How many digital are in the repeating cycle of 17/27
To determine the length of the repeating cycle of a fraction in decimal form, we can perform long division.
When we divide 17 by 27, the remainder after the first division is 17. We then multiply the remainder by 10 and continue the process.
```
0.6296296296296296
---------
27| 17.000000000000000
- 16
----------
10.000000000000000
- 9
---------------
100.000000000000000
- 81
---------------
190.000000000000000
- 189
---------------
1000.000000000000000
- 972
---------------
280.000000000000000
- 270
---------------
1000.000000000000000
- 972
---------------
280.000000000000000
- 270
---------------
100
```
We can observe a repeating cycle starting after the first few digits (17). The cycle in this case is 627 with a length of 3 digits. Therefore, the repeating cycle of 17/27 in decimal form has a length of 3 digits.
To find the repeating cycle of the decimal representation of a fraction, you need to perform long division. Let's calculate the decimal representation of 17/27 step by step:
1. Divide 17 by 27:
- The quotient is 0.
- Write down a decimal point: 0.
2. Multiply the quotient (0) by 27 to get the next dividend:
- 0 * 27 = 0.
3. Bring down the next digit from the numerator (17):
- 0 remainder 17.
4. Divide 17 by 27:
- The quotient is 0.
- Write down the next digit after the decimal point: 00.
5. Multiply the quotient (00) by 27 to get the next dividend:
- 0 * 27 = 0.
6. Bring down the next digit from the numerator (17):
- 0 remainder 17.
7. Divide 17 by 27:
- The quotient is 0.
- Write down the next digit after the decimal point: 000.
8. Multiply the quotient (000) by 27 to get the next dividend:
- 0 * 27 = 0.
9. Bring down the next digit from the numerator (17):
- 0 remainder 17.
This process repeats indefinitely, with the remainder always being 17. Therefore, the repeating cycle of the decimal representation for 17/27 is 0.629629... where the digit 629 repeats.
To determine the number of digits in the repeating cycle of a fraction, you need to perform long division. Here's how you can do it for the fraction 17/27:
1. Begin by dividing 17 by 27 using long division:
0.6296296...
------------
27 | 17.0000000
2. The first division result is 0.62 (rounded to two decimal places).
3. Subtract the product of the division result (0.62) and the divisor (27) from the dividend (17) to get the remainder:
0.6296296...
------------
27 | 17.0000000
-16.83
------
321
4. Bring down a 0 from the decimal part to continue the division:
0.6296296...
------------
27 | 17.0000000
-16.83
------
3210
5. Divide 3210 by 27:
0.6296296...
------------
27 | 17.0000000
-16.83
------
3210
-2700
------
510
6. The next division result is 0.62 (rounded to two decimal places).
7. Repeat steps 4-6 to continue the division until the pattern repeats.
8. As you can see, the division pattern of 17/27 is 0.6296296... where 629 repeats. Therefore, there are three digits in the repeating cycle: 6, 2, and 9.