What is 82.222... as a Fraction?

82.2 equals 7409 as a fraction.

How do you turn 82.2 repeating into a fraction?

Detailed Answer:

Step 1: To convert 82.2 repeating into a fraction, begin writing this simple equation:

n = 82.2 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (2), so multiply both sides by 10¹ = 10.

10 × n = 822.2 (equation 2)

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

10 × n = 822.2
 1 × n = 82.2
 9 × n = 740

n = 7409 (answer)

82.2 = 7409 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 7409 is also equal to 8229 when expressed as a mixed number.

The repeating decimal 82.2 (vinculum notation) has a repeated block length of 1. It is also represented as 82.222... (ellipsis notation) which equals approximately 82.22222 (decimal approximation)(*).

The recurring decimal 82.2 can be written as a ratio of two integers having 740 as the numerator and 9 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.