What is 20.141616... as a Fraction?

20.1416 equals 997014950 as a fraction.

How do you turn 20.1416 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.1416 repeating into a fraction, begin writing this simple equation:

n = 20.1416 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (16) and 2 digits in the non-repeating part (14). Multiply both sides by 10⁴ = 10000.

10000 × n = 201416.16 (equation 2)

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

10000 × n = 201416.16
  100 × n = 20.1416
 9900 × n = 199402

1994029900 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 = 1994029900 = 199402 ÷ 29900 ÷ 2 = 997014950. So,

20.1416 = 997014950 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 997014950 is also equal to 207014950 when expressed as a mixed number.

The repeating decimal 20.1416 (vinculum notation) has a repeated block length of 2. It is also represented as 20.14161616... (ellipsis notation) which equals approximately 20.141616161616 (decimal approximation)(*).

The recurring decimal 20.1416 can be written as a ratio of two integers having 99701 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.