What is 2.14285714285714285714285714285714285714... as a Fraction?

2.14285714 equals 157 as a fraction.

How do you turn 2.14285714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.14285714 repeating into a fraction, begin writing this simple equation:

n = 2.14285714 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (285714) and 2 digits in the non-repeating part (14). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 214285714.285714 (equation 2)

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

100000000 × n = 214285714.285714
      100 × n = 2.14285714
 99999900 × n = 214285500

21428550099999900 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 14285700 (the GCF - greatest common factor).

n = 21428550099999900 = 214285500 ÷ 1428570099999900 ÷ 14285700 = 157. So,

2.14285714 = 157 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 157 is also equal to 217 when expressed as a mixed number.

The repeating decimal 2.14285714 (vinculum notation) has a repeated block length of 6. It is also represented as 2.14285714285714285714... (ellipsis notation) which equals approximately 2.14285714285714285714285714285714 (decimal approximation)(*).

The recurring decimal 2.14285714 can be written as a ratio of two integers having 15 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.