What is 2.12181818181818... as a Fraction?

2.1218 equals 1167550 as a fraction.

How do you turn 2.1218 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.1218 repeating into a fraction, begin writing this simple equation:

n = 2.1218 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (18) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁴ = 10000.

10000 × n = 21218.18 (equation 2)

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

10000 × n = 21218.18
  100 × n = 2.1218
 9900 × n = 21006

210069900 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 18 (the GCF - greatest common factor).

n = 210069900 = 21006 ÷ 189900 ÷ 18 = 1167550. So,

2.1218 = 1167550 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 1167550 is also equal to 267550 when expressed as a mixed number.

The repeating decimal 2.1218 (vinculum notation) has a repeated block length of 2. It is also represented as 2.12181818... (ellipsis notation) which equals approximately 2.121818181818 (decimal approximation)(*).

The recurring decimal 2.1218 can be written as a ratio of two integers having 1167 as the numerator and 550 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.