What is 10.10076923076923076923076923076923076923... as a Fraction?

10.10076923 equals 131311300 as a fraction.

How do you turn 10.10076923 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.10076923 repeating into a fraction, begin writing this simple equation:

n = 10.10076923 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (076923) and 2 digits in the non-repeating part (10). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 1010076923.076923 (equation 2)

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

100000000 × n = 1010076923.076923
      100 × n = 10.10076923
 99999900 × n = 1010075913

101007591399999900 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 76923 (the GCF - greatest common factor).

n = 101007591399999900 = 1010075913 ÷ 7692399999900 ÷ 76923 = 131311300. So,

10.10076923 = 131311300 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 131311300 is also equal to 101311300 when expressed as a mixed number.

The repeating decimal 10.10076923 (vinculum notation) has a repeated block length of 6. It is also represented as 10.10076923076923076923... (ellipsis notation) which equals approximately 10.10076923076923076923076923076923 (decimal approximation)(*).

The recurring decimal 10.10076923 can be written as a ratio of two integers having 13131 as the numerator and 1300 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.