Elementary Algebra

Lynn Marecek

ISBN #9780998625713

1st Edition

6,645 Questions

32,192 Students Helped

Homework Questions

Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

This chapter emphasizes the importance of a systematic approach to solving word problems by reading carefully, identifying unknown quantities, translating words into algebraic equations, and using step-by-step algebraic techniques. The learned strategies are applicable to various types of problems, including number problems, problems dealing with consecutive integers, and percent applications. Maintaining a positive attitude is highlighted as essential in overcoming the challenges often associated with word problems.

Learning Objectives

Develop a positive attitude toward word problems and use it as a tool for success.

Apply a systematic problem-solving strategy to translate everyday scenarios into algebraic expressions and equations.

Solve single-variable and two-variable word problems, including number problems and consecutive integer problems.

Interpret and solve percent applications by converting percentages into decimals and forming appropriate equations.

Key Concepts

CONCEPT

DEFINITION

Word Problem

A real-world scenario described in words that requires a translation into mathematical expressions or equations in order to be solved.

Variable

A symbol (often a letter) used to represent an unknown quantity in algebra.

Translate

The process of converting words or phrases from a problem into an algebraic expression or equation.

Consecutive Integers

A sequence of integers where each number is one more than the previous number (e.g., 7, 8, 9).

Percent Applications

Problems that involve percentages which are converted into decimals and used in forming equations to find unknown quantities.

Example Problems

Example 1

In the following exercises, prepare the lists described. List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

Example 2

In the following exercises, prepare the lists described. List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.

Example 3

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question. Two-thirds of the children in the fourth-grade class are girls. If there are 20 girls, what is the total number of children in the class?

Example 4

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question. Three-fifths of the members of the school choir are women. If there are 24 women, what is the total number of choir members?

Example 5

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question. Zachary has 25 country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

Step-by-Step Explanations

QUESTION

How do you determine the original price of an item bought on sale for half of its original price?

STEP-BY-STEP ANSWER:

Step 1: Read the problem carefully and understand the information given.
Step 2: Identify that you need to find the original price. Assign a variable (e.g., p) to represent the original price.
Step 3: Translate the statement ‘bought at half the original price’ into an equation: Sale Price = 1/2 × p.
Step 4: Substitute the given sale price into the equation (e.g., if sale price is $18, then 18 = 1/2 × p).
Step 5: Solve the equation by multiplying both sides by 2 to isolate p.
Step 6: Check the answer: verifying that half of the computed original price equals the sale price.
Final Answer: The original price is $36.

Using a Problem-Solving Strategy

QUESTION

How do you solve for two consecutive integers whose sum is a given number?

STEP-BY-STEP ANSWER:

Step 1: Read the problem and decide that you need two consecutive integers.
Step 2: Let n represent the first integer. Then the second integer is n + 1.
Step 3: Set up the equation based on the problem’s statement (e.g., n + (n + 1) = sum).
Step 4: Combine like terms and solve for n.
Step 5: Once n is found, compute n + 1 to find the consecutive integer.
Step 6: Check the solution by summing both numbers to ensure they equal the given sum.
Final Answer: The two integers satisfy the equation (for example, if the sum is 47, n = 23 and n + 1 = 24).

Solving Consecutive Integer Problems

Common Mistakes

Failing to read the problem thoroughly before attempting to solve it.
Incorrect translation of word phrases into algebraic expressions or equations due to misinterpretation of key terms such as 'less than' or 'more than'.
Neglecting to check whether the final solution fits the original problem scenario.
Using more variables than necessary when the problem can be resolved with a single variable.
Errors in converting percentages to decimals when solving percent applications.