What is 13.31515... as a Fraction?

13.315 equals 2197165 as a fraction.

How do you turn 13.315 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.315 repeating into a fraction, begin writing this simple equation:

n = 13.315 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (15) and 1 digit in the non-repeating part (3). Multiply both sides by 10³ = 1000.

1000 × n = 13315.15 (equation 2)

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

1000 × n = 13315.15
  10 × n = 13.315
 990 × n = 13182

13182990 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 6 (the GCF - greatest common factor).

n = 13182990 = 13182 ÷ 6990 ÷ 6 = 2197165. So,

13.315 = 2197165 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 2197165 is also equal to 1352165 when expressed as a mixed number.

The repeating decimal 13.315 (vinculum notation) has a repeated block length of 2. It is also represented as 13.3151515... (ellipsis notation) which equals approximately 13.31515151515 (decimal approximation)(*).

The recurring decimal 13.315 can be written as a ratio of two integers having 2197 as the numerator and 165 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.