What is 165.12777... as a Fraction?

165.127 equals 29723180 as a fraction.

How do you turn 165.127 repeating into a fraction?

Detailed Answer:

Step 1: To convert 165.127 repeating into a fraction, begin writing this simple equation:

n = 165.127 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (7) and 2 digits in the non-repeating part (12). Multiply both sides by 10³ = 1000.

1000 × n = 165127.7 (equation 2)

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

1000 × n = 165127.7
 100 × n = 165.127
 900 × n = 148615

148615900 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 = 148615900 = 148615 ÷ 5900 ÷ 5 = 29723180. So,

165.127 = 29723180 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 29723180 is also equal to 16523180 when expressed as a mixed number.

The repeating decimal 165.127 (vinculum notation) has a repeated block length of 1. It is also represented as 165.12777... (ellipsis notation) which equals approximately 165.1277777 (decimal approximation)(*).

The recurring decimal 165.127 can be written as a ratio of two integers having 29723 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.