What is 13.21676767... as a Fraction?

13.2167 equals 654234950 as a fraction.

How do you turn 13.2167 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.2167 repeating into a fraction, begin writing this simple equation:

n = 13.2167 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (67) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁴ = 10000.

10000 × n = 132167.67 (equation 2)

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

10000 × n = 132167.67
  100 × n = 13.2167
 9900 × n = 130846

1308469900 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 = 1308469900 = 130846 ÷ 29900 ÷ 2 = 654234950. So,

13.2167 = 654234950 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 654234950 is also equal to 1310734950 when expressed as a mixed number.

The repeating decimal 13.2167 (vinculum notation) has a repeated block length of 2. It is also represented as 13.21676767... (ellipsis notation) which equals approximately 13.216767676767 (decimal approximation)(*).

The recurring decimal 13.2167 can be written as a ratio of two integers having 65423 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.