What is 12.13153153153... as a Fraction?

12.13153 equals 6733555 as a fraction.

How do you turn 12.13153 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.13153 repeating into a fraction, begin writing this simple equation:

n = 12.13153 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (153) and 2 digits in the non-repeating part (13). Multiply both sides by 10⁵ = 100000.

100000 × n = 1213153.153 (equation 2)

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

100000 × n = 1213153.153
   100 × n = 12.13153
 99900 × n = 1211940

121194099900 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 180 (the GCF - greatest common factor).

n = 121194099900 = 1211940 ÷ 18099900 ÷ 180 = 6733555. So,

12.13153 = 6733555 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 6733555 is also equal to 1273555 when expressed as a mixed number.

The repeating decimal 12.13153 (vinculum notation) has a repeated block length of 3. It is also represented as 12.13153153153... (ellipsis notation) which equals approximately 12.13153153153153153 (decimal approximation)(*).

The recurring decimal 12.13153 can be written as a ratio of two integers having 6733 as the numerator and 555 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.