Math Problem Statement

Write 0. 32 overbar as a fraction in simplest form.

Solution

To express 0.320.\overline{32} (the repeating decimal 0.3232320.323232\ldots) as a fraction, we can follow these steps:

Step 1: Let x=0.32x = 0.\overline{32}

We define xx as the repeating decimal: x=0.323232x = 0.323232\ldots

Step 2: Eliminate the repeating decimal

Since the repeating part is 2 digits long ("32"), we multiply both sides of the equation by 100 (because the repeat length is 2 digits): 100x=32.323232100x = 32.323232\ldots

Step 3: Subtract the original equation from the new equation

Now subtract x=0.323232x = 0.323232\ldots from 100x=32.323232100x = 32.323232\ldots: 100xx=32.3232320.323232100x - x = 32.323232\ldots - 0.323232\ldots 99x=3299x = 32

Step 4: Solve for xx

Now, solve for xx by dividing both sides of the equation by 99: x=3299x = \frac{32}{99}

Final Answer:

The fraction equivalent of 0.320.\overline{32} is: 3299\frac{32}{99}

This fraction is already in its simplest form because 32 and 99 have no common factors other than 1.


Would you like to see more examples, or need further clarification? Here are some related questions:

  1. How can we convert 0.60.\overline{6} into a fraction?
  2. What is the fractional representation of 0.1428570.\overline{142857}?
  3. How would you convert 0.1230.\overline{123} into a fraction?
  4. What is the general method for converting repeating decimals to fractions?
  5. What do we mean when a decimal is called "repeating"?

Tip: If the decimal repeats for nn digits, multiply by 10n10^n to eliminate the repeating part before subtracting.

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Math Problem Analysis

Mathematical Concepts

Repeating Decimals
Fractions
Algebra

Formulas

x = 0.323232...
100x = 32.323232...
x = 32/99

Theorems

Converting repeating decimals to fractions

Suitable Grade Level

Grades 7-9