What is 123.333... as a Fraction?

123.3 equals 3703 as a fraction.

How do you turn 123.3 repeating into a fraction?

Detailed Answer:

Step 1: To convert 123.3 repeating into a fraction, begin writing this simple equation:

n = 123.3 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3), so multiply both sides by 10¹ = 10.

10 × n = 1233.3 (equation 2)

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

10 × n = 1233.3
 1 × n = 123.3
 9 × n = 1110

11109 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 = 11109 = 1110 ÷ 39 ÷ 3 = 3703. So,

123.3 = 3703 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 3703 is also equal to 12313 when expressed as a mixed number.

The repeating decimal 123.3 (vinculum notation) has a repeated block length of 1. It is also represented as 123.333... (ellipsis notation) which equals approximately 123.33333 (decimal approximation)(*).

The recurring decimal 123.3 can be written as a ratio of two integers having 370 as the numerator and 3 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.