What is 13.590909090... as a Fraction?

13.590 equals 29922 as a fraction.

How do you turn 13.590 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.590 repeating into a fraction, begin writing this simple equation:

n = 13.590 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (90) and 1 digit in the non-repeating part (5). Multiply both sides by 10³ = 1000.

1000 × n = 13590.90 (equation 2)

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

1000 × n = 13590.90
  10 × n = 13.590
 990 × n = 13455

13455990 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 45 (the GCF - greatest common factor).

n = 13455990 = 13455 ÷ 45990 ÷ 45 = 29922. So,

13.590 = 29922 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 29922 is also equal to 131322 when expressed as a mixed number.

The repeating decimal 13.590 (vinculum notation) has a repeated block length of 2. It is also represented as 13.5909090... (ellipsis notation) which equals approximately 13.59090909090 (decimal approximation)(*).

The recurring decimal 13.590 can be written as a ratio of two integers having 299 as the numerator and 22 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.