What is 1.2285714285714285714285714285714285714... as a Fraction?

1.2285714 equals 4335 as a fraction.

How do you turn 1.2285714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.2285714 repeating into a fraction, begin writing this simple equation:

n = 1.2285714 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (285714) and 1 digit in the non-repeating part (2). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 12285714.285714 (equation 2)

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

10000000 × n = 12285714.285714
      10 × n = 1.2285714
 9999990 × n = 12285702

122857029999990 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 285714 (the GCF - greatest common factor).

n = 122857029999990 = 12285702 ÷ 2857149999990 ÷ 285714 = 4335. So,

1.2285714 = 4335 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 4335 is also equal to 1835 when expressed as a mixed number.

The repeating decimal 1.2285714 (vinculum notation) has a repeated block length of 6. It is also represented as 1.2285714285714285714... (ellipsis notation) which equals approximately 1.2285714285714285714285714285714 (decimal approximation)(*).

The recurring decimal 1.2285714 can be written as a ratio of two integers having 43 as the numerator and 35 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.