What is 3.7428571428571428571428571428571428571... as a Fraction?

3.7428571 equals 13135 as a fraction.

How do you turn 3.7428571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.7428571 repeating into a fraction, begin writing this simple equation:

n = 3.7428571 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (428571) and 1 digit in the non-repeating part (7). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 37428571.428571 (equation 2)

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

10000000 × n = 37428571.428571
      10 × n = 3.7428571
 9999990 × n = 37428534

374285349999990 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 285714 (the GCF - greatest common factor).

n = 374285349999990 = 37428534 ÷ 2857149999990 ÷ 285714 = 13135. So,

3.7428571 = 13135 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 13135 is also equal to 32635 when expressed as a mixed number.

The repeating decimal 3.7428571 (vinculum notation) has a repeated block length of 6. It is also represented as 3.7428571428571428571... (ellipsis notation) which equals approximately 3.7428571428571428571428571428571 (decimal approximation)(*).

The recurring decimal 3.7428571 can be written as a ratio of two integers having 131 as the numerator and 35 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.