What is 340.133... as a Fraction?

340.133 equals 510215 as a fraction.

How do you turn 340.133 repeating into a fraction?

Detailed Answer:

Step 1: To convert 340.133 repeating into a fraction, begin writing this simple equation:

n = 340.133 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3) and 2 digits in the non-repeating part (13). Multiply both sides by 10³ = 1000.

1000 × n = 340133.3 (equation 2)

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

1000 × n = 340133.3
 100 × n = 340.133
 900 × n = 306120

306120900 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 60 (the GCF - greatest common factor).

n = 306120900 = 306120 ÷ 60900 ÷ 60 = 510215. So,

340.133 = 510215 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 510215 is also equal to 340215 when expressed as a mixed number.

The repeating decimal 340.133 (vinculum notation) has a repeated block length of 1. It is also represented as 340.13333... (ellipsis notation) which equals approximately 340.1333333 (decimal approximation)(*).

The recurring decimal 340.133 can be written as a ratio of two integers having 5102 as the numerator and 15 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.