What is 15.131515... as a Fraction?

15.1315 equals 249671650 as a fraction.

How do you turn 15.1315 repeating into a fraction?

Detailed Answer:

Step 1: To convert 15.1315 repeating into a fraction, begin writing this simple equation:

n = 15.1315 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (15) and 2 digits in the non-repeating part (13). Multiply both sides by 10⁴ = 10000.

10000 × n = 151315.15 (equation 2)

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

10000 × n = 151315.15
  100 × n = 15.1315
 9900 × n = 149802

1498029900 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 = 1498029900 = 149802 ÷ 69900 ÷ 6 = 249671650. So,

15.1315 = 249671650 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 249671650 is also equal to 152171650 when expressed as a mixed number.

The repeating decimal 15.1315 (vinculum notation) has a repeated block length of 2. It is also represented as 15.13151515... (ellipsis notation) which equals approximately 15.131515151515 (decimal approximation)(*).

The recurring decimal 15.1315 can be written as a ratio of two integers having 24967 as the numerator and 1650 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.