What is 15.1412121212... as a Fraction?

15.1412 equals 249831650 as a fraction.

How do you turn 15.1412 repeating into a fraction?

Detailed Answer:

Step 1: To convert 15.1412 repeating into a fraction, begin writing this simple equation:

n = 15.1412 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (12) and 2 digits in the non-repeating part (14). Multiply both sides by 10⁴ = 10000.

10000 × n = 151412.12 (equation 2)

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

10000 × n = 151412.12
  100 × n = 15.1412
 9900 × n = 149898

1498989900 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 6 (the GCF - greatest common factor).

n = 1498989900 = 149898 ÷ 69900 ÷ 6 = 249831650. So,

15.1412 = 249831650 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 249831650 is also equal to 152331650 when expressed as a mixed number.

The repeating decimal 15.1412 (vinculum notation) has a repeated block length of 2. It is also represented as 15.14121212... (ellipsis notation) which equals approximately 15.141212121212 (decimal approximation)(*).

The recurring decimal 15.1412 can be written as a ratio of two integers having 24983 as the numerator and 1650 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.