What is 12.40999... as a Fraction?

12.409 equals 1241100 as a fraction.

How do you turn 12.409 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.409 repeating into a fraction, begin writing this simple equation:

n = 12.409 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (9) and 2 digits in the non-repeating part (40). Multiply both sides by 10³ = 1000.

1000 × n = 12409.9 (equation 2)

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

1000 × n = 12409.9
 100 × n = 12.409
 900 × n = 11169

11169900 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 9 (the GCF - greatest common factor).

n = 11169900 = 11169 ÷ 9900 ÷ 9 = 1241100. So,

12.409 = 1241100 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 1241100 is also equal to 1241100 when expressed as a mixed number.

The repeating decimal 12.409 (vinculum notation) has a repeated block length of 1. It is also represented as 12.40999... (ellipsis notation) which equals approximately 12.4099999 (decimal approximation)(*).

The recurring decimal 12.409 can be written as a ratio of two integers having 1241 as the numerator and 100 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.