What is 82.12333... as a Fraction?

82.123 equals 24637300 as a fraction.

How do you turn 82.123 repeating into a fraction?

Detailed Answer:

Step 1: To convert 82.123 repeating into a fraction, begin writing this simple equation:

n = 82.123 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3) and 2 digits in the non-repeating part (12). Multiply both sides by 10³ = 1000.

1000 × n = 82123.3 (equation 2)

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

1000 × n = 82123.3
 100 × n = 82.123
 900 × n = 73911

73911900 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 3 (the GCF - greatest common factor).

n = 73911900 = 73911 ÷ 3900 ÷ 3 = 24637300. So,

82.123 = 24637300 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 24637300 is also equal to 8237300 when expressed as a mixed number.

The repeating decimal 82.123 (vinculum notation) has a repeated block length of 1. It is also represented as 82.12333... (ellipsis notation) which equals approximately 82.1233333 (decimal approximation)(*).

The recurring decimal 82.123 can be written as a ratio of two integers having 24637 as the numerator and 300 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.