What is 13.221616161616... as a Fraction?

13.2216 equals 654474950 as a fraction.

How do you turn 13.2216 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.2216 repeating into a fraction, begin writing this simple equation:

n = 13.2216 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (16) and 2 digits in the non-repeating part (22). Multiply both sides by 10⁴ = 10000.

10000 × n = 132216.16 (equation 2)

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

10000 × n = 132216.16
  100 × n = 13.2216
 9900 × n = 130894

1308949900 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 2 (the GCF - greatest common factor).

n = 1308949900 = 130894 ÷ 29900 ÷ 2 = 654474950. So,

13.2216 = 654474950 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 654474950 is also equal to 1310974950 when expressed as a mixed number.

The repeating decimal 13.2216 (vinculum notation) has a repeated block length of 2. It is also represented as 13.22161616... (ellipsis notation) which equals approximately 13.221616161616 (decimal approximation)(*).

The recurring decimal 13.2216 can be written as a ratio of two integers having 65447 as the numerator and 4950 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.