ALGEBRAIC WORD PROBLEMS

Concept: An equation is a mathematical statement that shows that two quantities are equal. When an equation contains a variable, we use algebra to find the value of the variable. To solve an equation means to find the number which replaces the variable to make the mathematical statement true.

The basic strategy in solving an equation with a variable is to isolate the variable, that is, get the variable alone on one side of the equation, and the constant terms (numbers) on the other side of the equation. We do this by combining like terms and performing inverse (opposite) operations. Addition and subtraction are inverse operations. Multiplication and division are inverse operations.

Word problems use stories and sentences that contain numbers, and ask you to find the answer to a question. Listed below are some basic expressions often used in algebraic word problems, and their meanings. These are followed by 5 basic steps with which you can usually solve algebraic word problems..

Word Expressions | Algebraic Expression | Meaning
A number increased by four | n + 4 or 4 + n | Add
Nine more than a number | n + 9 or 9 + n | Add
Eight less than a number | n - 8 | Subtract
Eight decreased by a number | 8 - n | Subtract
Four times a number | 4n | Multiply
The product of 10 and a number | 10n | Multiply
A number divided by 5 or five divided by a number |  | Divide
One-fourth of a number |  | Divide

Steps for Solving Algebraic Word Problems

Step 1 | Read the word problem carefully and completely. Determine what the word problem is asking you to find out.
Step 2 | Determine what information given in the word problem is needed to solve the problem, and what information is not needed to solve the problem.
Step 3 | Determine which mathematical and algebraic operations to use in solving the word problem. Often more than one operation will be needed.
Step 4 | Set up the appropriate operations, and solve the problem.
Step 5 | Re-read the word problem to make sure that the solution makes sense, and that you answered the question that was asked. Write the final answer.

Example 1 Fourteen percent were belligerent while the rest were merely eristic. If 4,300 were eristic, how many were belligerent?

Step 1 | Read the word problem carefully and completely. Determine what the word problem is asking you to find out. | How many were belligerent?
Step 2 | Determine what information given in the word problem is needed to solve the problem, and what information is not needed to solve the problem. | No unnecessary information in this word problem.
Step 3 | Determine which mathematical and algebraic operations to use in solving the word problem. Often more than one operation will be needed. | Use proportion set-up.
Step 4 | Set up the appropriate operations, and solve the problem. |
| If 14% were belligerent, what percent were eristic? 86% | 100% 14% = 86%
| If 86% were eristic, and 4300 were eristic, how many were belligerent? | 4300 = 86% X = 14%
|  | (14 x 4300) / 86 = 700
Step 5 | Re-read the word problem to make sure that the solution makes sense, and that you answered the question that was asked. Write the final answer. | How many were belligerent? 700

Example 2 The sum of 9 and five times a certain number is 33 more than the number. What is the number?

Step 1 | Read the word problem carefully and completely. Determine what the word problem is asking you to find out. | What is the number?
Step 2 | Determine what information given in the word problem is needed to solve the problem, and what information is not needed to solve the problem. | No unnecessary information in this word problem.
Step 3 | Determine which mathematical and algebraic operations to use in solving the word problem. Often more than one operation will be needed. |
| The sum of 9 and The word sum tells us that we will be adding 9 to something. We will be adding 9 to five times a certain number. We dont know what that certain number is yet, so let us call that number X. Five times that certain number is expressed mathematically as 5X. So we will be adding 9 to 5X. | 5X + 9
| is 33 more than the number. The word is will be written mathematically as =. | 5X + 9 =
| 33 more than the number. The number is still expressed mathematically as X. More than tells us to add. So we will be adding 33 to X. | 5X + 9 = X + 33
Step 4 | Set up the appropriate operations, and solve the problem. | 5X + 9 = X + 33 5X + 9 = X + 33 - 9 - 9 0 24 5X = X + 24 - X - X 4X 0 4X = 24 4X = 24 4 4 X = 6
Step 5 | Re-read the word problem to make sure that the solution makes sense, and that you answered the question that was asked. Write the final answer. | What is the number? 6

Example 3 A number (n) multiplied by 6 is equal to the same number added to 45. What is the number (n)?

Step 1 | Read the word problem carefully and completely. Determine what the word problem is asking you to find out. | What is the number (n)?
Step 2 | Determine what information given in the word problem is needed to solve the problem, and what information is not needed to solve the problem. | No unnecessary information in this word problem.
Step 3 | Determine which mathematical and algebraic operations to use in solving the word problem. Often more than one operation will be needed. |
| A number (n) We dont know what the number is yet, so we call it n. | n
| A number (n) multiplied by 6 We will multiply n by 6. | 6n
| is equal to This phrase is written mathematically as =. | 6n =
| the same number added to 45. The same number is n. We will add n to 45. | 6n = 45 + n
Step 4 | Set up the appropriate operations, and solve the problem. | 6n = 45 + n 6n = 45 + n -n -n 5n 0 5n = 45 5n = 45 5 5 n = 9
Step 5 | Re-read the word problem to make sure that the solution makes sense, and that you answered the question that was asked. Write the final answer. | What is the number (n)? 9

Example 4 When the sum of 7 and a certain number is increased by 14, the result is the same as the product of 4 and the number. What is the number?

Step 1 | Read the word problem carefully and completely. Determine what the word problem is asking you to find out. | What is the number?
Step 2 | Determine what information given in the word problem is needed to solve the problem, and what information is not needed to solve the problem. | No unnecessary information in this word problem.
Step 3 | Determine which mathematical and algebraic operations to use in solving the word problem. Often more than one operation will be needed. |
| the sum of This phrase tells us that we will be adding. The sum of 7 and a certain number tells us to add 7 and a certain number. We dont know what that number is yet, so we will call it n. | 7 + n
| When the sum of 7 and a certain number is increased by 14 The word increased tells us to add. We will be adding 14 to (7 + n) | (7 + n) + 14
| The phrase the result is the same as is written mathematically as = . | (7 + n) + 14 =
| The phrase the product tells us we will be multiplying. |
| The phrase the product of 4 and the number is mathematically written as 4 x n, or 4n. | (7 + n) + 14 = 4n
Step 4 | Set up the appropriate operations, and solve the problem. | (7 + n) + 14 = 4n 21 + n = 4n 21 + n = 4n -n -n 0 3n 21 = 3n 21 = 3n 3 3 7 = n
Step 5 | Re-read the word problem to make sure that the solution makes sense, and that you answered the question that was asked. Write the final answer. | What is the number? 7