What is 1.22857285728572857... as a Fraction?

1.22857 equals 2456919998 as a fraction.

How do you turn 1.22857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.22857 repeating into a fraction, begin writing this simple equation:

n = 1.22857 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (2857) and 1 digit in the non-repeating part (2). Multiply both sides by 10⁵ = 100000.

100000 × n = 122857.2857 (equation 2)

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

100000 × n = 122857.2857
    10 × n = 1.22857
 99990 × n = 122845

12284599990 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 5 (the GCF - greatest common factor).

n = 12284599990 = 122845 ÷ 599990 ÷ 5 = 2456919998. So,

1.22857 = 2456919998 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 2456919998 is also equal to 1457119998 when expressed as a mixed number.

The repeating decimal 1.22857 (vinculum notation) has a repeated block length of 4. It is also represented as 1.2285728572857... (ellipsis notation) which equals approximately 1.228572857285728572857 (decimal approximation)(*).

The recurring decimal 1.22857 can be written as a ratio of two integers having 24569 as the numerator and 19998 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.