What is 83.33333... as a Fraction?

83.333 equals 2503 as a fraction.

How do you turn 83.333 repeating into a fraction?

Detailed Answer:

Step 1: To convert 83.333 repeating into a fraction, begin writing this simple equation:

n = 83.333 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3) and 2 digits in the non-repeating part (33). Multiply both sides by 10³ = 1000.

1000 × n = 83333.3 (equation 2)

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

1000 × n = 83333.3
 100 × n = 83.333
 900 × n = 75000

75000900 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 300 (the GCF - greatest common factor).

n = 75000900 = 75000 ÷ 300900 ÷ 300 = 2503. So,

83.333 = 2503 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 2503 is also equal to 8313 when expressed as a mixed number.

The repeating decimal 83.333 (vinculum notation) has a repeated block length of 1. It is also represented as 83.33333... (ellipsis notation) which equals approximately 83.3333333 (decimal approximation)(*).

The recurring decimal 83.333 can be written as a ratio of two integers having 250 as the numerator and 3 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.