What is 57.106... as a Fraction?

57.106 equals 428375 as a fraction.

How do you turn 57.106 repeating into a fraction?

Detailed Answer:

Step 1: To convert 57.106 repeating into a fraction, begin writing this simple equation:

n = 57.106 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 2 digits in the non-repeating part (10). Multiply both sides by 10³ = 1000.

1000 × n = 57106.6 (equation 2)

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

1000 × n = 57106.6
 100 × n = 57.106
 900 × n = 51396

51396900 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 12 (the GCF - greatest common factor).

n = 51396900 = 51396 ÷ 12900 ÷ 12 = 428375. So,

57.106 = 428375 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 428375 is also equal to 57875 when expressed as a mixed number.

The repeating decimal 57.106 (vinculum notation) has a repeated block length of 1. It is also represented as 57.10666... (ellipsis notation) which equals approximately 57.1066666 (decimal approximation)(*).

The recurring decimal 57.106 can be written as a ratio of two integers having 4283 as the numerator and 75 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.