Chris McMullen - Algebra Word Problems Practice Workbook with Full Solutions

Asingle written word can make a big difference in the solution. Its a commonmistake for students to focus so much on the numbers that they dont notice avery important word. As you read the problem, circle or underline what youbelieve will be key information: numbers like 12 years, $3.75, or 25% written numbers like two, one-third, or none key words that relate to mathematical operations liketotal, increased by, or tripled what you are solving for, like Annas age or thenumber of apples in the cart Identify the Given Information The information given in the problem is used to solve for thedesired unknown, so the first step is to gather the information that you know.You can do this by circling or underlining the numerical information in theproblem, or you could make a table of this information. First identify all of the numbers like 3 bananas or 5days. Beware that some numbers are stated using words, likewriting five instead of 5, a dozen instead of 12, or doubled instead of 2times. The number zero is often disguised.

Forexample, if a problem states there are no grapes left, this is equivalent to statingthat there are zero grapes. What Are You Solving for? Read carefully to determine what the problem isasking you to find. Some problems ask a question like, What is Julies age?or How far did Pat walk? Other problems state the question in a sentencelike, Determine the number of apples in the barrel. Some questions ask formore than one answer, like How old are Liz and Tim? Indicate What Each Unknown Represents In the beginning of the solution, it helps to writea phrase like the example below in order to remind you what each unknownrepresents. The unknown should usually represent what you are trying to solvefor. x = the original number of cookies Multiple Unknowns If there are two (or more) unknowns, try to let onevariable represent the smallest unknown. x = the original number of cookies Multiple Unknowns If there are two (or more) unknowns, try to let onevariable represent the smallest unknown.

For example, suppose that Melissa isthree years older than Doug. In this case, Doug is younger, so you could let x representDougs age: x = Dougs age x + 3 = Melissas age If there are two (or more) unknowns, but it isdifficult or inconvenient to express both unknowns in terms of a singlevariable (as we did above), it is possible to use two different variables. Ifyou use multiple variables, you will need to write down more than one equation.If there are two variables, you will need two equations. x = bananas y = oranges 3x 2y = 14 4x + 5y = 57 Relating the Unknowns to the Given Information Write down an equation to help you solve for thevariable. (If there are two different variables, you will need to write downtwo different equations.) The language in the problem helps you relate thevariables to the given information. Translate the words into symbols by lookingfor words that relate to mathematical operations.

Note that the examples in thefollowing tables are designed to help with common expressions, but do notaccount for every possible way for the English language to describe eachmathematical operation: You need to think about the wording of every problem. Beware of Possible Extraneous Information Occasionally, a problem includes extraneousinformation that isnt needed to solve a problem. Although most problems giveyou only the information that is needed, it is a good habit to ask, Whichinformation is needed to solve the problem? Remember that a rare problem mayinclude numbers that arent relevant to the solution. Be Confident and Determined Successful students know that a solution exists.They are determined to figure it out. Working with Integers The following features are common in word problems: Represent two consecutive integers with x and (x + 1). Athird consecutive integer would equal (x + 2), and so on.

Represent two consecutive even or odd integers with x and(x + 2). (The two numbers will have a difference of 2 whether they are both oddor both even.) If there is a third consecutive even or odd integer, that equals(x + 4). To solve for the digits of a two-digit number, multiplythe tens digit by 10 and the units digit by 1. For example, if the problemstates that the units digit is 5 times the tens digit, let the tens digit equalx, the units digit equals 5x, and the number equals 10(x) + 1(5x) = 10x + 5x =15x. Suppose that you solve the problem and obtain x = 1. In this example, thetens digit is 1, and the units digit is 5.

The number is 10(1) + 1(5) = 15. To solve for the digits of a three-digit number, multiplythe hundreds digit by 100, the tens digit by 10, and the units digit by 1. Forexample, if the problem states that the tens digit is twice the units digit andthat the hundreds digit is triple the tens digit, let the units digit equal x,the tens digit equals 2x, the hundreds digit equals 3(2x) = 6x, and the numberequals 100(6x) + 10(2x) + 1(x) = 600x + 20x + x = 621x. Suppose that you solvethe problem and obtain x = 1. In this example, the units digit is 1, the tensdigit is 2, and the hundreds digit is 6. The number is 100(6) + 10(2) + 1(1) =621.

To reverse the digits of a two-digit number,swap the place of the tens and units digit. For example, if a problem statesthat the units digit is x and the tens digit is x + 2, the number is 10(x + 2)+ 1(x) = 10x + 20 + x = 11x + 20 and the reversed number is 10(x) + 1(x + 2) =10x + x + 2 = 11x + 2. Suppose that you solve the problem and obtain x = 5. Inthis example, the units digit is 5, the tens digit is 7, the number is 10(7) +1(5) = 75 and the reversed number is 10(5) + 1(7) = 57. Observe that 75 and 57indeed have their digits reversed. Sum, Product, Difference, and Ratio If you know the sum, product, difference, or ratio of twonumbers, but arent told what either number equals, let the following examplesserve as a guide: If the sum of two numbers equals 42 (for example), let onenumber be x and the other number will be (42 x).