Question

express 17/19 as a repeating decimal and mark repeating digits with a bar

Answer 1

To express \( \frac{17}{19} \) as a repeating decimal, we can perform the division:

1. Divide 17 by 19. Since 17 is less than 19, we start with 0 and add a decimal point.
2. Multiply 17 by 10 to get 170.
3. Divide 170 by 19, which goes 8 times (since \( 19 \times 8 = 152 \)).
4. Subtract 152 from 170 to get a remainder of 18.
5. Bring down a 0 to get 180.
6. Divide 180 by 19, which goes 9 times (since \( 19 \times 9 = 171 \)).
7. Subtract 171 from 180 to get a remainder of 9.
8. Bring down a 0 to get 90.
9. Divide 90 by 19, which goes 4 times (since \( 19 \times 4 = 76 \)).
10. Subtract 76 from 90 to get a remainder of 14.
11. Bring down a 0 to get 140.
12. Divide 140 by 19, which goes 7 times (since \( 19 \times 7 = 133 \)).
13. Subtract 133 from 140 to get a remainder of 7.
14. Bring down a 0 to get 70.
15. Divide 70 by 19, which goes 3 times (since \( 19 \times 3 = 57 \)).
16. Subtract 57 from 70 to get a remainder of 13.
17. Bring down a 0 to get 130.
18. Divide 130 by 19, which goes 6 times (since \( 19 \times 6 = 114 \)).
19. Subtract 114 from 130 to get a remainder of 16.
20. Bring down a 0 to get 160.
21. Divide 160 by 19, which goes 8 times (since \( 19 \times 8 = 152 \)).
22. Subtract 152 from 160 to get a remainder of 8.
23. Bring down a 0 to get 80.
24. Divide 80 by 19, which goes 4 times (since \( 19 \times 4 = 76 \)).
25. Subtract 76 from 80 to get a remainder of 4.
26. Bring down a 0 to get 40.
27. Divide 40 by 19, which goes 2 times (since \( 19 \times 2 = 38 \)).
28. Subtract 38 from 40 to get a remainder of 2.
29. Bring down a 0 to get 20.
30. Divide 20 by 19, which goes 1 time (since \( 19 \times 1 = 19 \)).
31. Subtract 19 from 20 to get a remainder of 1.
32. Bring down a 0 to get 10.
33. Divide 10 by 19, which goes 0 times.
34. Now bring down another 0 to get 100.
35. Divide 100 by 19, which goes 5 times (since \( 19 \times 5 = 95 \)).
36. Subtract 95 from 100 to get a remainder of 5.
37. Bring down a 0 to get 50.
38. Divide 50 by 19, which goes 2 times (since \( 19 \times 2 = 38 \)).
39. Subtract 38 from 50 to get a remainder of 12.
40. Bring down a 0 to get 120.
41. Divide 120 by 19, which goes 6 times (since \( 19 \times 6 = 114 \)).
42. Subtract 114 from 120 to get a remainder of 6.
43. Bring down a 0 to get 60.
44. Divide 60 by 19, which goes 3 times (since \( 19 \times 3 = 57 \)).
45. Subtract 57 from 60 to get a remainder of 3.
46. Bring down a 0 to get 30.
47. Divide 30 by 19, which goes 1 time (since \( 19 \times 1 = 19 \)).
48. Subtract 19 from 30 to get a remainder of 11.
49. Bring down a 0 to get 110.
50. Divide 110 by 19, which goes 5 times (since \( 19 \times 5 = 95 \)).
51. Subtract 95 from 110 to get a remainder of 15.
52. Bring down a 0 to get 150.
53. Divide 150 by 19, which goes 7 times (since \( 19 \times 7 = 133 \)).
54. Subtract 133 from 150 to get a remainder of 17.

At this point, the decimal repeats because we return to the original value of 17 with new digits.

We find that \( \frac{17}{19} = 0.\overline{894736842105263157} \).

Thus, the repeating decimal representation of \( \frac{17}{19} \) is:

\[
0.\overline{894736842105263157}
\]