What is 51.10555... as a Fraction?

51.105 equals 9199180 as a fraction.

How do you turn 51.105 repeating into a fraction?

Detailed Answer:

Step 1: To convert 51.105 repeating into a fraction, begin writing this simple equation:

n = 51.105 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (5) and 2 digits in the non-repeating part (10). Multiply both sides by 10³ = 1000.

1000 × n = 51105.5 (equation 2)

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

1000 × n = 51105.5
 100 × n = 51.105
 900 × n = 45995

45995900 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 5 (the GCF - greatest common factor).

n = 45995900 = 45995 ÷ 5900 ÷ 5 = 9199180. So,

51.105 = 9199180 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 9199180 is also equal to 5119180 when expressed as a mixed number.

The repeating decimal 51.105 (vinculum notation) has a repeated block length of 1. It is also represented as 51.10555... (ellipsis notation) which equals approximately 51.1055555 (decimal approximation)(*).

The recurring decimal 51.105 can be written as a ratio of two integers having 9199 as the numerator and 180 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.