0.857142857 As A Fraction

Repeating Decimal to Fraction Conversion:

\[ 0.\overline{857142} = \frac{6}{7} \]

Fraction Result:

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1. What is 0.857142857 as a Fraction?

The repeating decimal 0.857142857... is exactly equal to the fraction 6/7. This is a repeating decimal with a 6-digit repeating sequence (857142).

2. Conversion Method

To convert 0.857142857... to a fraction:

\[ x = 0.\overline{857142} \] \[ 1,000,000x = 857,142.\overline{857142} \] \[ 1,000,000x - x = 857,142 \] \[ 999,999x = 857,142 \] \[ x = \frac{857,142}{999,999} = \frac{6}{7} \]

3. Mathematical Proof

Verification by division:

\[ 6 \div 7 = 0.\overline{857142} \]

The division exactly reproduces the original repeating decimal.

4. Applications

Practical Uses: This conversion is useful in probability calculations, ratio representations, and when exact fractional values are preferred over decimal approximations.

5. Frequently Asked Questions (FAQ)

Q1: Why does 6/7 produce this repeating decimal?
A: Because 7 doesn't divide evenly into powers of 10, creating a repeating sequence of 6 digits (the maximum possible for denominator 7).

Q2: How can I recognize other fractions that produce similar patterns?
A: Fractions with denominators that are prime numbers (other than 2 or 5) will produce repeating decimals. The length of the repeating sequence is related to the denominator.

Q3: Is 0.857142857... exactly equal to 6/7?
A: Yes, they are mathematically identical representations of the same value.

Q4: Can this be simplified further?
A: No, 6/7 is already in its simplest form as the numerator and denominator share no common divisors other than 1.

Q5: How would this be expressed as a percentage?
A: 6/7 ≈ 85.7142857...%, with the same repeating decimal pattern.