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 on fractions lays a solid foundation for understanding parts of a whole through visual models and number lines. It covers the conversion between improper fractions and mixed numbers, the identification of equivalent and simplified fractions using the Equivalent Fractions Property, and practical operations such as multiplying and dividing fractions. Mastering these concepts not only aids in mathematical problem-solving but also has real-world applications in areas such as baking, carpentry, and music.

Learning Objectives

1

Understand the meaning of fractions and how they represent parts of a whole.

2

Model fractions using visual aids such as fraction circles, tiles, and number lines.

3

Convert between improper fractions and mixed numbers, and find equivalent fractions using the Equivalent Fractions Property.

4

Simplify fractions by eliminating common factors and use them in multiplication and division operations.

5

Apply fraction concepts to solve real-world problems in contexts like baking, carpentry, and music measures.

Key Concepts

CONCEPT

DEFINITION

Fraction

A number written in the form a/b where a (the numerator) represents parts of a whole and b (the denominator) indicates the total number of equal parts; b ≠ 0.

Proper Fraction

A fraction where the numerator is less than the denominator, representing a value less than one.

Improper Fraction

A fraction in which the numerator is greater than or equal to the denominator, representing a value of one or greater.

Mixed Number

A number that combines a whole number with a proper fraction, such as 1 1/3.

Equivalent Fractions

Different fractions that represent the same value, which can be obtained by multiplying or dividing the numerator and denominator by the same nonzero number.

Simplified Fraction

A fraction in which the numerator and denominator have no common factors other than 1.

Property of One

The principle that any nonzero number divided by itself equals one (a/a = 1, for a ≠ 0).

Reciprocal

The multiplicative inverse of a fraction; for a fraction a/b, its reciprocal is b/a (provided a ≠ 0).

Example Problems

Example 1

In the following exercises, name the fraction of each figure that is shaded.

Example 2

In the following exercises, name the fraction of each figure that is shaded.

Example 3

In the following exercises, shade parts of circles or squares to model the following fractions. $$\frac{1}{2}$$

Example 4

In the following exercises, shade parts of circles or squares to model the following fractions. $$\frac{1}{3}$$

Example 5

In the following exercises, shade parts of circles or squares to model the following fractions. $$\frac{3}{4}$$
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Step-by-Step Explanations

QUESTION

Convert the improper fraction 11/6 into a mixed number.

STEP-BY-STEP ANSWER:

Step 1: Divide the numerator (11) by the denominator (6). The quotient is 1.
Step 2: Multiply the quotient (1) by the denominator (6) to get 6.
Step 3: Subtract this product (6) from the numerator (11) to find the remainder, which is 5.
Step 4: Write the mixed number as the quotient with the remainder over the original denominator: 1 5/6.
Final Answer: 11/6 = 1 5/6.

Converting an Improper Fraction to a Mixed Number

QUESTION

Simplify the fraction 10/15.

STEP-BY-STEP ANSWER:

Step 1: Find the common factors of the numerator (10) and the denominator (15). A common factor is 5.
Step 2: Divide both the numerator and denominator by 5.
Step 3: 10 ÷ 5 = 2 and 15 ÷ 5 = 3.
Final Answer: 10/15 simplifies to 2/3.

Simplifying a Fraction

QUESTION

Multiply the fractions 2/3 and 3/5.

STEP-BY-STEP ANSWER:

Step 1: Multiply the numerators: 2 × 3 = 6.
Step 2: Multiply the denominators: 3 × 5 = 15.
Step 3: Write the result as 6/15.
Step 4: Simplify the fraction by dividing numerator and denominator by their greatest common divisor (3): 6 ÷ 3 = 2 and 15 ÷ 3 = 5.
Final Answer: 2/3 × 3/5 = 2/5.

Multiplying Fractions

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Common Mistakes

  • Confusing the numerator with the denominator when interpreting a fraction.
  • Neglecting to simplify fractions completely due to overlooking common factors.
  • Mixing up the process of converting improper fractions to mixed numbers or vice versa.
  • Failing to use equivalent fractions properly by not multiplying both the numerator and denominator by the same number.
  • Incorrectly identifying the number of parts when visualizing fractions on a number line.