Math Problem Statement

3.8 3 repeating

Solution

The decimal 3.83.\overline{8} (where 8 repeats indefinitely) can be expressed as a fraction. Let me show you how to convert it:

Step-by-step conversion of 3.83.\overline{8} to a fraction:

  1. Let x=3.8x = 3.\overline{8}.
  2. Multiply both sides of the equation by 10 to shift the repeating decimal: 10x=38.810x = 38.\overline{8}
  3. Subtract the original equation x=3.8x = 3.\overline{8} from this new equation: 10xx=38.83.810x - x = 38.\overline{8} - 3.\overline{8} This simplifies to: 9x=359x = 35
  4. Solve for xx: x=359x = \frac{35}{9}

Thus, 3.8=3593.\overline{8} = \frac{35}{9}.

Would you like more details on this process or have any further questions?


Here are 5 related questions:

  1. How do you convert repeating decimals with different patterns (e.g., 0.230.\overline{23}) to fractions?
  2. What is the process for converting non-repeating decimals to fractions?
  3. Can all repeating decimals be expressed as fractions?
  4. How can you recognize if a decimal is terminating or repeating?
  5. How do you simplify fractions after converting from repeating decimals?

Tip: A repeating decimal is always a rational number, meaning it can be written as a fraction of two integers!

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Math Problem Analysis

Mathematical Concepts

Algebra
Repeating Decimals
Fractions

Formulas

x = 3.\overline{8}
10x = 38.\overline{8}
10x - x = 9x
x = \frac{35}{9}

Theorems

Rational Numbers Theorem (Repeating decimals can be expressed as fractions)

Suitable Grade Level

Grades 7-9