What is 37.14666... as a Fraction?

37.146 equals 278675 as a fraction.

How do you turn 37.146 repeating into a fraction?

Detailed Answer:

Step 1: To convert 37.146 repeating into a fraction, begin writing this simple equation:

n = 37.146 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 2 digits in the non-repeating part (14). Multiply both sides by 10³ = 1000.

1000 × n = 37146.6 (equation 2)

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

1000 × n = 37146.6
 100 × n = 37.146
 900 × n = 33432

33432900 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 12 (the GCF - greatest common factor).

n = 33432900 = 33432 ÷ 12900 ÷ 12 = 278675. So,

37.146 = 278675 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 278675 is also equal to 371175 when expressed as a mixed number.

The repeating decimal 37.146 (vinculum notation) has a repeated block length of 1. It is also represented as 37.14666... (ellipsis notation) which equals approximately 37.1466666 (decimal approximation)(*).

The recurring decimal 37.146 can be written as a ratio of two integers having 2786 as the numerator and 75 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.