What is 1.2857142857142857142857142857142857142... as a Fraction?

1.2857142 equals 97 as a fraction.

How do you turn 1.2857142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.2857142 repeating into a fraction, begin writing this simple equation:

n = 1.2857142 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (857142) and 1 digit in the non-repeating part (2). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 12857142.857142 (equation 2)

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

10000000 × n = 12857142.857142
      10 × n = 1.2857142
 9999990 × n = 12857130

128571309999990 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 = 128571309999990 = 12857130 ÷ 14285709999990 ÷ 1428570 = 97. So,

1.2857142 = 97 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 97 is also equal to 127 when expressed as a mixed number.

The repeating decimal 1.2857142 (vinculum notation) has a repeated block length of 6. It is also represented as 1.2857142857142857142... (ellipsis notation) which equals approximately 1.2857142857142857142857142857142 (decimal approximation)(*).

The recurring decimal 1.2857142 can be written as a ratio of two integers having 9 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.