What is 10.38571857185718571... as a Fraction?

10.38571 equals 17307816665 as a fraction.

How do you turn 10.38571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.38571 repeating into a fraction, begin writing this simple equation:

n = 10.38571 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (8571) and 1 digit in the non-repeating part (3). Multiply both sides by 10⁵ = 100000.

100000 × n = 1038571.8571 (equation 2)

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

100000 × n = 1038571.8571
    10 × n = 10.38571
 99990 × n = 1038468

103846899990 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 6 (the GCF - greatest common factor).

n = 103846899990 = 1038468 ÷ 699990 ÷ 6 = 17307816665. So,

10.38571 = 17307816665 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 17307816665 is also equal to 10642816665 when expressed as a mixed number.

The repeating decimal 10.38571 (vinculum notation) has a repeated block length of 4. It is also represented as 10.3857185718571... (ellipsis notation) which equals approximately 10.385718571857185718571 (decimal approximation)(*).

The recurring decimal 10.38571 can be written as a ratio of two integers having 173078 as the numerator and 16665 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.