What is 3.74285742857428574285742857... as a Fraction?

3.742857 equals 37428299999 as a fraction.

How do you turn 3.742857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.742857 repeating into a fraction, begin writing this simple equation:

n = 3.742857 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (42857) and 1 digit in the non-repeating part (7). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 3742857.42857 (equation 2)

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

1000000 × n = 3742857.42857
     10 × n = 3.742857
 999990 × n = 3742820

3742820999990 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 10 (the GCF - greatest common factor).

n = 3742820999990 = 3742820 ÷ 10999990 ÷ 10 = 37428299999. So,

3.742857 = 37428299999 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 37428299999 is also equal to 37428599999 when expressed as a mixed number.

The repeating decimal 3.742857 (vinculum notation) has a repeated block length of 5. It is also represented as 3.7428574285742857... (ellipsis notation) which equals approximately 3.74285742857428574285742857 (decimal approximation)(*).

The recurring decimal 3.742857 can be written as a ratio of two integers having 374282 as the numerator and 99999 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.