Solving Equations with Decimals

Introduction

• Equations with decimals can be treated similarly to equations with fractions.
• Multiplying by a common denominator can eliminate the decimals.

Example 1: Multiplying to Eliminate Decimals

Given Equation:

• ( \frac{1}{10}x + \frac{16}{10} = \frac{32}{10} - \frac{3}{10} )

Steps:

1. Multiply by 10:
   • Move all decimals one place to the right:
     • (1x + 16 = 32 - 3x)
2. Combine Like Terms:
   • Rearrange to have variable on one side:
     • (4x = 16)
3. Solve for x:
   • Divide both sides by 4:
     • (x = 4)

Example 2: Handling Smaller Decimals

Given Equation:

• (\frac{5}{100}(4-x) + \frac{1}{10} = \frac{32}{100})

Steps:

1. Common Denominator is 100:
   • Multiply entire equation by 100:
2. Handling Multiplication:
   • First, multiply (100 \times 0.05 \times (4-x)):
     • Simplify: (5 \times (4-x))
3. Moving Decimals:
   • Adjust each term by moving decimals two places right:
     • Get terms like (10x) and (32)
4. Distribute and Combine:
   • Distribute and combine any like terms.
5. Isolating x:
   • Subtract 20 and divide by 5:
     • (\frac{12}{5} = x)

Important Tips

• Use this method only for equations, not expressions.
• Always look for the equal sign to determine if you can treat both sides equally by multiplying.