What is 17.78333... as a Fraction?

17.783 equals 106760 as a fraction.

How do you turn 17.783 repeating into a fraction?

Detailed Answer:

Step 1: To convert 17.783 repeating into a fraction, begin writing this simple equation:

n = 17.783 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3) and 2 digits in the non-repeating part (78). Multiply both sides by 10³ = 1000.

1000 × n = 17783.3 (equation 2)

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

1000 × n = 17783.3
 100 × n = 17.783
 900 × n = 16005

16005900 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 15 (the GCF - greatest common factor).

n = 16005900 = 16005 ÷ 15900 ÷ 15 = 106760. So,

17.783 = 106760 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 106760 is also equal to 174760 when expressed as a mixed number.

The repeating decimal 17.783 (vinculum notation) has a repeated block length of 1. It is also represented as 17.78333... (ellipsis notation) which equals approximately 17.7833333 (decimal approximation)(*).

The recurring decimal 17.783 can be written as a ratio of two integers having 1067 as the numerator and 60 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.