## Using a Problem-solving Strategy for Word Problems

In earlier tutorials, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then you’ve increased your math vocabulary as you learned about more algebraic procedures, and you’ve had more practice translating from words into algebra.

You have also translated word sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated.

Now we’ll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We’ll demonstrate the strategy as we solve the following problem.

## Example

Pete bought a shirt on sale for \(\text{\$18},\) which is one-half the original price. What was the original price of the shirt?

### Solution

Step 1. **Read** the problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don’t understand, look them up in a dictionary or on the Internet.

*In this problem, do you understand what is being discussed? Do you understand every word?*

Step 2. **Identify** what you are looking for. It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!

*In this problem, the words “what was the original price of the shirt” tell you that what you are looking for: the original price of the shirt.*

Step 3. **Name** what you are looking for. Choose a variable to represent that quantity. You can use any letter for the variable, but it may help to choose one that helps you remember what it represents.

*Let \(p=\) the original price of the shirt*

Step 4. **Translate** into an equation. It may help to first restate the problem in one sentence, with all the important information. Then translate the sentence into an equation.

Step 5. **Solve** the equation using good algebra techniques. Even if you know the answer right away, using algebra will better prepare you to solve problems that do not have obvious answers.

Write the equation. | |

Multiply both sides by 2. | |

Simplify. |

Step 6. **Check** the answer in the problem and make sure it makes sense.

*We found that*\(p=36,\)*which means the original price was*\(\text{\$36}.\)*Does*\(\text{\$36}\)*make sense in the problem? Yes, because*\(18\)*is one-half of*\(36,\)*and the shirt was on sale at half the original price.*

Step 7. **Answer** the question with a complete sentence.

*The problem asked “What was the original price of the shirt?” The answer to the question is: “The original price of the shirt was*\(\text{\$36}.”\)

If this were a homework exercise, our work might look like this:

In Mathematics 106, we learned how to translate and solve basic percent equations and used them to solve sales tax and commission applications. In the next example, we will apply our **Problem Solving Strategy** to more applications of percent.

## Example

Nga’s car insurance premium increased by \(\text{\$60},\) which was \(\text{8%}\) of the original cost. What was the original cost of the premium?

### Solution

Step 1. Read the problem. Remember, if there are words you don’t understand, look them up. | |

Step 2. Identify what you are looking for. | the original cost of the premium |

Step 3. Name. Choose a variable to represent the original cost of premium. | Let \(c=\text{the original cost}\) |

Step 4. Translate. Restate as one sentence. Translate into an equation. | |

Step 5. Solve the equation. | |

Divide both sides by 0.08. | |

Simplify. | \(c=750\) |

Step 6. Check: Is our answer reasonable? Yes, a \$750 premium on auto insurance is reasonable. Now let’s check our algebra. Is 8% of 750 equal to 60? | |

Step 7. Answer the question. | The original cost of Nga’s premium was \$750. |