What is 83.21888... as a Fraction?

83.218 equals 74897900 as a fraction.

How do you turn 83.218 repeating into a fraction?

Detailed Answer:

Step 1: To convert 83.218 repeating into a fraction, begin writing this simple equation:

n = 83.218 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (8) and 2 digits in the non-repeating part (21). Multiply both sides by 10³ = 1000.

1000 × n = 83218.8 (equation 2)

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

1000 × n = 83218.8
 100 × n = 83.218
 900 × n = 74897

n = 74897900 (answer)

83.218 = 74897900 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 74897900 is also equal to 83197900 when expressed as a mixed number.

The repeating decimal 83.218 (vinculum notation) has a repeated block length of 1. It is also represented as 83.21888... (ellipsis notation) which equals approximately 83.2188888 (decimal approximation)(*).

The recurring decimal 83.218 can be written as a ratio of two integers having 74897 as the numerator and 900 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.