 = 5x$$ $$-2x = -300$$ $$x = 150$$ Step 5) State the answer using a nice clear sentence

Since x is 150, we know the distance is 150 miles. We will state our answer as:

The distance from Mary's house to the state fair is 150 miles.

Step 6) Check the result by reading back through the problem

We know that it takes two hours longer to travel to the state fair on the highway versus the interstate. If Mary drives at 30 miles per hour for 5 hours, she will drive 150 miles. If she drives two hours less or 3 hours at 50 miles per hour, she will also drive 150 miles.

Example 3: Solve each word problem

A local brewery has a vat which can be completely filled with beer from the inlet pipe in 60 minutes. The outlet pipe can completely drain a full vat in 80 minutes. If for some reason, both pipes are left open, how long would it take to completely fill an empty vat?

Step 1) Read the problem carefully and determine what you are asked to find

We are asked to find how long it will take to fill the empty vat, given that the inlet and outlet pipes are both on.

Step 2) Assign a variable to represent the unknown

Let x = the number of minutes it will take to fill the vat

Step 3) Write out an equation which describes the given situation

In this case, we have two competing forces. The inlet pipe is pumping beer into the vat and the outlet pipe is pumping beer out. Let's think about what happens in one unit of time or one minute: $$\frac{1}{60} - \frac{1}{80} = \frac{4 - 3}{240} = \frac{1}{240}$$ In one minute, 1/240 of the job is complete. In other words, the vat is 1/240 of the way full.

Since x is the number of minutes it will take to fill the vat, we can multiply 1/240 by x and set this equal to 1. $$\frac{1}{240}x = 1$$ The fractional amount of the job that is completed after 1 minute is multiplied by the number of minutes it takes to complete the job. The result is 1 completed job or a full vat.

Step 4) Solve the equation

$$\frac{1}{240}x = 1$$ Multiply both sides by the reciprocal: $$x = 240$$ Step 5) State the answer using a nice clear sentence

Since x is 240, it will take 240 minutes or 4 hours to complete the job. We will state our answer as:

It will take 240 minutes or 4 hours to completely fill the vat with beer.

Step 6) Check the result by reading back through the problem

We know each minute, 1/240 of the job is done. If we continue for 240 minutes, we will have one completed job.

### Six-Step Method for Solving Word Problems with Rational Expressions

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

### Solving a Proportion Problem

In some cases, we will see word problems that involve setting up and solving a proportion. Let's look at an example.Example 1: Solve each word problem

Jason's car uses 20 gallons of gas to travel 400 miles. If Jason currently has 7 gallons of gas in his car, how much gas is needed to travel 200 miles?

Step 1) Read the problem carefully and determine what you are asked to find

We are asked to find how many additional gallons of gasoline are needed to travel 200 miles.

Step 2) Assign a variable to represent the unknown

Let x = the number of additional gallons of gasoline needed to travel 200 miles.

Step 3) Write out an equation which describes the given situation

We will set up a proportion: $$\frac{400 \hspace{.2em} miles}{20 \hspace{.2em} gallons} = \frac{200 \hspace{.2em} miles}{(x + 7) \hspace{.2em} gallons}$$ Step 4) Solve the equation

We can multiply both sides by the LCD, which is 20(x + 7), or we can just cross multiply. $$400(x + 7) = 200 \cdot 20$$ $$400x + 2800 = 4000$$ $$400x = 1200$$ $$x = 3$$ Step 5) State the answer using a nice clear sentence

Since x is 3, this tells us that Jason needs an additional 3 gallons of gas. We will state our answer as:

Jason needs an additional 3 gallons of gas, to travel 200 miles.

Step 6) Check the result by reading back through the problem

We know the car gets 20 miles to the gallon. We get this from dividing 400 by 20. If Jason wants to travel 200 miles, he will need 10 gallons of gas. We get this by dividing 200 by 20. Since he started out with 7 gallons of gas, he needs 3 additional gallons to obtain a total of 10.

### Motion Word Problems with Rational Expressions

Earlier in our course, we learned how to use the distance formula when working with motion word problems. Recall the distance formula relates the distance traveled (d) to the rate of speed (r) times the time traveled (t).d = rt

Let's look at an example.

