Book cover for Prealgebra

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

ISBN #9781938168994

1st Edition

5,729 Questions

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210,912 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter emphasizes the importance of a systematic problem-solving strategy when approaching word problems. It illustrates how to translate everyday situations and verbal descriptions into algebraic equations using variables. The step-by-step method includes reading, identifying what is needed, assigning variables, translating, solving, checking, and finally, answering in complete sentences. In addition, the chapter connects these algebraic skills with practical applications, such as money calculations and geometric measurements, reinforcing that these skills are valuable across many real-world scenarios.

Learning Objectives

1

Apply a systematic problem solving strategy to tackle word problems in various contexts.

2

Translate verbal descriptions into algebraic expressions and equations effectively.

3

Solve number problems and money applications using algebraic techniques.

4

Check and validate solutions to ensure they are reasonable in real-life contexts.

5

Develop a positive attitude and confidence when approaching challenging word problems.

Key Concepts

CONCEPT

DEFINITION

Problem Solving Strategy

A systematic method that involves reading the problem, identifying what is asked, assigning variables, translating the words into equations, solving, checking, and answering in complete sentences.

Word Problem

A problem expressed in words rather than mathematical symbols that requires interpretation and translation into an algebraic expression or equation.

Variable

A symbol, typically a letter, used to represent an unknown quantity in an equation.

Algebraic Equation

A statement that equates two expressions using variables and constants, often derived from translating a word problem.

Consecutive Integers

Integers that follow one after the other in order, with each being one more than the previous (e.g., n, n+1, n+2).

Money Applications

Real-life problems that involve calculations related to monetary values, such as coin counts, sales discounts, and tax computations.

Example Problems

Example 1

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 2

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 3

Zachary has 25 country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

Example 4

One-fourth of the candies in a bag of are red. If there are 23 red candies, how many candies are in the bag?

Example 5

There are 16 girls in a school club. The number of girls is 4 more than twice the number of boys. Find the number of boys in the club.
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Step-by-Step Explanations

QUESTION

Pete bought a shirt on sale for $18, which is one-half the original price. What was the original price of the shirt?

STEP-BY-STEP ANSWER:

Step 1: Read the problem carefully to understand that you need to find the original price.
Step 2: Identify the unknown quantity; here, it is the original price, which we call p.
Step 3: Since $18 is one-half of the original price, set up the equation: 18 = (1/2)p.
Step 4: Solve the equation by multiplying both sides by 2: p = 18 * 2.
Step 5: Simplify the multiplication: p = 36.
Step 6: Check the answer: Half of 36 is indeed 18, which matches the problem.
Final Answer: The original price of the shirt was $36.

Original Price (Example 9.1)

QUESTION

Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought 11 apples. How many bananas did he bring?

STEP-BY-STEP ANSWER:

Step 1: Read the problem to note that we need to find the number of bananas.
Step 2: Identify the unknown quantity, and choose b for the number of bananas.
Step 3: Translate the relationship: the number of apples = 2b + 3.
Step 4: Substitute the known value for apples: 2b + 3 = 11.
Step 5: Solve the equation by subtracting 3 from both sides: 2b = 8.
Step 6: Divide both sides by 2 to find b: b = 4.
Step 7: Check the answer by verifying that twice 4 plus 3 equals 11.
Final Answer: Yash brought 4 bananas to the picnic.

Bananas at a Picnic (Example 9.2)

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Common Mistakes

  • Failing to fully read the problem and missing key details.
  • Choosing a variable that does not clearly represent the unknown quantity.
  • Incorrectly translating verbal phrases into algebraic expressions (e.g., reversing subtraction terms).
  • Skipping the step of checking the solution to ensure it makes sense in context.
  • Overlooking negative signs or percentage values when forming equations.