What is 21.67142714271427142... as a Fraction?

21.67142 equals 108346349995 as a fraction.

How do you turn 21.67142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.67142 repeating into a fraction, begin writing this simple equation:

n = 21.67142 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (7142) and 1 digit in the non-repeating part (6). Multiply both sides by 10⁵ = 100000.

100000 × n = 2167142.7142 (equation 2)

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

100000 × n = 2167142.7142
    10 × n = 21.67142
 99990 × n = 2166926

216692699990 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 2 (the GCF - greatest common factor).

n = 216692699990 = 2166926 ÷ 299990 ÷ 2 = 108346349995. So,

21.67142 = 108346349995 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 108346349995 is also equal to 213356849995 when expressed as a mixed number.

The repeating decimal 21.67142 (vinculum notation) has a repeated block length of 4. It is also represented as 21.6714271427142... (ellipsis notation) which equals approximately 21.671427142714271427142 (decimal approximation)(*).

The recurring decimal 21.67142 can be written as a ratio of two integers having 1083463 as the numerator and 49995 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.