What is 10.121414... as a Fraction?

10.1214 equals 501014950 as a fraction.

How do you turn 10.1214 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.1214 repeating into a fraction, begin writing this simple equation:

n = 10.1214 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (14) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁴ = 10000.

10000 × n = 101214.14 (equation 2)

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

10000 × n = 101214.14
  100 × n = 10.1214
 9900 × n = 100202

1002029900 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 2 (the GCF - greatest common factor).

n = 1002029900 = 100202 ÷ 29900 ÷ 2 = 501014950. So,

10.1214 = 501014950 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 501014950 is also equal to 106014950 when expressed as a mixed number.

The repeating decimal 10.1214 (vinculum notation) has a repeated block length of 2. It is also represented as 10.12141414... (ellipsis notation) which equals approximately 10.121414141414 (decimal approximation)(*).

The recurring decimal 10.1214 can be written as a ratio of two integers having 50101 as the numerator and 4950 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.