Which digit will appear in the 534th place after the decimal point in the decimal representation of 5/13?
5/13 = .384615384615..
so the repeat is 384615 , namely 6 digits long
534/6 = 89 with Remainder 0 , making 534 a multiple of 6
so the 6th digit will be a 5
and the 12th digit will be a 5
and the 18th digit will be a 5
...
and the 528th digit will be a 5
and the 534th digit will be a 5
Well, if I had a penny for every time someone asked me about decimal places... I'd have a few extra pennies! Now, let's calculate that 534th digit. To avoid clowning around, let's use some math tricks. The decimal representation of 5/13 is approximately 0.384615384... Notice any pattern yet? It repeats after 6 digits. So, if we divide 534 by 6, we get 89 with a remainder of 0. Since our pattern repeats, the digit that appears in the 534th place after the decimal point is the same as the digit in the 0th place, which is... *drumroll* 3. Ta-da!
To find the digit that appears in the 534th place after the decimal point in the decimal representation of 5/13, we need to perform long division. Here's how you can proceed:
Step 1: Divide 5 by 13. The quotient is 0, so write 0 above the bar.
0.
______
13 | 5
Step 2: Move the decimal point of 5 to the right to make it 50. Write it next to the 5.
0.
______
13 | 50
Step 3: Divide 50 by 13. The quotient is 3, so write 3 above the bar.
0. 3
______
13 | 50
Step 4: Multiply 3 by 13 to get 39. Write it below the 50.
0. 3
______
13 | 50
- 39
Step 5: Subtract 39 from 50 to get 11. Bring down the next digit (which is 0 since we have reached the end).
0. 3
______
13 | 50
- 39
11
Step 6: Divide 110 (the new number obtained by bringing down 0) by 13. The quotient is 8, so write 8 above the bar.
0. 38
_______
13 | 500
- 39
11
- 0
-----------
Step 7: Multiply 8 by 13 to get 104. Write it below the 110.
0. 38
_______
13 | 500
- 39
11
- 0
-----------
110
Step 8: Subtract 104 from 110 to get 6. Since 6 is less than 13, we know that the decimal representation is repeating. Bring down the next digit (which is 0 since we have reached the end).
0. 38
_______
13 | 500
- 39
11
- 0
-----------
110
- 104
_______
6
Step 9: Divide 60 (the new number obtained by bringing down 0) by 13. The quotient is 4, so write 4 above the bar.
0. 384
________
13 | 5000
- 39
11
- 0
-----------
110
- 104
_______
60
Step 10: Multiply 4 by 13 to get 52. Write it below the 60.
0. 3846
_________
13 | 50000
- 39
11
- 0
-----------
110
- 104
_______
606
- 52
_______
54
Step 11: Subtract 52 from 60 to get 8. Since 8 is less than 13, we know that the decimal representation is repeating.
Continuing this process, we can see that the decimal representation of 5/13 repeats the sequence 3846. Since we are looking for the digit in the 534th place after the decimal point, we count the sequence 3846 repeatedly until we reach the 534th position.
534 รท 4 = 133 remainder 2
This means that the repetition will occur 133 times fully, and we will have 2 digits left to count. Therefore, we count the first 2 digits of the repeating sequence, which are 3 and 8.
Thus, the digit that will appear in the 534th place after the decimal point in the decimal representation of 5/13 is 8.
To find the digit that appears in the 534th place after the decimal point in the decimal representation of 5/13, we can perform long division.
Let's start by dividing 5 by 13:
0. 3 8 4 6 1 5
________________
13 | 5. 0 0 0 0 0 0 0 0
- 3 9
-----
1 0
- 9
--
1
The division continues indefinitely, as there is a remainder of 1.
From our calculation, we can see that the decimal representation of 5/13 is 0.384615... with the digits 3, 8, 4, 6, 1, and 5 repeating. Since the decimal representation repeats after 6 digits, we can determine that the 534th place will be the same as the 534 % 6 = 0th place, which is 3.
Therefore, the digit that will appear in the 534th place after the decimal point in the decimal representation of 5/13 is 3.