0.8 Repeating As A Fraction Calculator

Convert the repeating decimal 0.8 (where 8 repeats infinitely) to a fraction using our calculator and step-by-step guide. Learn the formula, assumptions, and examples.

How to Convert 0.8 Repeating to a Fraction

Converting a repeating decimal to a fraction involves algebraic manipulation to eliminate the repeating pattern. Here's the step-by-step process for 0.8 repeating:

Let x = 0.888... (the repeating decimal)
Multiply both sides by 10 to shift the decimal point: 10x = 8.888...
Subtract the original equation from this new equation: 10x - x = 8.888... - 0.888...
Simplify: 9x = 8
Solve for x: x = 8/9

Note: This method works for any repeating decimal where the repeating pattern starts immediately after the decimal point.

Formula Used

For a repeating decimal 0.aaa... (where 'a' repeats infinitely):

x = 0.aaa...

Multiply by 10: 10x = a.aaa...

Subtract original: 9x = a

Therefore: x = a/9

For 0.8 repeating, a = 8, so the fraction is 8/9.

Worked Example

Let's convert 0.888... to a fraction step by step:

Let x = 0.888...
Multiply by 10: 10x = 8.888...
Subtract original: 10x - x = 8.888... - 0.888...
9x = 8
x = 8/9

The fraction 8/9 is equivalent to 0.888... and can be simplified to its simplest form.

Frequently Asked Questions

How do I convert a repeating decimal to a fraction?: Use the algebraic method shown in the guide, where you set the repeating decimal equal to a variable, multiply by 10 to shift the decimal, and then subtract the original equation to eliminate the repeating pattern.
What is 0.8 repeating as a fraction?: 0.8 repeating is equal to the fraction 8/9.
Can this method be used for other repeating decimals?: Yes, this method works for any repeating decimal where the repeating pattern starts immediately after the decimal point. For decimals with non-repeating digits before the repeating pattern, a slightly more complex method is needed.
Is 8/9 the simplest form of the fraction?: Yes, 8/9 is already in its simplest form since 8 and 9 have no common factors other than 1.