What is 164.5777... as a Fraction?

164.57 equals 740645 as a fraction.

How do you turn 164.57 repeating into a fraction?

Detailed Answer:

Step 1: To convert 164.57 repeating into a fraction, begin writing this simple equation:

n = 164.57 (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 = 16457.7 (equation 2)

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

100 × n = 16457.7
 10 × n = 164.57
 90 × n = 14812

1481290 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 = 1481290 = 14812 ÷ 290 ÷ 2 = 740645. So,

164.57 = 740645 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 740645 is also equal to 1642645 when expressed as a mixed number.

The repeating decimal 164.57 (vinculum notation) has a repeated block length of 1. It is also represented as 164.5777... (ellipsis notation) which equals approximately 164.577777 (decimal approximation)(*).

The recurring decimal 164.57 can be written as a ratio of two integers having 7406 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.