Lesson 85: Equations with Fractions and Decimals

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• Lesson Description
• Learning Objectives
• Topics Covered
• Key Terms
• Lesson Examples
• Practice Problems
• Tips & Common Mistakes
• Summary
• Challenge Problems
• Unit Navigation

Lesson Description

This lesson teaches students how to solve equations that include fractions and decimals. Techniques include clearing fractions using the least common denominator (LCD), converting decimals to fractions, and performing arithmetic with decimals to isolate variables and solve equations efficiently.

Learning Objectives

• Identify equations that contain fractions and decimals.
• Solve equations by clearing fractions using the least common denominator.
• Solve equations involving decimals using multiplication or division to simplify.
• Check solutions by substituting values back into the original equation.

Topics Covered

• Equations with fractions
• Equations with decimals
• Least common denominator (LCD) strategy
• Decimal multiplication and division in equations

Key Terms

• Fraction Equation: An equation that contains one or more fractions with the variable in the numerator or denominator.
• Decimal Equation: An equation that contains one or more decimal numbers.
• Least Common Denominator (LCD): The smallest number that all denominators divide into, used to clear fractions in equations.
• Clearing Fractions: Multiplying all terms by the LCD to eliminate fractions.

Lesson Examples

Example 1: Solving a Fraction Equation

1. Solve: (2/3)x + 5 = 11
2. Subtract 5 from both sides: (2/3)x = 6
3. Multiply both sides by 3/2: x = 6 * (3/2) = 9

Example 2: Solving a Decimal Equation

1. Solve: 0.4x - 2.2 = 1.8
2. Add 2.2 to both sides: 0.4x = 4.0
3. Divide both sides by 0.4: x = 4.0 / 0.4 = 10

Practice Problems

1. (3/4)x + 7 = 10
2. 5x - 0.6 = 2.4
3. (2/5)x - 3 = 1
4. 0.25x + 1.5 = 2.5

Tips & Common Mistakes

• Always multiply all terms by the LCD when clearing fractions.
• Convert decimals to fractions if it makes arithmetic easier.
• Check your solution by substituting it back into the original equation to verify correctness.

Summary

Equations with fractions and decimals can be solved efficiently by clearing fractions using the least common denominator or working carefully with decimal operations. Always verify solutions to maintain accuracy and confidence in solving these types of equations.

Challenge Problems

1. (5/6)x + 2 = 7
2. 0.3x - 1.5 = 0.6
3. (7/8)x - 3.5 = 4.25