What is 13.17142142142... as a Fraction?

13.17142 equals 526333996 as a fraction.

How do you turn 13.17142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.17142 repeating into a fraction, begin writing this simple equation:

n = 13.17142 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (142) and 2 digits in the non-repeating part (17). Multiply both sides by 10⁵ = 100000.

100000 × n = 1317142.142 (equation 2)

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

100000 × n = 1317142.142
   100 × n = 13.17142
 99900 × n = 1315825

131582599900 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 25 (the GCF - greatest common factor).

n = 131582599900 = 1315825 ÷ 2599900 ÷ 25 = 526333996. So,

13.17142 = 526333996 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 526333996 is also equal to 136853996 when expressed as a mixed number.

The repeating decimal 13.17142 (vinculum notation) has a repeated block length of 3. It is also represented as 13.17142142142... (ellipsis notation) which equals approximately 13.17142142142142142 (decimal approximation)(*).

The recurring decimal 13.17142 can be written as a ratio of two integers having 52633 as the numerator and 3996 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.