What is 2.512121212... as a Fraction?

2.512 equals 829330 as a fraction.

How do you turn 2.512 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.512 repeating into a fraction, begin writing this simple equation:

n = 2.512 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (12) and 1 digit in the non-repeating part (5). Multiply both sides by 10³ = 1000.

1000 × n = 2512.12 (equation 2)

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

1000 × n = 2512.12
  10 × n = 2.512
 990 × n = 2487

2487990 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 3 (the GCF - greatest common factor).

n = 2487990 = 2487 ÷ 3990 ÷ 3 = 829330. So,

2.512 = 829330 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 829330 is also equal to 2169330 when expressed as a mixed number.

The repeating decimal 2.512 (vinculum notation) has a repeated block length of 2. It is also represented as 2.5121212... (ellipsis notation) which equals approximately 2.51212121212 (decimal approximation)(*).

The recurring decimal 2.512 can be written as a ratio of two integers having 829 as the numerator and 330 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.