What is 5.234615346153461534615346153... as a Fraction?

5.2346153 equals 5234563999990 as a fraction.

How do you turn 5.2346153 repeating into a fraction?

Detailed Answer:

Step 1: To convert 5.2346153 repeating into a fraction, begin writing this simple equation:

n = 5.2346153 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (46153) and 2 digits in the non-repeating part (23). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 52346153.46153 (equation 2)

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

10000000 × n = 52346153.46153
     100 × n = 5.2346153
 9999900 × n = 52345630

523456309999900 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 10 (the GCF - greatest common factor).

n = 523456309999900 = 52345630 ÷ 109999900 ÷ 10 = 5234563999990. So,

5.2346153 = 5234563999990 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 5234563999990 is also equal to 5234613999990 when expressed as a mixed number.

The repeating decimal 5.2346153 (vinculum notation) has a repeated block length of 5. It is also represented as 5.23461534615346153... (ellipsis notation) which equals approximately 5.234615346153461534615346153 (decimal approximation)(*).

The recurring decimal 5.2346153 can be written as a ratio of two integers having 5234563 as the numerator and 999990 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.