What is 33.3333... as a Fraction?

33.333 equals 1003 as a fraction.

How do you turn 33.333 repeating into a fraction?

Detailed Answer:

Step 1: To convert 33.333 repeating into a fraction, begin writing this simple equation:

n = 33.333 (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 = 33333.3 (equation 2)

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

1000 × n = 33333.3
 100 × n = 33.333
 900 × n = 30000

30000900 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 = 30000900 = 30000 ÷ 300900 ÷ 300 = 1003. So,

33.333 = 1003 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 1003 is also equal to 3313 when expressed as a mixed number.

The repeating decimal 33.333 (vinculum notation) has a repeated block length of 1. It is also represented as 33.33333... (ellipsis notation) which equals approximately 33.3333333 (decimal approximation)(*).

The recurring decimal 33.333 can be written as a ratio of two integers having 100 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.