To convert the fraction [tex]\( \frac{41}{333} \)[/tex] into a repeating decimal, we will use the long division method. Let's work through this step-by-step.
1. Set up the division:
We are dividing 41 by 333. Since 41 is less than 333, the integer part of the quotient is 0. Therefore, we start with the decimal part.
[tex]\[
0.\overline{abc}
\][/tex]
2. Long division steps:
- Step 1: Multiply 41 by 10 to get 410. We now need to divide 410 by 333.
- Step 2: 333 goes into 410 (1 time). We write down 1 and calculate the new remainder.
[tex]\[
410 - 333 \times 1 = 410 - 333 = 77
\][/tex]
So, the first digit [tex]\(a\)[/tex] of the repeating decimal sequence is 1.
- Step 3: Multiply remainder 77 by 10 to get 770. We now need to divide 770 by 333.
- Step 4: 333 goes into 770 (2 times). We write down 2 and calculate the new remainder.
[tex]\[
770 - 333 \times 2 = 770 - 666 = 104
\][/tex]
So, the second digit [tex]\(b\)[/tex] of the repeating decimal sequence is 2.
- Step 5: Multiply remainder 104 by 10 to get 1040. We now need to divide 1040 by 333.
- Step 6: 333 goes into 1040 (3 times). We write down 3 and calculate the new remainder.
[tex]\[
1040 - 333 \times 3 = 1040 - 999 = 41
\][/tex]
So, the third digit [tex]\(c\)[/tex] of the repeating decimal sequence is 3.
3. Identify that the sequence repeats:
We are back to a remainder of 41, which is the original numerator of the fraction. This tells us that the sequence [tex]\(123\)[/tex] will now begin to repeat infinitely.
4. Conclusion:
The fraction [tex]\(\frac{41}{333}\)[/tex] can be written as a repeating decimal:
[tex]\[
z = 0.\overline{123123123\ldots}
\][/tex]
Therefore, the repeating decimal part is [tex]\(abc = 1, 2, 3\)[/tex].
[tex]\[
\boxed{a=1}, \boxed{b=2}, \text{ and } \boxed{c=3}.
\][/tex]
