What is 0.134615346153461534615346153... as a Fraction?

0.1346153 equals 67307499995 as a fraction.

How do you turn 0.1346153 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.1346153 repeating into a fraction, begin writing this simple equation:

n = 0.1346153 (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 = 1346153.46153 (equation 2)

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

10000000 × n = 1346153.46153
     100 × n = 0.1346153
 9999900 × n = 1346140

13461409999900 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 = 13461409999900 = 1346140 ÷ 209999900 ÷ 20 = 67307499995. So,

0.1346153 = 67307499995 as the lowest possible fraction.

The repeating decimal 0.1346153 (vinculum notation) has a repeated block length of 5. It is also represented as 0.13461534615346153... (ellipsis notation) which equals approximately 0.134615346153461534615346153 (decimal approximation)(*).

The recurring decimal 0.1346153 can be written as a ratio of two integers having 67307 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.