What is 3.224285742857428574285742857... as a Fraction?

3.2242857 equals 64485071999980 as a fraction.

How do you turn 3.2242857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.2242857 repeating into a fraction, begin writing this simple equation:

n = 3.2242857 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (42857) and 2 digits in the non-repeating part (22). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 32242857.42857 (equation 2)

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

10000000 × n = 32242857.42857
     100 × n = 3.2242857
 9999900 × n = 32242535

322425359999900 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 5 (the GCF - greatest common factor).

n = 322425359999900 = 32242535 ÷ 59999900 ÷ 5 = 64485071999980. So,

3.2242857 = 64485071999980 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 64485071999980 is also equal to 34485671999980 when expressed as a mixed number.

The repeating decimal 3.2242857 (vinculum notation) has a repeated block length of 5. It is also represented as 3.22428574285742857... (ellipsis notation) which equals approximately 3.224285742857428574285742857 (decimal approximation)(*).

The recurring decimal 3.2242857 can be written as a ratio of two integers having 6448507 as the numerator and 1999980 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.