What is 21.4384384384... as a Fraction?

21.4384 equals 7139333 as a fraction.

How do you turn 21.4384 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.4384 repeating into a fraction, begin writing this simple equation:

n = 21.4384 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (384) and 1 digit in the non-repeating part (4). Multiply both sides by 10⁴ = 10000.

10000 × n = 214384.384 (equation 2)

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

10000 × n = 214384.384
   10 × n = 21.4384
 9990 × n = 214170

2141709990 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 30 (the GCF - greatest common factor).

n = 2141709990 = 214170 ÷ 309990 ÷ 30 = 7139333. So,

21.4384 = 7139333 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 7139333 is also equal to 21146333 when expressed as a mixed number.

The repeating decimal 21.4384 (vinculum notation) has a repeated block length of 3. It is also represented as 21.4384384384... (ellipsis notation) which equals approximately 21.4384384384384384 (decimal approximation)(*).

The recurring decimal 21.4384 can be written as a ratio of two integers having 7139 as the numerator and 333 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.