What is 15.134615346153461534615346153... as a Fraction?

15.1346153 equals 7567232499995 as a fraction.

How do you turn 15.1346153 repeating into a fraction?

Detailed Answer:

Step 1: To convert 15.1346153 repeating into a fraction, begin writing this simple equation:

n = 15.1346153 (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 = 151346153.46153 (equation 2)

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

10000000 × n = 151346153.46153
     100 × n = 15.1346153
 9999900 × n = 151344640

1513446409999900 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 = 1513446409999900 = 151344640 ÷ 209999900 ÷ 20 = 7567232499995. So,

15.1346153 = 7567232499995 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 7567232499995 is also equal to 1567307499995 when expressed as a mixed number.

The repeating decimal 15.1346153 (vinculum notation) has a repeated block length of 5. It is also represented as 15.13461534615346153... (ellipsis notation) which equals approximately 15.134615346153461534615346153 (decimal approximation)(*).

The recurring decimal 15.1346153 can be written as a ratio of two integers having 7567232 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.