What is 165.17777... as a Fraction?

165.177 equals 743345 as a fraction.

How do you turn 165.177 repeating into a fraction?

Detailed Answer:

Step 1: To convert 165.177 repeating into a fraction, begin writing this simple equation:

n = 165.177 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (7) and 2 digits in the non-repeating part (17). Multiply both sides by 10³ = 1000.

1000 × n = 165177.7 (equation 2)

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

1000 × n = 165177.7
 100 × n = 165.177
 900 × n = 148660

148660900 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 20 (the GCF - greatest common factor).

n = 148660900 = 148660 ÷ 20900 ÷ 20 = 743345. So,

165.177 = 743345 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 743345 is also equal to 165845 when expressed as a mixed number.

The repeating decimal 165.177 (vinculum notation) has a repeated block length of 1. It is also represented as 165.17777... (ellipsis notation) which equals approximately 165.1777777 (decimal approximation)(*).

The recurring decimal 165.177 can be written as a ratio of two integers having 7433 as the numerator and 45 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.