What is 16.230606... as a Fraction?

16.2306 equals 535613300 as a fraction.

How do you turn 16.2306 repeating into a fraction?

Detailed Answer:

Step 1: To convert 16.2306 repeating into a fraction, begin writing this simple equation:

n = 16.2306 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (06) and 2 digits in the non-repeating part (23). Multiply both sides by 10⁴ = 10000.

10000 × n = 162306.06 (equation 2)

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

10000 × n = 162306.06
  100 × n = 16.2306
 9900 × n = 160683

1606839900 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 = 1606839900 = 160683 ÷ 39900 ÷ 3 = 535613300. So,

16.2306 = 535613300 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 535613300 is also equal to 167613300 when expressed as a mixed number.

The repeating decimal 16.2306 (vinculum notation) has a repeated block length of 2. It is also represented as 16.23060606... (ellipsis notation) which equals approximately 16.230606060606 (decimal approximation)(*).

The recurring decimal 16.2306 can be written as a ratio of two integers having 53561 as the numerator and 3300 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.