What is 2.35757... as a Fraction?

2.357 equals 389165 as a fraction.

How do you turn 2.357 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.357 repeating into a fraction, begin writing this simple equation:

n = 2.357 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (57) and 1 digit in the non-repeating part (3). Multiply both sides by 10³ = 1000.

1000 × n = 2357.57 (equation 2)

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

1000 × n = 2357.57
  10 × n = 2.357
 990 × n = 2334

2334990 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 6 (the GCF - greatest common factor).

n = 2334990 = 2334 ÷ 6990 ÷ 6 = 389165. So,

2.357 = 389165 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 389165 is also equal to 259165 when expressed as a mixed number.

The repeating decimal 2.357 (vinculum notation) has a repeated block length of 2. It is also represented as 2.3575757... (ellipsis notation) which equals approximately 2.35757575757 (decimal approximation)(*).

The recurring decimal 2.357 can be written as a ratio of two integers having 389 as the numerator and 165 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.