What is 165.2777... as a Fraction?

165.27 equals 297518 as a fraction.

How do you turn 165.27 repeating into a fraction?

Detailed Answer:

Step 1: To convert 165.27 repeating into a fraction, begin writing this simple equation:

n = 165.27 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (7) and 1 digit in the non-repeating part (2). Multiply both sides by 10² = 100.

100 × n = 16527.7 (equation 2)

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

100 × n = 16527.7
 10 × n = 165.27
 90 × n = 14875

1487590 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 = 1487590 = 14875 ÷ 590 ÷ 5 = 297518. So,

165.27 = 297518 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 297518 is also equal to 165518 when expressed as a mixed number.

The repeating decimal 165.27 (vinculum notation) has a repeated block length of 1. It is also represented as 165.2777... (ellipsis notation) which equals approximately 165.277777 (decimal approximation)(*).

The recurring decimal 165.27 can be written as a ratio of two integers having 2975 as the numerator and 18 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.