What is 67.1966... as a Fraction?

67.196 equals 20159300 as a fraction.

How do you turn 67.196 repeating into a fraction?

Detailed Answer:

Step 1: To convert 67.196 repeating into a fraction, begin writing this simple equation:

n = 67.196 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 2 digits in the non-repeating part (19). Multiply both sides by 10³ = 1000.

1000 × n = 67196.6 (equation 2)

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

1000 × n = 67196.6
 100 × n = 67.196
 900 × n = 60477

60477900 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 3 (the GCF - greatest common factor).

n = 60477900 = 60477 ÷ 3900 ÷ 3 = 20159300. So,

67.196 = 20159300 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 20159300 is also equal to 6759300 when expressed as a mixed number.

The repeating decimal 67.196 (vinculum notation) has a repeated block length of 1. It is also represented as 67.19666... (ellipsis notation) which equals approximately 67.1966666 (decimal approximation)(*).

The recurring decimal 67.196 can be written as a ratio of two integers having 20159 as the numerator and 300 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.