Example 2: Solve each word problem

Mary likes to travel to the next town to see the state fair. When she leaves her house, she can take one of two routes, the highway, or the interstate. Mary averages 30 miles per hour when she drives on the highway to get to the state fair. She averages 50 miles per hour when she takes the interstate. If both routes are exactly the same length, and she saves two hours by driving on the interstate, how far away is the state fair?

Step 1) Read the problem carefully and determine what you are asked to find

We are asked to find the distance in miles from her house to the state fair.

Step 2) Assign a variable to represent the unknown

Let x = The distance in miles from Mary's house to the fair.

Step 3) Write out an equation which describes the given situation

Let's organize our information in a table.

Trip | Rate (mph) | Time (hours) | Distance (miles) |
---|---|---|---|

Highway | 30 | x/30 | x |

Interstate | 50 | x/50 | x |

d = rt

t = d/r

Time traveled is equal to the distance divided by the rate of speed. In each case, we took the distance of x and divided by the rate of speed.

To set up an equation, we think about what information is given in the problem. We are told that it takes two hours longer on the highway than on the interstate. This means the time it takes to drive on the interstate plus 2 hours will be equal to or the same as the time it takes to drive on the highway. $$\frac{x}{50} + 2 = \frac{x}{30}$$ Step 4) Solve the equation

$$\frac{x}{50} + 2 = \frac{x}{30}$$ We will begin by multiplying each side by the LCD. The LCD is 150. $$150 \cdot \frac{x}{50} + 150 \cdot 2 = 150 \cdot \frac{x}{30}$$ $$\require{cancel}3\cancel{150} \cdot \frac{x}{\cancel{50}} + 300 = 5\cancel{150} \cdot \frac{x}{\cancel{30}}$$ $$3x + 300 = 5x$$ $$-2x = -300$$ $$x = 150$$ Step 5) State the answer using a nice clear sentence

Since x is 150, we know the distance is 150 miles. We will state our answer as:

The distance from Mary's house to the state fair is 150 miles.

Step 6) Check the result by reading back through the problem

We know that it takes two hours longer to travel to the state fair on the highway versus the interstate. If Mary drives at 30 miles per hour for 5 hours, she will drive 150 miles. If she drives two hours less or 3 hours at 50 miles per hour, she will also drive 150 miles.

### Rate of Work Word Problems

A very common word problem in Algebra involves rates of work. If a job can be completed in t units of time, then 1/t of the job is completed per unit of time. As an example, if a job can be completed in 10 minutes, then after 1 minute, 1/10 of the job is completed. Let's look at an example.Example 3: Solve each word problem

A local brewery has a vat which can be completely filled with beer from the inlet pipe in 60 minutes. The outlet pipe can completely drain a full vat in 80 minutes. If for some reason, both pipes are left open, how long would it take to completely fill an empty vat?

Step 1) Read the problem carefully and determine what you are asked to find

We are asked to find how long it will take to fill the empty vat, given that the inlet and outlet pipes are both on.

Step 2) Assign a variable to represent the unknown

Let x = the number of minutes it will take to fill the vat

Step 3) Write out an equation which describes the given situation

In this case, we have two competing forces. The inlet pipe is pumping beer into the vat and the outlet pipe is pumping beer out. Let's think about what happens in one unit of time or one minute: $$\frac{1}{60} - \frac{1}{80} = \frac{4 - 3}{240} = \frac{1}{240}$$ In one minute, 1/240 of the job is complete. In other words, the vat is 1/240 of the way full.

Since x is the number of minutes it will take to fill the vat, we can multiply 1/240 by x and set this equal to 1. $$\frac{1}{240}x = 1$$ The fractional amount of the job that is completed after 1 minute is multiplied by the number of minutes it takes to complete the job. The result is 1 completed job or a full vat.

Step 4) Solve the equation

$$\frac{1}{240}x = 1$$ Multiply both sides by the reciprocal: $$x = 240$$ Step 5) State the answer using a nice clear sentence

Since x is 240, it will take 240 minutes or 4 hours to complete the job. We will state our answer as:

It will take 240 minutes or 4 hours to completely fill the vat with beer.

Step 6) Check the result by reading back through the problem

We know each minute, 1/240 of the job is done. If we continue for 240 minutes, we will have one completed job.

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