To convert the fraction \( \frac{21}{99} \) to a decimal using long division, we will go through the long division process step by step:
1. Write down the dividend (21) and the divisor (99).
2. Determine how many times 99 can go into 21. Since 21 is less than 99, it cannot go into 21 even once. In this case, we will need to introduce a decimal point and add a zero to the right of 21, making it 210.
21.0
---
99 ) 21.000
3. Now, how many times does 99 go into 210? It goes into it 2 times, because 2 x 99 = 198. Write 2 above the line, after the decimal point.
21.0
---2
99 ) 21.000
-198
4. Subtract 198 from 210 to get the remainder. 210 - 198 = 12. Bring down the next zero to the right of the remainder to prepare for the next step of division. This makes our new number 120.
21.000
---2
99 ) 21.000
-198
120
5. Determine how many times 99 goes into 120. It goes once, because 1 x 99 = 99.
21.000
---2.1
99 ) 21.000
-198
120
-99
6. Subtract 99 from 120 to get the new remainder. 120 - 99 = 21. Bring down the next zero to the right of the remainder to prepare for the next step of division. This makes our new number 210 (again).
21.000
---2.1
99 ) 21.000
-198
120
-99
210
7. Repeat step 5 and step 6. You will find this process repeating endlessly since 99 goes into 210 twice, leaving a remainder of 21, which leads to adding another zero and getting 210 again. In decimal terms, this means the fraction \( \frac{21}{99} \) is a repeating decimal where the digit pattern "21" repeats indefinitely.
So, the decimal representation of \( \frac{21}{99} \) is 0.212121..., and usually, it is denoted by putting a line (vinculum) over the repeating part:
0.\overline{21}