What is 2.5714285714285714285714285714285714285... as a Fraction?

2.5714285 equals 187 as a fraction.

How do you turn 2.5714285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.5714285 repeating into a fraction, begin writing this simple equation:

n = 2.5714285 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (714285) and 1 digit in the non-repeating part (5). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 25714285.714285 (equation 2)

Step 3: Now subtract equation 1 from equation 2 to cancel the repeating block (or repetend) out.

10000000 × n = 25714285.714285
      10 × n = 2.5714285
 9999990 × n = 25714260

257142609999990 could be the answer, but it still can be put in the simplest form, i.e., reduced.

To simplify this fraction, divide the numerator and denominator by 1428570 (the GCF - greatest common factor).

n = 257142609999990 = 25714260 ÷ 14285709999990 ÷ 1428570 = 187. So,

2.5714285 = 187 as the lowest possible fraction.

As the numerator is greater than the denominator, we have an improper fraction, so we can also express it as a mixed number, thus 187 is also equal to 247 when expressed as a mixed number.

The repeating decimal 2.5714285 (vinculum notation) has a repeated block length of 6. It is also represented as 2.5714285714285714285... (ellipsis notation) which equals approximately 2.5714285714285714285714285714285 (decimal approximation)(*).

The recurring decimal 2.5714285 can be written as a ratio of two integers having 18 as the numerator and 7 as the denominator. So, it is a rational number (named after ratio). It can be shown that a number is rational if its decimal representation is repeating or terminating.

(*) At present, there is no single universally accepted notation or phrasing for repeating decimals.