What is 333.3333... as a Fraction?

333.333 equals 10003 as a fraction.

How do you turn 333.333 repeating into a fraction?

Detailed Answer:

Step 1: To convert 333.333 repeating into a fraction, begin writing this simple equation:

n = 333.333 (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 = 333333.3 (equation 2)

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

1000 × n = 333333.3
 100 × n = 333.333
 900 × n = 300000

300000900 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 = 300000900 = 300000 ÷ 300900 ÷ 300 = 10003. So,

333.333 = 10003 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 10003 is also equal to 33313 when expressed as a mixed number.

The repeating decimal 333.333 (vinculum notation) has a repeated block length of 1. It is also represented as 333.33333... (ellipsis notation) which equals approximately 333.3333333 (decimal approximation)(*).

The recurring decimal 333.333 can be written as a ratio of two integers having 1000 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.