Converting 0.83333 To A Fraction: A Step-By-Step Guide

Converting a decimal like 0.83333 to a fraction involves understanding its repeating nature and applying mathematical techniques to express it accurately.

Understanding Repeating Decimals

The decimal 0.83333 has a repeating digit ‘3’ after the decimal point, often denoted as 0.8̅3. This indicates that the digit ‘3’ repeats indefinitely.

Steps to Convert 0.83333 to a Fraction

1. Set Up the Equation

   Let $x=0.83333\dots$

2. Multiply to Eliminate the Repeating Part

   Since one digit repeats, multiply both sides by 10:

   $10x=8.33333\dots$

3. Subtract the Original Equation

   Subtract the original $x$ from this new equation:

   $10x-x=8.33333\dots -0.83333\dots$

   This simplifies to:

   $9x=7.5$

4. Solve for $x$

   Divide both sides by 9:

   x=7.59x = \frac{7.5}{9}x=97.5​

5. Simplify the Fraction

   To eliminate the decimal in the numerator, multiply both numerator and denominator by 10:

   x=7590x = \frac{75}{90}x=9075​

   Find the greatest common divisor (GCD) of 75 and 90, which is 15, and divide both by this number:

   x=75÷1590÷15=56x = \frac{75 \div 15}{90 \div 15} = \frac{5}{6}x=90÷1575÷15​=65​

Therefore, 0.83333…=560.83333\ldots = \frac{5}{6}0.83333…=65​.

Verification

To confirm, divide 5 by 6:

$5÷6=0.83333\dots$

This matches the original decimal, verifying the accuracy of the conversion.

Conclusion

Converting repeating decimals to fractions requires recognizing the repeating sequence and applying algebraic methods to express it as a fraction. In this case, 0.83333 repeating is equivalent to 56\frac{5}{6}65​.

FAQ

1. What does a repeating decimal mean?
   A repeating decimal has one or more digits that repeat infinitely after the decimal point.
2. How do you denote a repeating decimal?
   A repeating decimal is denoted with a bar over the repeating digit(s), e.g., 0.8̅3.
3. Can all repeating decimals be converted to fractions?
   Yes, every repeating decimal can be expressed as a fraction.
4. Is 0.83333 the same as 0.8333?
   No, 0.83333 has an additional repeating ‘3’, making it closer to 56\frac{5}{6}65​.
5. Why is it important to convert decimals to fractions?
   Converting decimals to fractions can simplify calculations and provide exact values.