What is 2.15999... as a Fraction?

2.159 equals 5425 as a fraction.

How do you turn 2.159 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.159 repeating into a fraction, begin writing this simple equation:

n = 2.159 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (9) and 2 digits in the non-repeating part (15). Multiply both sides by 10³ = 1000.

1000 × n = 2159.9 (equation 2)

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

1000 × n = 2159.9
 100 × n = 2.159
 900 × n = 1944

1944900 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 36 (the GCF - greatest common factor).

n = 1944900 = 1944 ÷ 36900 ÷ 36 = 5425. So,

2.159 = 5425 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 5425 is also equal to 2425 when expressed as a mixed number.

The repeating decimal 2.159 (vinculum notation) has a repeated block length of 1. It is also represented as 2.15999... (ellipsis notation) which equals approximately 2.1599999 (decimal approximation)(*).

The recurring decimal 2.159 can be written as a ratio of two integers having 54 as the numerator and 25 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.