### Learning Objectives

- Define a process for problem solving
- Translate words into algebraic expressions and equations
- Define a process for solving word problems

Word problems can be tricky. Often it takes a bit of practice to convert an English sentence into a mathematical sentence, which is one of the first steps to solving word problems. In the table below, words or phrases commonly associated with mathematical operators are categorized. Word problems often contain these or similar words, so it’s good to see what mathematical operators are associated with them.

How much will it cost?

Addition [latex]+[/latex] | Subtraction [latex]-[/latex] | Multiplication [latex]\times[/latex] | Variable ? | Equals [latex]=[/latex] |
---|---|---|---|---|

More than | Less than | Double | A number | Is |

Together | In the past | Product | Often, a value for which no information is given. | The same as |

Sum | slower than | times | After how many hours? | |

Total | the remainder of | |||

In the future | difference | |||

faster than |

Some examples follow:

- [latex]x\text{ is }5[/latex] becomes [latex]x=5[/latex]
- Three more than a number becomes [latex]x+3[/latex]
- Four less than a number becomes [latex]x-4[/latex]
- Double the cost becomes [latex]2\cdot\text{ cost }[/latex]
- Groceries and gas together for the week cost $250 means [latex]\text{ groceries }+\text{ gas }=250[/latex]
- The difference of 9 and a number becomes [latex]9-x[/latex]. Notice how 9 is first in the sentence and the expression

Let’s practice translating a few more English phrases into algebraic expressions.

### Example

Translate the table into algebraic expressions:

some number | the sum of the number and 3 | twice the sum of the number and 3 |

a length | double the length | double the length, decreased by 6 |

a cost | the difference of the cost and 20 | 2 times the difference of the cost and 20 |

some quantity | the difference of 5 and the quantity | the difference of 5 and the quantity, divided by 2 |

an amount of time | triple the amount of time | triple the amount of time, increased by 5 |

a distance | the sum of [latex]-4[/latex] and the distance | the sum of [latex]-4[/latex] and the twice the distance |

In this example video, we show how to translate more words into mathematical expressions.

The power of algebra is how it can help you model real situations in order to answer questions about them.

Here are some steps to translate problem situations into algebraic equations you can solve. Not *every* word problem fits perfectly into these steps, but they will help you get started.

- Read and understand the problem.
- Determine the constants and variables in the problem.
- Translate words into algebraic expressions and equations.
- Write an equation to represent the problem.
- Solve the equation.
- Check and interpret your answer. Sometimes writing a sentence helps.

### Example

Twenty-eight less than five times a certain number is 232. What is the number?

In the video that follows, we show another example of how to translate a sentence into a mathematical expression using a problem solving method.

Another type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other, such as 3, 4, 5. If we are looking for several consecutive numbers it is important to first identify what they look like with variables before we set up the equation.

For example, let’s say I want to know the next consecutive integer after 4. In mathematical terms, we would add 1 to 4 to get 5. We can generalize this idea as follows: the consecutive integer of any number, *x*, is [latex]x+1[/latex]. If we continue this pattern we can define any number of consecutive integers from any starting point. The following table shows how to describe four consecutive integers using algebraic notation.

First | [latex]x[/latex] |

Second | [latex]x+1[/latex] |

Third | [latex]x+2[/latex] |

Fourth | [latex]x+3[/latex] |

We apply the idea of consecutive integers to solving a word problem in the following example.

### Example

The sum of three consecutive integers is 93. What are the integers?

In the following video we show another example of a consecutive integer problem.