Please help.I need to find two- hundredth digit in the decimal representation of each of the following fractions
5/6
4/11
1/13
Thanks.
To find the two-hundredth digit in the decimal representation of a fraction, you can divide the numerator by the denominator and examine the resulting decimal representation.
1. Let's start with the fraction 5/6:
To find the decimal representation, divide the numerator (5) by the denominator (6):
5 ÷ 6 = 0.833333...
Now, we need to determine the two-hundredth digit in the decimal representation. Since the fraction did not terminate and continues with repeating 3s, we know that the pattern repeats. To find the two-hundredth digit, we can look for any repeating pattern in the decimal representation.
In this case, we can see that the pattern repeats every 1 digit. Therefore, the two-hundredth digit in the decimal representation of 5/6 is the same as the 100th digit, which is 3.
2. Let's move on to the fraction 4/11:
Divide the numerator (4) by the denominator (11):
4 ÷ 11 = 0.363636...
Just like with the previous fraction, we need to find the repeating pattern in the decimal representation. Here, we can see that the pattern "36" repeats every 2 digits. Therefore, the two-hundredth digit in the decimal representation of 4/11 is 6.
3. Lastly, consider the fraction 1/13:
Divide the numerator (1) by the denominator (13):
1 ÷ 13 = 0.076923...
In this case, the repeating pattern is "076923," which repeats every 6 digits. Therefore, the two-hundredth digit in the decimal representation of 1/13 is 9.
So, the two-hundredth digits for the given fractions are:
1. For 5/6, the two-hundredth digit is 3.
2. For 4/11, the two-hundredth digit is 6.
3. For 1/13, the two-hundredth digit is 9.
5/6 = 0.83333...
4/11 = .363636...
1/13 = .076923 076923 ...
Look at the length of each repeating group. Divide that into 200 to see how many repetitions there are, and what remainder.
For instance
13/70 = .1857 142857 142857 ...
There are 4 leading digits, so that leaves 196. The periodic part has length 6. 196/6 = 32 remainder 4.
So, the 200th digit is
4 + 32*6 + 4
The 4th digit of the repeating part is 8. So, the 200th digit is 8.