What is 166.5777... as a Fraction?

166.57 equals 749645 as a fraction.

How do you turn 166.57 repeating into a fraction?

Detailed Answer:

Step 1: To convert 166.57 repeating into a fraction, begin writing this simple equation:

n = 166.57 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (7) and 1 digit in the non-repeating part (5). Multiply both sides by 10² = 100.

100 × n = 16657.7 (equation 2)

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

100 × n = 16657.7
 10 × n = 166.57
 90 × n = 14992

1499290 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 = 1499290 = 14992 ÷ 290 ÷ 2 = 749645. So,

166.57 = 749645 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 749645 is also equal to 1662645 when expressed as a mixed number.

The repeating decimal 166.57 (vinculum notation) has a repeated block length of 1. It is also represented as 166.5777... (ellipsis notation) which equals approximately 166.577777 (decimal approximation)(*).

The recurring decimal 166.57 can be written as a ratio of two integers having 7496 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.