What is 13.1999... as a Fraction?

13.199 equals 665 as a fraction.

How do you turn 13.199 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.199 repeating into a fraction, begin writing this simple equation:

n = 13.199 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (9) and 2 digits in the non-repeating part (19). Multiply both sides by 10³ = 1000.

1000 × n = 13199.9 (equation 2)

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

1000 × n = 13199.9
 100 × n = 13.199
 900 × n = 11880

11880900 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 180 (the GCF - greatest common factor).

n = 11880900 = 11880 ÷ 180900 ÷ 180 = 665. So,

13.199 = 665 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 665 is also equal to 1315 when expressed as a mixed number.

The repeating decimal 13.199 (vinculum notation) has a repeated block length of 1. It is also represented as 13.19999... (ellipsis notation) which equals approximately 13.1999999 (decimal approximation)(*).

The recurring decimal 13.199 can be written as a ratio of two integers having 66 as the numerator and 5 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.