What is 5.71212... as a Fraction?

5.712 equals 37766 as a fraction.

How do you turn 5.712 repeating into a fraction?

Detailed Answer:

Step 1: To convert 5.712 repeating into a fraction, begin writing this simple equation:

n = 5.712 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (12) and 1 digit in the non-repeating part (7). Multiply both sides by 10³ = 1000.

1000 × n = 5712.12 (equation 2)

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

1000 × n = 5712.12
  10 × n = 5.712
 990 × n = 5655

5655990 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 15 (the GCF - greatest common factor).

n = 5655990 = 5655 ÷ 15990 ÷ 15 = 37766. So,

5.712 = 37766 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 37766 is also equal to 54766 when expressed as a mixed number.

The repeating decimal 5.712 (vinculum notation) has a repeated block length of 2. It is also represented as 5.7121212... (ellipsis notation) which equals approximately 5.71212121212 (decimal approximation)(*).

The recurring decimal 5.712 can be written as a ratio of two integers having 377 as the numerator and 66 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.