What is 10.16428571428571428571428571428571428571... as a Fraction?

10.16428571 equals 1423140 as a fraction.

How do you turn 10.16428571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.16428571 repeating into a fraction, begin writing this simple equation:

n = 10.16428571 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (428571) and 2 digits in the non-repeating part (16). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 1016428571.428571 (equation 2)

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

100000000 × n = 1016428571.428571
      100 × n = 10.16428571
 99999900 × n = 1016427555

101642755599999900 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 714285 (the GCF - greatest common factor).

n = 101642755599999900 = 1016427555 ÷ 71428599999900 ÷ 714285 = 1423140. So,

10.16428571 = 1423140 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 1423140 is also equal to 1023140 when expressed as a mixed number.

The repeating decimal 10.16428571 (vinculum notation) has a repeated block length of 6. It is also represented as 10.16428571428571428571... (ellipsis notation) which equals approximately 10.16428571428571428571428571428571 (decimal approximation)(*).

The recurring decimal 10.16428571 can be written as a ratio of two integers having 1423 as the numerator and 140 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.