What is 10.138571857185718571... as a Fraction?

10.138571 equals 1689593166650 as a fraction.

How do you turn 10.138571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.138571 repeating into a fraction, begin writing this simple equation:

n = 10.138571 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (8571) and 2 digits in the non-repeating part (13). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 10138571.8571 (equation 2)

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

1000000 × n = 10138571.8571
    100 × n = 10.138571
 999900 × n = 10137558

10137558999900 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 = 10137558999900 = 10137558 ÷ 6999900 ÷ 6 = 1689593166650. So,

10.138571 = 1689593166650 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 1689593166650 is also equal to 1023093166650 when expressed as a mixed number.

The repeating decimal 10.138571 (vinculum notation) has a repeated block length of 4. It is also represented as 10.13857185718571... (ellipsis notation) which equals approximately 10.1385718571857185718571 (decimal approximation)(*).

The recurring decimal 10.138571 can be written as a ratio of two integers having 1689593 as the numerator and 166650 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.