What is 10.24285742857428574285742857... as a Fraction?

10.242857 equals 2048551199998 as a fraction.

How do you turn 10.242857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.242857 repeating into a fraction, begin writing this simple equation:

n = 10.242857 (equation 1)

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

1000000 × n = 10242857.42857 (equation 2)

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

1000000 × n = 10242857.42857
     10 × n = 10.242857
 999990 × n = 10242755

10242755999990 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 5 (the GCF - greatest common factor).

n = 10242755999990 = 10242755 ÷ 5999990 ÷ 5 = 2048551199998. So,

10.242857 = 2048551199998 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 2048551199998 is also equal to 1048571199998 when expressed as a mixed number.

The repeating decimal 10.242857 (vinculum notation) has a repeated block length of 5. It is also represented as 10.2428574285742857... (ellipsis notation) which equals approximately 10.24285742857428574285742857 (decimal approximation)(*).

The recurring decimal 10.242857 can be written as a ratio of two integers having 2048551 as the numerator and 199998 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.