What is 0.612 repeating (meaning the repeating line is over 612) as a fraction
The Correct Answer and Explanation is:
Correct Answer:
The repeating decimal 0.612612612… (with 612 repeating) is equal to the fraction 612/999. This can be simplified to 68/111.
Explanation
Repeating decimals can be written as fractions by using algebraic methods. In this case, the repeating decimal is 0.612612612…, where the group “612” repeats indefinitely.
Let us set:
x = 0.612612612…
Since the repeating part has three digits, we multiply x by 1000 to shift the decimal point three places to the right:
1000x = 612.612612…
Now subtract the original value of x from this new equation:
1000x – x = 612.612612… – 0.612612…
This simplifies to:
999x = 612
Now divide both sides by 999:
x = 612 / 999
This is the fraction form of the repeating decimal. Next, simplify the fraction by finding the greatest common divisor (GCD) of 612 and 999. We use the Euclidean Algorithm:
- 999 ÷ 612 = 1 remainder 387
- 612 ÷ 387 = 1 remainder 225
- 387 ÷ 225 = 1 remainder 162
- 225 ÷ 162 = 1 remainder 63
- 162 ÷ 63 = 2 remainder 36
- 63 ÷ 36 = 1 remainder 27
- 36 ÷ 27 = 1 remainder 9
- 27 ÷ 9 = 3 remainder 0
So, the GCD is 9.
Now divide numerator and denominator by 9:
612 ÷ 9 = 68
999 ÷ 9 = 111
Therefore, the simplified fraction is:
68/111
So, 0.612612612… = 68/111
This method works for any repeating decimal and is particularly useful for converting repeating patterns into exact fractional form.
