What is 13.45666... as a Fraction?

13.456 equals 4037300 as a fraction.

How do you turn 13.456 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.456 repeating into a fraction, begin writing this simple equation:

n = 13.456 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 2 digits in the non-repeating part (45). Multiply both sides by 10³ = 1000.

1000 × n = 13456.6 (equation 2)

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

1000 × n = 13456.6
 100 × n = 13.456
 900 × n = 12111

12111900 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 = 12111900 = 12111 ÷ 3900 ÷ 3 = 4037300. So,

13.456 = 4037300 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 4037300 is also equal to 13137300 when expressed as a mixed number.

The repeating decimal 13.456 (vinculum notation) has a repeated block length of 1. It is also represented as 13.45666... (ellipsis notation) which equals approximately 13.4566666 (decimal approximation)(*).

The recurring decimal 13.456 can be written as a ratio of two integers having 4037 as the numerator and 300 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.