What is 15.12142142142... as a Fraction?

15.12142 equals 1510639990 as a fraction.

How do you turn 15.12142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 15.12142 repeating into a fraction, begin writing this simple equation:

n = 15.12142 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (142) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁵ = 100000.

100000 × n = 1512142.142 (equation 2)

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

100000 × n = 1512142.142
   100 × n = 15.12142
 99900 × n = 1510630

151063099900 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 10 (the GCF - greatest common factor).

n = 151063099900 = 1510630 ÷ 1099900 ÷ 10 = 1510639990. So,

15.12142 = 1510639990 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 1510639990 is also equal to 1512139990 when expressed as a mixed number.

The repeating decimal 15.12142 (vinculum notation) has a repeated block length of 3. It is also represented as 15.12142142142... (ellipsis notation) which equals approximately 15.12142142142142142 (decimal approximation)(*).

The recurring decimal 15.12142 can be written as a ratio of two integers having 151063 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.