What is 13.134615346153461534615346153... as a Fraction?

13.1346153 equals 6567242499995 as a fraction.

How do you turn 13.1346153 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.1346153 repeating into a fraction, begin writing this simple equation:

n = 13.1346153 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (46153) and 2 digits in the non-repeating part (13). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 131346153.46153 (equation 2)

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

10000000 × n = 131346153.46153
     100 × n = 13.1346153
 9999900 × n = 131344840

1313448409999900 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 20 (the GCF - greatest common factor).

n = 1313448409999900 = 131344840 ÷ 209999900 ÷ 20 = 6567242499995. So,

13.1346153 = 6567242499995 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 6567242499995 is also equal to 1367307499995 when expressed as a mixed number.

The repeating decimal 13.1346153 (vinculum notation) has a repeated block length of 5. It is also represented as 13.13461534615346153... (ellipsis notation) which equals approximately 13.134615346153461534615346153 (decimal approximation)(*).

The recurring decimal 13.1346153 can be written as a ratio of two integers having 6567242 as the numerator and 499995 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.