What is 2.121616... as a Fraction?

2.1216 equals 52512475 as a fraction.

How do you turn 2.1216 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.1216 repeating into a fraction, begin writing this simple equation:

n = 2.1216 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (16) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁴ = 10000.

10000 × n = 21216.16 (equation 2)

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

10000 × n = 21216.16
  100 × n = 2.1216
 9900 × n = 21004

210049900 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 4 (the GCF - greatest common factor).

n = 210049900 = 21004 ÷ 49900 ÷ 4 = 52512475. So,

2.1216 = 52512475 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 52512475 is also equal to 23012475 when expressed as a mixed number.

The repeating decimal 2.1216 (vinculum notation) has a repeated block length of 2. It is also represented as 2.12161616... (ellipsis notation) which equals approximately 2.121616161616 (decimal approximation)(*).

The recurring decimal 2.1216 can be written as a ratio of two integers having 5251 as the numerator and 2475 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.