What is 10.162857128571285712857128571... as a Fraction?

10.1628571 equals 203255111999980 as a fraction.

How do you turn 10.1628571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.1628571 repeating into a fraction, begin writing this simple equation:

n = 10.1628571 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (28571) and 2 digits in the non-repeating part (16). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 101628571.28571 (equation 2)

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

10000000 × n = 101628571.28571
     100 × n = 10.1628571
 9999900 × n = 101627555

1016275559999900 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 = 1016275559999900 = 101627555 ÷ 59999900 ÷ 5 = 203255111999980. So,

10.1628571 = 203255111999980 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 203255111999980 is also equal to 103257111999980 when expressed as a mixed number.

The repeating decimal 10.1628571 (vinculum notation) has a repeated block length of 5. It is also represented as 10.16285712857128571... (ellipsis notation) which equals approximately 10.162857128571285712857128571 (decimal approximation)(*).

The recurring decimal 10.1628571 can be written as a ratio of two integers having 20325511 as the numerator and 1999980 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.