what is 0.612 ( 12 repeating) as a fraction
The Correct Answer and Explanation is:
Correct Answer:
0.612 (with 12 repeating) = 203⁄330
Explanation
To convert the repeating decimal 0.612121212… to a fraction, we break it into two parts: the non-repeating part and the repeating part.
We are given:
- Non-repeating part: 0.6
- Repeating part: 121212…
Step 1: Let x = 0.612121212…
Step 2: Multiply x by 1000 to move the decimal point past the repeating section:
1000x = 612.121212…
Step 3: Multiply x by 10 to align the repeating parts:
10x = 6.121212…
Now subtract the two equations:
markdownCopyEdit1000x = 612.121212...
- 10x = 6.121212...
---------------------
990x = 606
Step 4: Solve for x:
iniCopyEditx = 606 / 990
Step 5: Simplify the fraction. We need the greatest common divisor (GCD) of 606 and 990.
606 and 990 are both divisible by 6:
- 606 ÷ 6 = 101
- 990 ÷ 6 = 165
Now the fraction becomes:
iniCopyEditx = 101 / 165
But 101 and 165 are both divisible by 1 only (101 is a prime number), so this is fully simplified.
Oops! This is from subtracting 1000x – 10x. But we didn’t set up the decimal correctly.
Let’s fix it more precisely:
Let x = 0.612121212…
To isolate the repeating part “12”, multiply x by 100:
100x = 61.212121…
Now subtract x:
CopyEdit100x - x = 61.212121... - 0.612121...
99x = 60.6
Now convert 60.6 to a fraction:
60.6 = 606 / 10
So:
rubyCopyEdit99x = 606 / 10
=> x = (606 / 10) / 99 = 606 / (10 * 99) = 606 / 990
=> Simplify 606/990 = 203 / 330
Final Answer: 203⁄330
