What is 13.111333... as a Fraction?

13.1113 equals 196671500 as a fraction.

How do you turn 13.1113 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.1113 repeating into a fraction, begin writing this simple equation:

n = 13.1113 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3) and 3 digits in the non-repeating part (111). Multiply both sides by 10⁴ = 10000.

10000 × n = 131113.3 (equation 2)

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

10000 × n = 131113.3
 1000 × n = 13.1113
 9000 × n = 118002

1180029000 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 6 (the GCF - greatest common factor).

n = 1180029000 = 118002 ÷ 69000 ÷ 6 = 196671500. So,

13.1113 = 196671500 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 196671500 is also equal to 131671500 when expressed as a mixed number.

The repeating decimal 13.1113 (vinculum notation) has a repeated block length of 1. It is also represented as 13.111333... (ellipsis notation) which equals approximately 13.11133333 (decimal approximation)(*).

The recurring decimal 13.1113 can be written as a ratio of two integers having 19667 as the numerator and 1500 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.