What is 16.2101010101... as a Fraction?

16.2101 equals 8024495 as a fraction.

How do you turn 16.2101 repeating into a fraction?

Detailed Answer:

Step 1: To convert 16.2101 repeating into a fraction, begin writing this simple equation:

n = 16.2101 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (01) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁴ = 10000.

10000 × n = 162101.01 (equation 2)

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

10000 × n = 162101.01
  100 × n = 16.2101
 9900 × n = 160480

1604809900 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 20 (the GCF - greatest common factor).

n = 1604809900 = 160480 ÷ 209900 ÷ 20 = 8024495. So,

16.2101 = 8024495 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 8024495 is also equal to 16104495 when expressed as a mixed number.

The repeating decimal 16.2101 (vinculum notation) has a repeated block length of 2. It is also represented as 16.21010101... (ellipsis notation) which equals approximately 16.210101010101 (decimal approximation)(*).

The recurring decimal 16.2101 can be written as a ratio of two integers having 8024 as the numerator and 495 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.