What is 165.19777... as a Fraction?

165.197 equals 74339450 as a fraction.

How do you turn 165.197 repeating into a fraction?

Detailed Answer:

Step 1: To convert 165.197 repeating into a fraction, begin writing this simple equation:

n = 165.197 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (7) and 2 digits in the non-repeating part (19). Multiply both sides by 10³ = 1000.

1000 × n = 165197.7 (equation 2)

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

1000 × n = 165197.7
 100 × n = 165.197
 900 × n = 148678

148678900 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 2 (the GCF - greatest common factor).

n = 148678900 = 148678 ÷ 2900 ÷ 2 = 74339450. So,

165.197 = 74339450 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 74339450 is also equal to 16589450 when expressed as a mixed number.

The repeating decimal 165.197 (vinculum notation) has a repeated block length of 1. It is also represented as 165.19777... (ellipsis notation) which equals approximately 165.1977777 (decimal approximation)(*).

The recurring decimal 165.197 can be written as a ratio of two integers having 74339 as the numerator and 450 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.