What is 1.6818181818181... as a Fraction?

1.681 equals 3722 as a fraction.

How do you turn 1.681 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.681 repeating into a fraction, begin writing this simple equation:

n = 1.681 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (81) and 1 digit in the non-repeating part (6). Multiply both sides by 10³ = 1000.

1000 × n = 1681.81 (equation 2)

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

1000 × n = 1681.81
  10 × n = 1.681
 990 × n = 1665

1665990 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 45 (the GCF - greatest common factor).

n = 1665990 = 1665 ÷ 45990 ÷ 45 = 3722. So,

1.681 = 3722 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 3722 is also equal to 11522 when expressed as a mixed number.

The repeating decimal 1.681 (vinculum notation) has a repeated block length of 2. It is also represented as 1.6818181... (ellipsis notation) which equals approximately 1.68181818181 (decimal approximation)(*).

The recurring decimal 1.681 can be written as a ratio of two integers having 37 as the numerator and 22 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.