What is 8.216161616... as a Fraction?

8.216 equals 4067495 as a fraction.

How do you turn 8.216 repeating into a fraction?

Detailed Answer:

Step 1: To convert 8.216 repeating into a fraction, begin writing this simple equation:

n = 8.216 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (16) and 1 digit in the non-repeating part (2). Multiply both sides by 10³ = 1000.

1000 × n = 8216.16 (equation 2)

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

1000 × n = 8216.16
  10 × n = 8.216
 990 × n = 8134

8134990 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 = 8134990 = 8134 ÷ 2990 ÷ 2 = 4067495. So,

8.216 = 4067495 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 4067495 is also equal to 8107495 when expressed as a mixed number.

The repeating decimal 8.216 (vinculum notation) has a repeated block length of 2. It is also represented as 8.2161616... (ellipsis notation) which equals approximately 8.21616161616 (decimal approximation)(*).

The recurring decimal 8.216 can be written as a ratio of two integers having 4067 as the numerator and 495 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.