What is 4.20153153153... as a Fraction?

4.20153 equals 4663711100 as a fraction.

How do you turn 4.20153 repeating into a fraction?

Detailed Answer:

Step 1: To convert 4.20153 repeating into a fraction, begin writing this simple equation:

n = 4.20153 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (153) and 2 digits in the non-repeating part (20). Multiply both sides by 10⁵ = 100000.

100000 × n = 420153.153 (equation 2)

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

100000 × n = 420153.153
   100 × n = 4.20153
 99900 × n = 419733

41973399900 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 9 (the GCF - greatest common factor).

n = 41973399900 = 419733 ÷ 999900 ÷ 9 = 4663711100. So,

4.20153 = 4663711100 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 4663711100 is also equal to 4223711100 when expressed as a mixed number.

The repeating decimal 4.20153 (vinculum notation) has a repeated block length of 3. It is also represented as 4.20153153153... (ellipsis notation) which equals approximately 4.20153153153153153 (decimal approximation)(*).

The recurring decimal 4.20153 can be written as a ratio of two integers having 46637 as the numerator and 11100 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.