What is 13.10076923076923076923076923076923076923... as a Fraction?

13.10076923 equals 170311300 as a fraction.

How do you turn 13.10076923 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.10076923 repeating into a fraction, begin writing this simple equation:

n = 13.10076923 (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 = 1310076923.076923 (equation 2)

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

100000000 × n = 1310076923.076923
      100 × n = 13.10076923
 99999900 × n = 1310075613

131007561399999900 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 = 131007561399999900 = 1310075613 ÷ 7692399999900 ÷ 76923 = 170311300. So,

13.10076923 = 170311300 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 170311300 is also equal to 131311300 when expressed as a mixed number.

The repeating decimal 13.10076923 (vinculum notation) has a repeated block length of 6. It is also represented as 13.10076923076923076923... (ellipsis notation) which equals approximately 13.10076923076923076923076923076923 (decimal approximation)(*).

The recurring decimal 13.10076923 can be written as a ratio of two integers having 17031 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.