Math Problem Statement

Rewrite as a simplified fraction.

[\large{3.\overline{2} = {?}}]

Solution

To express 3.23.\overline{2} as a simplified fraction, we can follow these steps:

  1. Let x=3.2x = 3.\overline{2}, where the repeating part is 22.

  2. Multiply both sides of the equation by 10 to move the decimal point: 10x=32.210x = 32.\overline{2}

  3. Now subtract the original equation (x=3.2x = 3.\overline{2}) from this new equation: 10xx=32.23.210x - x = 32.\overline{2} - 3.\overline{2} 9x=299x = 29

  4. Solve for xx by dividing both sides by 9: x=299x = \frac{29}{9}

Thus, 3.23.\overline{2} as a simplified fraction is 299\frac{29}{9}.

Would you like any more details on the steps? Here are 5 more questions for practice:

  1. Rewrite 1.61.\overline{6} as a fraction.
  2. Express 2.42.\overline{4} as a fraction.
  3. Convert 0.30.\overline{3} to a fraction.
  4. How do you rewrite 4.74.\overline{7} as a fraction?
  5. Simplify 5.85.\overline{8} as a fraction.

Tip: When repeating decimals are converted to fractions, they often simplify to a ratio of two integers by manipulating algebraic expressions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Repeating Decimals
Fractions

Formulas

Let x = repeating decimal form
Multiply by powers of 10 to shift the decimal
Subtract equations to eliminate the repeating part
Solve for x and simplify

Theorems

Algebraic manipulation of repeating decimals
Conversion of repeating decimals to fractions

Suitable Grade Level

Grades 8-10