What is 22.23555... as a Fraction?

22.235 equals 5003225 as a fraction.

How do you turn 22.235 repeating into a fraction?

Detailed Answer:

Step 1: To convert 22.235 repeating into a fraction, begin writing this simple equation:

n = 22.235 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (5) and 2 digits in the non-repeating part (23). Multiply both sides by 10³ = 1000.

1000 × n = 22235.5 (equation 2)

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

1000 × n = 22235.5
 100 × n = 22.235
 900 × n = 20012

20012900 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 = 20012900 = 20012 ÷ 4900 ÷ 4 = 5003225. So,

22.235 = 5003225 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 5003225 is also equal to 2253225 when expressed as a mixed number.

The repeating decimal 22.235 (vinculum notation) has a repeated block length of 1. It is also represented as 22.23555... (ellipsis notation) which equals approximately 22.2355555 (decimal approximation)(*).

The recurring decimal 22.235 can be written as a ratio of two integers having 5003 as the numerator and 225 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.