com Improve Your Math Fluency monkeyphysicsblog.com CONTENTS INTRODUCTION 1 STRATEGIES AND TIPS 2 ALGEBRA REFRESHER 3 EXAMPLES 4 WORD PROBLEMS ABOUT THE AUTHOR www.com INTRODUCTION This workbook is designed to help practice solving standard word problems. Every problem is fully solved using algebra: Simply turn the page to check the solution. The first chapter offers some tips for solving word problems, the second chapter provides a quick refresher of essential algebra skills, and the third chapter includes several examples to help serve as a guide for how to solve algebra word problems. A variety of problems are included, such as: • age problems • problems with integers • relating the digits of a number • fractions, decimals, and percentages • average values • ratios and proportions • problems with money • simple interest problems • rate problems • two moving objects • mixture problems • people working together • problems with levers • perimeter and area May you (or your students) find this workbook useful and become more fluent with algebra word problems.com 1 STRATEGIES AND TIPS Read the Problem Carefully First read the entire problem.
Be sure to read every word. A single written word can make a big difference in the solution. It’s a common mistake for students to focus so much on the numbers that they don’t notice a very important word. As you read the problem, circle or underline what you believe 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 like total, increased by, or tripled • what you are solving for, like Anna’s age or the number of apples in the cart Identify the Given Information The information given in the problem is used to solve for the desired 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 the problem, or you could make a table of this information. • First identify all of the numbers like 3 bananas or 5 days. • Beware that some numbers are stated using words, like writing “five” instead of 5, a “dozen” instead of 12, or “doubled” instead of 2 times. • The number “zero” is often disguised.
For example, if a problem states “there are no grapes left,” this is equivalent to stating that there are zero grapes. What Are You Solving for? Read carefully to determine what the problem is asking you to find. Some problems ask a question like, “What is Julie’s age?” or “How far did Pat walk?” Other problems state the question in a sentence like, “Determine the www.com number of apples in the barrel.” Some questions ask for more than one answer, like “How old are Liz and Tim?” www.com Indicate What Each Unknown Represents In the beginning of the solution, it helps to write a phrase like the example below in order to remind you what each unknown represents. The unknown should usually represent what you are trying to solve for.
That way, your solution will be complete once you solve for the unknown. x = the original number of cookies Multiple Unknowns If there are two (or more) unknowns, try to let one variable represent the smallest unknown. For example, suppose that Melissa is three years older than Doug. In this case, Doug is younger, so you could let x represent Doug’s age: x = Doug’s age x + 3 = Melissa’s age If there are two (or more) unknowns, but it is difficult or inconvenient to express both unknowns in terms of a single variable (as we did above), it is possible to use two different variables.
If you 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 www.com Relating the Unknowns to the Given Information Write down an equation to help you solve for the variable. (If there are two different variables, you will need to write down two different equations.) The language in the problem helps you relate the variables to the given information.
Translate the words into symbols by looking for words that relate to mathematical operations. Note that the examples in the following tables are designed to help with common expressions, but do not account for every possible way for the English language to describe each mathematical operation: You need to think about the wording of every problem.com Beware of Possible Extraneous Information Occasionally, a problem includes extraneous information that isn’t needed to solve a problem. Although most problems give you only the information that is needed, it is a good habit to ask, “Which information is needed to solve the problem?” Remember that a rare problem may include numbers that aren’t relevant to the solution. Be Confident and Determined Successful students know that a solution exists.
They are determined to figure it out.com Working with Integers The following features are common in word problems: • Represent two consecutive integers with x and (x + 1). A third 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 odd or 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, multiply the tens digit by 10 and the units digit by 1. For example, if the problem states that the units digit is 5 times the tens digit, let the tens digit equal x, 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, the tens 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, multiply the hundreds digit by 100, the tens digit by 10, and the units digit by 1. For example, if the problem states that the tens digit is twice the units digit and that 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 number equals 100(6x) + 10(2x) + 1(x) = 600x + 20x + x = 621x. Suppose that you solve the problem and obtain x = 1.
In this example, the units digit is 1, the tens digit 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 states that 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. In this 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 57 indeed have their digits reversed.com Sum, Product, Difference, and Ratio If you know the sum, product, difference, or ratio of two numbers, but aren’t told what either number equals, let the following examples serve as a guide: • If the sum of two numbers equals 42 (for example), let one number be x and the other number will be (42 – x). • If the product of two numbers equals 36 (for example), let one number be x and the other number will be 36/x.
• If the difference between two numbers is 5 (for example), let one number be x and the larger number will be (x + 5). • If the ratio of two numbers is 3 (for example), let one number be x and the larger number will be 3x.com Fractions, Decimals, and Percentages Following are some tips for dealing with fractions, decimals, and percentages: • Divide by 100 to convert a percent into a decimal.4 • When there are decimals in an equation, multiply the entire equation by the power of 10 needed in order to remove all of the decimals.24x + x = 6 multiply by 100 24x + 100x = 600 • When there are fractions in an equation, multiply the entire equation by the lowest common denominator. x/2 – 1/x = 1/3 multiply by 6x 3x2 – 6 = 2x • The phrases “increased by” or “decreased by” are compared to 100% (or 1). * If x increases by 20%, this means 1.
* If x decreases by 1/4, this means 3x/4 or 0.com Ratios and Proportions A ratio expresses a fixed relationship in the form of a fraction. For example, if there are 300 girls and 200 boys in a particular school, the ratio of girls to boys attending that school is 3 to 2. We could express this ratio with a colon (3:2), as a fraction (3/2), as a decimal (since 3/2 = 1.5), or as a percent (150%). When a problem gives you the ratio, but not the quantity of each, if you let x be the quantity represented by the denominator, multiply the ratio by x to get the quantity represented by the numerator.
For example, if the ratio of white cars to black cars is 5:4, if you let x represent the number of black cars (since black corresponds to the denominator), the number of white cars will be 1. A proportion expresses an equality between two ratios. For example, if the ratio of apples to oranges is 4:3 and there are 64 apples, if we let x represent the number of oranges, we can use the following proportion to solve for x. Check for consistency when setting up a proportion: On both sides of the following equation, apples are on top and oranges are on the bottom.
4/3 = 64/x Cross multiply in order to remove the variable from the denominator. As a check, note that 64/48 = 4/3, such that the ratio of apples to oranges is indeed 4:3.com Money and Interest For problems with money expressed in decimals, like $3.25, after you write down the equation, if you multiply both sides of the equation by 100, it will remove all of the decimals from the problem. See the example below.97 = 12x 225x – 4297 = 1200x For problems that involve US coins, it is often convenient to express the money in terms of cents. For example, 5x + 10y is the amount of cents contained in x nickels and y dimes, since each nickel is worth 5 cents and each dime is worth 10 cents.
For problems that involve simple interest calculations, note that the interest (I) is equal to the principal (P) times the interest rate (r) in decimal form. In the formula below, note that P is multiplying r.com I = Pr For example, suppose that a student invests $500 in a savings account that earns interest at a rate of 3%. The principal is the original amount invested: P = $500. Divide the interest rate by 100% to convert it into a decimal: r = (3%)/(100%) = 0.
Use the formula above to determine the interest earned.03) = $15 If the account earns 3% interest per year, after one year, the new balance will be $515 (add the original principal to the interest to determine this).