What is 10.74285742857428574285742857... as a Fraction?

10.742857 equals 107427599999 as a fraction.

How do you turn 10.742857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.742857 repeating into a fraction, begin writing this simple equation:

n = 10.742857 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (42857) and 1 digit in the non-repeating part (7). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 10742857.42857 (equation 2)

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

1000000 × n = 10742857.42857
     10 × n = 10.742857
 999990 × n = 10742750

10742750999990 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 10 (the GCF - greatest common factor).

n = 10742750999990 = 10742750 ÷ 10999990 ÷ 10 = 107427599999. So,

10.742857 = 107427599999 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 107427599999 is also equal to 107428599999 when expressed as a mixed number.

The repeating decimal 10.742857 (vinculum notation) has a repeated block length of 5. It is also represented as 10.7428574285742857... (ellipsis notation) which equals approximately 10.74285742857428574285742857 (decimal approximation)(*).

The recurring decimal 10.742857 can be written as a ratio of two integers having 1074275 as the numerator and 99999 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.