0.83333 as a fraction

2 min read 16-01-2025

Meta Description: Learn how to convert the repeating decimal 0.83333... into a fraction. This comprehensive guide provides a step-by-step process, explanations, and real-world examples to help you master this essential math skill. Understand the method behind converting repeating decimals and solidify your understanding of fractions.

Understanding Repeating Decimals

The number 0.83333... is a repeating decimal. This means the digit 3 repeats infinitely. Converting repeating decimals to fractions requires a specific process. We'll break it down step-by-step.

How to Convert 0.83333... to a Fraction

Here's the method for converting 0.83333... into its fractional equivalent:

Step 1: Set up an equation.

Let x = 0.83333...

Step 2: Multiply to shift the repeating part.

We need to manipulate the equation so the repeating part lines up. Since the repeating digit is 3, we'll multiply both sides by 10:

10x = 8.33333...

Step 3: Subtract the original equation.

Subtract the original equation (x = 0.83333...) from the equation in Step 2:

10x - x = 8.33333... - 0.83333...

This simplifies to:

9x = 7.5

Step 4: Solve for x.

Divide both sides of the equation by 9:

x = 7.5 / 9

Step 5: Simplify the fraction.

The fraction 7.5/9 isn't in its simplest form. To simplify, we can multiply both the numerator and the denominator by 2 to remove the decimal:

x = (7.5 * 2) / (9 * 2) = 15/18

Now, simplify further by finding the greatest common divisor (GCD) of 15 and 18, which is 3:

x = 15/3 / 18/3 = 5/6

Therefore, 0.83333... is equal to 5/6.

Checking Your Work

To verify our answer, you can divide 5 by 6 using a calculator. You'll get 0.83333..., confirming our conversion is correct.

Why This Method Works

This method works because subtracting the original equation from the multiplied equation cancels out the infinitely repeating decimal part, leaving a manageable equation to solve.

Other Examples of Converting Repeating Decimals

Let's look at a couple more examples to solidify your understanding:

0.6666...: This is a simpler example. Let x = 0.6666... Multiplying by 10 gives 10x = 6.6666... Subtracting the original equation gives 9x = 6, so x = 6/9 = 2/3.
0.142857142857...: This repeating decimal has a longer repeating block. The process is the same, just requiring more steps and possibly a larger GCD to simplify the fraction.

Conclusion

Converting repeating decimals like 0.83333... to fractions is a valuable skill in mathematics. By following the step-by-step process outlined above, you can confidently convert any repeating decimal into its fractional equivalent. Remember to always simplify your fraction to its lowest terms. Now you can tackle those repeating decimals with confidence!