What is 3.131515... as a Fraction?

3.1315 equals 51671650 as a fraction.

How do you turn 3.1315 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.1315 repeating into a fraction, begin writing this simple equation:

n = 3.1315 (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 = 31315.15 (equation 2)

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

10000 × n = 31315.15
  100 × n = 3.1315
 9900 × n = 31002

310029900 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 = 310029900 = 31002 ÷ 69900 ÷ 6 = 51671650. So,

3.1315 = 51671650 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 51671650 is also equal to 32171650 when expressed as a mixed number.

The repeating decimal 3.1315 (vinculum notation) has a repeated block length of 2. It is also represented as 3.13151515... (ellipsis notation) which equals approximately 3.131515151515 (decimal approximation)(*).

The recurring decimal 3.1315 can be written as a ratio of two integers having 5167 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.