What is 20.85333... as a Fraction?

20.853 equals 156475 as a fraction.

How do you turn 20.853 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.853 repeating into a fraction, begin writing this simple equation:

n = 20.853 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3) and 2 digits in the non-repeating part (85). Multiply both sides by 10³ = 1000.

1000 × n = 20853.3 (equation 2)

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

1000 × n = 20853.3
 100 × n = 20.853
 900 × n = 18768

18768900 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 = 18768900 = 18768 ÷ 12900 ÷ 12 = 156475. So,

20.853 = 156475 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 156475 is also equal to 206475 when expressed as a mixed number.

The repeating decimal 20.853 (vinculum notation) has a repeated block length of 1. It is also represented as 20.85333... (ellipsis notation) which equals approximately 20.8533333 (decimal approximation)(*).

The recurring decimal 20.853 can be written as a ratio of two integers having 1564 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.