13.714285714285714 As A Fraction

Repeating Decimal to Fraction Conversion:

\[ 13.\overline{714285} = \frac{96}{7} \]

Calculation Steps:

1. Let x = 13.714285714285714...
2. The repeating part "714285" has 6 digits
3. Multiply both sides by 10^6: 1000000x = 13714285.714285714285...
4. Subtract original equation: 1000000x - x = 13714285.714285... - 13.714285...
5. 999999x = 13714272
6. x = 13714272 / 999999
7. Simplify fraction: divide numerator and denominator by 142857
8. Final result: 96/7

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1. What is 13.714285714285714 as a Fraction?

The repeating decimal 13.714285714285714... (where "714285" repeats infinitely) is exactly equal to the fraction 96/7. This is a precise mathematical equivalence, not an approximation.

2. Conversion Method

The standard algebraic method for converting repeating decimals to fractions:

\[ \begin{align*} 1)&\ x = 13.\overline{714285} \\ 2)&\ 10^6x = 13714285.\overline{714285} \\ 3)&\ 999999x = 13714272 \\ 4)&\ x = \frac{13714272}{999999} = \frac{96}{7} \end{align*} \]

Key Steps:

Identify the repeating pattern (6 digits in this case)
Multiply by 10^n where n is the pattern length
Subtract the original equation to eliminate the repeating part
Solve for x and simplify the resulting fraction

3. Verification

Proof: Dividing 96 by 7 indeed gives 13.714285714285714...

\[ 96 \div 7 = 13.\overline{714285} \]

4. Applications

Uses: This conversion is important in:

Exact mathematical calculations where floating-point approximations are insufficient
Number theory studies of repeating decimals
Computer algorithms that need precise fractional representations

5. Frequently Asked Questions (FAQ)

Q1: Why does 96/7 produce this repeating pattern?
A: Because 7 is a prime number that doesn't divide 10, creating a 6-digit repeating cycle in its decimal expansion.

Q2: How can I recognize such repeating decimals?
A: Fractions with denominator 7 always produce the same 6-digit cycle (142857, 285714, etc.), just starting at different points.

Q3: Is this exact or an approximation?
A: The fraction 96/7 is exactly equal to 13.714285... with the pattern repeating infinitely.

Q4: Can all repeating decimals be converted to fractions?
A: Yes, all repeating decimals are rational numbers and can be expressed as fractions of integers.

Q5: What's special about the 714285 repeating pattern?
A: It's one of the cyclic permutations of 142857, the repeating unit for fractions with denominator 7.