### Unit 3: Word Problems

In this unit, you will apply the skill of solving equations you mastered in Unit 2 to solve various types of word problems. When you encounter a word problem, you have to remember to read it carefully and think critically about what quantity you are asked to find, what quantities are known, and what the relationship is between them. This will help you set up an equation that will give you an answer to the problem. For example, if you know the discounted price of an item and need to find the original price, remember that the percent of the discount is taken from this original price, which is something you do not know!

You probably already know how to solve some of the problems covered in this unit, such as percent problems or uniform motion problems that can be solved by one arithmetic operation. However, you will now approach these types of problems from the algebraic point of view, which will enable you to move on to more complex problems. In these problems, you cannot just add/subtract and multiply/divide the quantities given and arrive at the answer. These problems can only be solved by setting up an equation.

**Completing this unit should take you approximately 8 hours.**

Upon successful completion of this unit, you will be able to:

- translate a verbal expression into a variable expression;
- use the Basic Percent Equation to solve problems involving percents;
- apply the Basic Percent Equation to problems involving mixtures, markups, and discounts;
- use the Uniform Motion Equation to solve problems involving uniform motion; and
- create equations in one variable and use them to solve problems.

### 3.1: Translating English to Math

### 3.1.1: Mathematical Symbols and Expressions for Common Words and Phrases

Read the section titled "Translating Word to Symbols" through "Sample Set A." Also, copy Table 1 into your notes for future reference. You will use this table in the next assignment.

Complete the exercise set. Use the article in the previous assignment to help you translate these verbal expressions into algebraic expressions.

### 3.1.2: Translating Verbal Expression into Mathematical

Watch these videos and take notes. In these videos, Dr. Sousa explains two examples of translating a given real-life situation to algebraic language. As you watch, pay attention to how the keywords such as

*more*and*less*translate into mathematical operations.Scroll down to Practice Set A. Using Sample Set A as a guide, complete exercises 1-7 and check your solutions (located below each problem). Developing the skill of translating verbal expressions to mathematical is the first step to solving word problems.

### 3.2: Translating Sentences into Equations

### 3.2.1: Number Problems

Read the section titled "Five Step Method." This method will guide you through solving problems in this and subsequent subunits of Unit 3. Read Examples 1 and 2 in Sample Set A, and try Exercises 1 and 2 from Practice Set A on your own. The solutions to the practice problems are shown directly below each problem.

### 3.2.2: Consecutive Integer Problems

Watch these videos and take notes. The first video will introduce you to the word problems involving consecutive integers, or integers that follow one another. Pay attention to the explanation of how to express one consecutive integer in terms of another. Note the difference between the problems: the first one is about consecutive integers, whereas the second one is about consecutive odd integers. The second video is another problem involving consecutive odd integers.

Scroll down to Example 3 in Sample Set A. Review the algebraic method of solving problems involving consecutive, odd, or even integers. Then, complete exercises 4, 34, 36, and 38. Click on the "Show Solutions" link next to each problem to check your answer.

Complete the exercise set. It provides more practice solving problems involving consecutive integers. Keep in mind that after you have solved your equation, you still have to answer the question in the problem.

### 3.2.3: General Statement Problems

Watch these videos and take notes. The first video is another example of solving a word problem using a linear equation. In the second video, you may skip the first four minutes (problems 1 and 2) and focus on problems 3 and 4. Note that while these problems are seemingly about different situations, the same basic approach is used.

Watch this video and take notes. This is an example of using algebraic solution to a real-life situation. Watch how each piece of information given about the shelves is translated into algebraic expression and then used to create an equation.

Complete the exercise set. Solving these word problems will help you apply your algebraic and critical thinking skills to various real-life situations.

### 3.3: Basic Uniform Motion Problems

### 3.3.1: Applying Uniform Motion Equation

Watch this video and take notes. In this video, the uniform motion equation is used to find speed and distance. A motion is

*uniform*when the speed, or rate, of the motion is constant. For example, if a car moves with the speed of 30 mph without accelerating or slowing down, then it is in uniform motion. The distance that the car travels during a given time can be found according to the uniform motion equation:*distance = rate × time*.

### 3.3.2: Problems Involving Objects Moving in Opposite Directions

Work through the problems on this page on your own and then review the solutions. This page contains examples of two common types of motion problems in which the object travel in the opposite directions, either towards each other or away from each other.

### 3.3.3: Uniform Motion Problems

Read this brief example of uniform motion problems, and then attempt the practice problems on the second page. When you have finished, you may check your answers against the answer key.

### 3.4: Creating Equations to Solve Problems

### 3.4.1: Value Mixture Problems

Read this page. The problems you are going to solve here all have to do with mixtures of two different types of a product, costing a different price. Click on the "new problem" button at the end of the page to try a practice problem, and check your answer. Continue this process by clicking on "new problem," and solve five problems. You can also create a worksheet of five problems by clicking on the button at the bottom of the page titled "Click Here for a Randomly Generated Worksheet and Answers." The answers will be provided at the end of the worksheet.

Watch this video and take notes. This is another example of a mixture problem, where the unit price of the mixture of two products is unknown.

### 3.4.2: Percent Mixture Problems

### 3.4.2.1: Definition of a Percent, Basic Percent Equation and Basic Percent Problems

Read this page. Watch the "How to Solve Percent Equations" video embedded in the text to review how to solve basic percent problems by setting up equations, if you need help. Then, complete practice problems 16-30. Use the embedded "Percent Problems" video (skipping the first five minutes) as a guide, if you need help with these problems. Once you have completed the practice problems, check your answers against the answer key.

Complete the exercise set. This will help you assess your proficiency in solving problems involving percents, which you have studied in Arithmetic. You will review the concept and application of percents again in the next assignment.

### 3.4.2.2: Solving Percent Mixture Problems

Watch these videos and take notes. While different in context, algebraically the first problem is similar to mixture problems earlier. Note the steps Sal Khan takes to choose a variable, translate all the information given into algebraic expressions, and set up an equation. The second video is another example of a percent mixture problem. This time, the percent concentration in one part of the mixture is unknown.

This app will guide you one step at a time through setting up the table you need to solve a typical percent mixture problem. This page focuses only on the type of problems in which amounts of both solutions being mixed are unknown. Try to go through at least three problems to achieve proficiency.

This page will help you solve various mixture problems (including the ones involving percents) one step at a time. Try to solve at least five problems or more, if necessary. You can also click on "More information on Money and Mixing Problems" at the bottom of the page to see more examples of mixture and percent mixture problems.

### 3.4.3: More Uniform Motion Problems

Watch these videos and take notes. In the first problem, two objects (cyclists, in this case) are traveling in opposite directions toward each other. Note how Dr. Sousa uses the table to organize all the information given in the problem.

In the second problem, there is only one traveler, but she travels to another city and then back. Again, a table is useful to organize all the information given. Note that the task here is to find the distance, but the variable chosen to be denoted, x, is the time it takes to complete a one-way trip. You will find that this is a convenient approach in most problems where time is not given directly.

In the third example, two people travel in the same direction and one has to catch up to the other. Note that the key information you need to set up the equation is not given in the problem explicitly: both people will have traveled the same distance by the time one overtakes the other.

This page will guide you through solving various uniform motion problems one step at a time. Try to solve at least five problems or more, if needed. You can also click on "Information on Distance-Rate-Time Problems" at the bottom of the page and scroll down to the section titled "Motion Problems" to see one more solved example of a uniform motion problem.