What is 13.113888... as a Fraction?

13.1138 equals 4721360 as a fraction.

How do you turn 13.1138 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.1138 repeating into a fraction, begin writing this simple equation:

n = 13.1138 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (8) and 3 digits in the non-repeating part (113). Multiply both sides by 10⁴ = 10000.

10000 × n = 131138.8 (equation 2)

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

10000 × n = 131138.8
 1000 × n = 13.1138
 9000 × n = 118025

1180259000 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 = 1180259000 = 118025 ÷ 259000 ÷ 25 = 4721360. So,

13.1138 = 4721360 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 4721360 is also equal to 1341360 when expressed as a mixed number.

The repeating decimal 13.1138 (vinculum notation) has a repeated block length of 1. It is also represented as 13.113888... (ellipsis notation) which equals approximately 13.11388888 (decimal approximation)(*).

The recurring decimal 13.1138 can be written as a ratio of two integers having 4721 as the numerator and 360 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.