What is 77.10666... as a Fraction?

77.106 equals 578375 as a fraction.

How do you turn 77.106 repeating into a fraction?

Detailed Answer:

Step 1: To convert 77.106 repeating into a fraction, begin writing this simple equation:

n = 77.106 (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 = 77106.6 (equation 2)

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

1000 × n = 77106.6
 100 × n = 77.106
 900 × n = 69396

69396900 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 = 69396900 = 69396 ÷ 12900 ÷ 12 = 578375. So,

77.106 = 578375 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 578375 is also equal to 77875 when expressed as a mixed number.

The repeating decimal 77.106 (vinculum notation) has a repeated block length of 1. It is also represented as 77.10666... (ellipsis notation) which equals approximately 77.1066666 (decimal approximation)(*).

The recurring decimal 77.106 can be written as a ratio of two integers having 5783 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.