What is 65.3166... as a Fraction?

65.316 equals 391960 as a fraction.

How do you turn 65.316 repeating into a fraction?

Detailed Answer:

Step 1: To convert 65.316 repeating into a fraction, begin writing this simple equation:

n = 65.316 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 2 digits in the non-repeating part (31). Multiply both sides by 10³ = 1000.

1000 × n = 65316.6 (equation 2)

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

1000 × n = 65316.6
 100 × n = 65.316
 900 × n = 58785

58785900 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 15 (the GCF - greatest common factor).

n = 58785900 = 58785 ÷ 15900 ÷ 15 = 391960. So,

65.316 = 391960 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 391960 is also equal to 651960 when expressed as a mixed number.

The repeating decimal 65.316 (vinculum notation) has a repeated block length of 1. It is also represented as 65.31666... (ellipsis notation) which equals approximately 65.3166666 (decimal approximation)(*).

The recurring decimal 65.316 can be written as a ratio of two integers having 3919 as the numerator and 60 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.