What is 21.5142142142... as a Fraction?

21.5142 equals 2149279990 as a fraction.

How do you turn 21.5142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.5142 repeating into a fraction, begin writing this simple equation:

n = 21.5142 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (142) and 1 digit in the non-repeating part (5). Multiply both sides by 10⁴ = 10000.

10000 × n = 215142.142 (equation 2)

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

10000 × n = 215142.142
   10 × n = 21.5142
 9990 × n = 214927

n = 2149279990 (answer)

21.5142 = 2149279990 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 2149279990 is also equal to 2151379990 when expressed as a mixed number.

The repeating decimal 21.5142 (vinculum notation) has a repeated block length of 3. It is also represented as 21.5142142142... (ellipsis notation) which equals approximately 21.5142142142142142 (decimal approximation)(*).

The recurring decimal 21.5142 can be written as a ratio of two integers having 214927 as the numerator and 9990 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.