Math Problem Statement

Convert the following repeating decimal to a fraction: 0.93̅

Solution

The problem asks to convert the repeating decimal 0.930.\overline{93} into a fraction.

Let's solve this step by step:

Step 1: Set up the repeating decimal

Let x=0.93x = 0.\overline{93} which means: x=0.93939393x = 0.93939393\ldots

Step 2: Eliminate the repeating part

To eliminate the repeating decimal, multiply both sides of the equation by 100 (since the repeating part consists of two digits): 100x=93.93939393100x = 93.93939393\ldots

Step 3: Subtract the two equations

Now subtract the original equation x=0.93x = 0.\overline{93} from 100x=93.93100x = 93.\overline{93}: 100xx=93.939393930.93939393100x - x = 93.93939393\ldots - 0.93939393\ldots 99x=9399x = 93

Step 4: Solve for xx

Now, divide both sides by 99: x=9399x = \frac{93}{99}

Step 5: Simplify the fraction

The greatest common divisor (GCD) of 93 and 99 is 3. So divide both the numerator and the denominator by 3: x=93÷399÷3=3133x = \frac{93 \div 3}{99 \div 3} = \frac{31}{33}

Thus, 0.93=31330.\overline{93} = \frac{31}{33}.

Would you like more details or have any questions?

Here are 5 related questions for practice:

  1. Convert 0.720.\overline{72} into a fraction.
  2. Express 0.470.\overline{47} as a fraction in simplest form.
  3. How would you convert 0.1230.\overline{123} to a fraction?
  4. What is the fractional equivalent of 0.580.\overline{58}?
  5. Simplify 0.190.\overline{19} as a fraction.

Tip: For repeating decimals, the number of digits in the repeating sequence determines how many powers of 10 to multiply by.

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Math Problem Analysis

Mathematical Concepts

Repeating Decimals
Fraction Conversion
Simplifying Fractions

Formulas

x = 0.93̅
100x - x = 93.93̅ - 0.93̅
x = 93/99

Theorems

Greatest Common Divisor (GCD)
Decimal to Fraction Conversion

Suitable Grade Level

Grades 7-10