What is 19.71212... as a Fraction?

19.712 equals 130166 as a fraction.

How do you turn 19.712 repeating into a fraction?

Detailed Answer:

Step 1: To convert 19.712 repeating into a fraction, begin writing this simple equation:

n = 19.712 (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 = 19712.12 (equation 2)

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

1000 × n = 19712.12
  10 × n = 19.712
 990 × n = 19515

19515990 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 = 19515990 = 19515 ÷ 15990 ÷ 15 = 130166. So,

19.712 = 130166 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 130166 is also equal to 194766 when expressed as a mixed number.

The repeating decimal 19.712 (vinculum notation) has a repeated block length of 2. It is also represented as 19.7121212... (ellipsis notation) which equals approximately 19.71212121212 (decimal approximation)(*).

The recurring decimal 19.712 can be written as a ratio of two integers having 1301 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.