What is 32.333333... as a Fraction?

32.333 equals 973 as a fraction.

How do you turn 32.333 repeating into a fraction?

Detailed Answer:

Step 1: To convert 32.333 repeating into a fraction, begin writing this simple equation:

n = 32.333 (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 = 32333.3 (equation 2)

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

1000 × n = 32333.3
 100 × n = 32.333
 900 × n = 29100

29100900 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 = 29100900 = 29100 ÷ 300900 ÷ 300 = 973. So,

32.333 = 973 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 973 is also equal to 3213 when expressed as a mixed number.

The repeating decimal 32.333 (vinculum notation) has a repeated block length of 1. It is also represented as 32.33333... (ellipsis notation) which equals approximately 32.3333333 (decimal approximation)(*).

The recurring decimal 32.333 can be written as a ratio of two integers having 97 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.