What is 1.371212... as a Fraction?

1.3712 equals 181132 as a fraction.

How do you turn 1.3712 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.3712 repeating into a fraction, begin writing this simple equation:

n = 1.3712 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (12) and 2 digits in the non-repeating part (37). Multiply both sides by 10⁴ = 10000.

10000 × n = 13712.12 (equation 2)

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

10000 × n = 13712.12
  100 × n = 1.3712
 9900 × n = 13575

135759900 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 75 (the GCF - greatest common factor).

n = 135759900 = 13575 ÷ 759900 ÷ 75 = 181132. So,

1.3712 = 181132 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 181132 is also equal to 149132 when expressed as a mixed number.

The repeating decimal 1.3712 (vinculum notation) has a repeated block length of 2. It is also represented as 1.37121212... (ellipsis notation) which equals approximately 1.371212121212 (decimal approximation)(*).

The recurring decimal 1.3712 can be written as a ratio of two integers having 181 as the numerator and 132 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.