What is 10.23131313... as a Fraction?

10.2313 equals 10129990 as a fraction.

How do you turn 10.2313 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.2313 repeating into a fraction, begin writing this simple equation:

n = 10.2313 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (13) and 2 digits in the non-repeating part (23). Multiply both sides by 10⁴ = 10000.

10000 × n = 102313.13 (equation 2)

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

10000 × n = 102313.13
  100 × n = 10.2313
 9900 × n = 101290

1012909900 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 = 1012909900 = 101290 ÷ 109900 ÷ 10 = 10129990. So,

10.2313 = 10129990 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 10129990 is also equal to 10229990 when expressed as a mixed number.

The repeating decimal 10.2313 (vinculum notation) has a repeated block length of 2. It is also represented as 10.23131313... (ellipsis notation) which equals approximately 10.231313131313 (decimal approximation)(*).

The recurring decimal 10.2313 can be written as a ratio of two integers having 10129 as the numerator and 990 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.