What is 10.23636363636363... as a Fraction?

10.2363 equals 56355 as a fraction.

How do you turn 10.2363 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.2363 repeating into a fraction, begin writing this simple equation:

n = 10.2363 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (63) and 2 digits in the non-repeating part (23). Multiply both sides by 10⁴ = 10000.

10000 × n = 102363.63 (equation 2)

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

10000 × n = 102363.63
  100 × n = 10.2363
 9900 × n = 101340

1013409900 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 180 (the GCF - greatest common factor).

n = 1013409900 = 101340 ÷ 1809900 ÷ 180 = 56355. So,

10.2363 = 56355 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 56355 is also equal to 101355 when expressed as a mixed number.

The repeating decimal 10.2363 (vinculum notation) has a repeated block length of 2. It is also represented as 10.23636363... (ellipsis notation) which equals approximately 10.236363636363 (decimal approximation)(*).

The recurring decimal 10.2363 can be written as a ratio of two integers having 563 as the numerator and 55 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.