What is 319.13333... as a Fraction?

319.133 equals 478715 as a fraction.

How do you turn 319.133 repeating into a fraction?

Detailed Answer:

Step 1: To convert 319.133 repeating into a fraction, begin writing this simple equation:

n = 319.133 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3) and 2 digits in the non-repeating part (13). Multiply both sides by 10³ = 1000.

1000 × n = 319133.3 (equation 2)

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

1000 × n = 319133.3
 100 × n = 319.133
 900 × n = 287220

287220900 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 60 (the GCF - greatest common factor).

n = 287220900 = 287220 ÷ 60900 ÷ 60 = 478715. So,

319.133 = 478715 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 478715 is also equal to 319215 when expressed as a mixed number.

The repeating decimal 319.133 (vinculum notation) has a repeated block length of 1. It is also represented as 319.13333... (ellipsis notation) which equals approximately 319.1333333 (decimal approximation)(*).

The recurring decimal 319.133 can be written as a ratio of two integers having 4787 as the numerator and 15 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.