What is 51.5666... as a Fraction?

51.56 equals 154730 as a fraction.

How do you turn 51.56 repeating into a fraction?

Detailed Answer:

Step 1: To convert 51.56 repeating into a fraction, begin writing this simple equation:

n = 51.56 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 1 digit in the non-repeating part (5). Multiply both sides by 10² = 100.

100 × n = 5156.6 (equation 2)

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

100 × n = 5156.6
 10 × n = 51.56
 90 × n = 4641

464190 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 3 (the GCF - greatest common factor).

n = 464190 = 4641 ÷ 390 ÷ 3 = 154730. So,

51.56 = 154730 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 154730 is also equal to 511730 when expressed as a mixed number.

The repeating decimal 51.56 (vinculum notation) has a repeated block length of 1. It is also represented as 51.5666... (ellipsis notation) which equals approximately 51.566666 (decimal approximation)(*).

The recurring decimal 51.56 can be written as a ratio of two integers having 1547 as the numerator and 30 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.