What is 10.112857128571285712857128571... as a Fraction?

10.1128571 equals 5056378499995 as a fraction.

How do you turn 10.1128571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.1128571 repeating into a fraction, begin writing this simple equation:

n = 10.1128571 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (28571) and 2 digits in the non-repeating part (11). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 101128571.28571 (equation 2)

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

10000000 × n = 101128571.28571
     100 × n = 10.1128571
 9999900 × n = 101127560

1011275609999900 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 20 (the GCF - greatest common factor).

n = 1011275609999900 = 101127560 ÷ 209999900 ÷ 20 = 5056378499995. So,

10.1128571 = 5056378499995 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 5056378499995 is also equal to 1056428499995 when expressed as a mixed number.

The repeating decimal 10.1128571 (vinculum notation) has a repeated block length of 5. It is also represented as 10.11285712857128571... (ellipsis notation) which equals approximately 10.112857128571285712857128571 (decimal approximation)(*).

The recurring decimal 10.1128571 can be written as a ratio of two integers having 5056378 as the numerator and 499995 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.