What is 3.77222... as a Fraction?

3.772 equals 679180 as a fraction.

How do you turn 3.772 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.772 repeating into a fraction, begin writing this simple equation:

n = 3.772 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (2) and 2 digits in the non-repeating part (77). Multiply both sides by 10³ = 1000.

1000 × n = 3772.2 (equation 2)

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

1000 × n = 3772.2
 100 × n = 3.772
 900 × n = 3395

3395900 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 5 (the GCF - greatest common factor).

n = 3395900 = 3395 ÷ 5900 ÷ 5 = 679180. So,

3.772 = 679180 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 679180 is also equal to 3139180 when expressed as a mixed number.

The repeating decimal 3.772 (vinculum notation) has a repeated block length of 1. It is also represented as 3.77222... (ellipsis notation) which equals approximately 3.7722222 (decimal approximation)(*).

The recurring decimal 3.772 can be written as a ratio of two integers having 679 as the numerator and 180 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.