What is 10.7076923076923076923076923076923076923... as a Fraction?

10.7076923 equals 69665 as a fraction.

How do you turn 10.7076923 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.7076923 repeating into a fraction, begin writing this simple equation:

n = 10.7076923 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (076923) and 1 digit in the non-repeating part (7). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 107076923.076923 (equation 2)

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

10000000 × n = 107076923.076923
      10 × n = 10.7076923
 9999990 × n = 107076816

1070768169999990 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 153846 (the GCF - greatest common factor).

n = 1070768169999990 = 107076816 ÷ 1538469999990 ÷ 153846 = 69665. So,

10.7076923 = 69665 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 69665 is also equal to 104665 when expressed as a mixed number.

The repeating decimal 10.7076923 (vinculum notation) has a repeated block length of 6. It is also represented as 10.7076923076923076923... (ellipsis notation) which equals approximately 10.7076923076923076923076923076923 (decimal approximation)(*).

The recurring decimal 10.7076923 can be written as a ratio of two integers having 696 as the numerator and 65 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.