What is 177.10777... as a Fraction?

177.107 equals 159397900 as a fraction.

How do you turn 177.107 repeating into a fraction?

Detailed Answer:

Step 1: To convert 177.107 repeating into a fraction, begin writing this simple equation:

n = 177.107 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (7) and 2 digits in the non-repeating part (10). Multiply both sides by 10³ = 1000.

1000 × n = 177107.7 (equation 2)

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

1000 × n = 177107.7
 100 × n = 177.107
 900 × n = 159397

n = 159397900 (answer)

177.107 = 159397900 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 159397900 is also equal to 17797900 when expressed as a mixed number.

The repeating decimal 177.107 (vinculum notation) has a repeated block length of 1. It is also represented as 177.10777... (ellipsis notation) which equals approximately 177.1077777 (decimal approximation)(*).

The recurring decimal 177.107 can be written as a ratio of two integers having 159397 as the numerator and 900 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.