How many digits are in the repeating cycle of 17/27?(1 point)
Express 713 as a repeating decimal. Mark the repeating digits with a bar.
Note: You will find the option to add a bar above text in the Geometry (∠
) keyboard.
(1 point)
The repeating cycle of a fraction can be found by noting that the remainder of the fraction when divided by the denominator remains the same for each subsequent division.
In this case, when we divide 17 by 27, the remainder is 17. We can continue the division as follows:
17 ÷ 27 = 0 remainder 17
170 ÷ 27 = 6 remainder 8
80 ÷ 27 = 2 remainder 26
260 ÷ 27 = 9 remainder 17
170 ÷ 27 = 6 remainder 8
...
and so on.
Since we have already encountered 17 again, we can conclude that the repeating cycle of 17/27 is 17. Therefore, there is only 1 digit in the repeating cycle.
To express 713 as a repeating decimal, we divide 713 by 1 to get 713.
713 ÷ 1 = 713
In this case, there is no repeating part in the decimal because it can be expressed as a terminating decimal. Therefore, we do not need to mark any repeating digits with a bar.
Express 7/13 as a repeating decimal. Mark the repeating digits with a bar.
Note: You will find the option to add a bar above text in the Geometry (∠
) keyboard.
(1 point)
To express 7/13 as a repeating decimal, we perform the long division:
0.
13 | 7.000
- 0
70
- 65
__
50
The remainder is 50. We can bring down another zero and continue the division:
0.5
13 | 7.000
- 0
70
- 65
__
50
- 39
__
110
The remainder is 110. We can bring down another zero and continue the division:
0.53
13 | 7.000
- 0
70
- 65
__
50
- 39
__
110
- 104
__
60
The remainder is 60. We can bring down another zero and continue the division:
0.538
13 | 7.000
- 0
70
- 65
__
50
- 39
__
110
- 104
__
60
- 52
__
80
The remainder is 80. We can bring down another zero and continue the division:
0.5384
13 | 7.0000
- 0
70
- 65
__
50
- 39
__
110
- 104
__
60
- 52
__
80
- 78
__
200
The remainder is 200. We can bring down another zero and continue the division:
0.53846
13 | 7.00000
- 0
70
- 65
__
50
- 39
__
110
- 104
__
60
- 52
__
80
- 78
__
200
- 195
____
50
The remainder is 50. We can bring down another zero and continue the division:
0.538461
13 | 7.000000
- 0
70
- 65
__
50
- 39
__
110
- 104
__
60
- 52
__
80
- 78
__
200
- 195
____
50
- 39
____
110
The remainder is 110, and we can notice that 110 was the same remainder we had before. Thus, the decimal repeats after the decimal point.
Hence, 7/13 as a repeating decimal is approximately 0.538461 with the digits 538461 repeating in a loop.
Convert 0.8¯¯¯ to a rational number.(1 point)
To convert the repeating decimal 0.8¯¯¯ to a rational number, we can let x equal the repeating decimal.
So, we have:
x = 0.8¯¯¯
Now, to eliminate the repeating part, we can multiply both sides of the equation by 10:
10x = 8.8¯¯¯
Next, to eliminate the repeating part again, we can subtract the original equation from the equation obtained when we multiplied both sides by 10:
10x - x = 8.8¯¯¯ - 0.8¯¯¯
Simplifying gives:
9x = 8
Finally, divide both sides of the equation by 9:
9x/9 = 8/9
x = 8/9
Therefore, 0.8¯¯¯ is a rational number equal to 8/9.
To convert 0.82¯¯¯¯¯ to a rational number, you would set x equal to 0.82¯¯¯¯¯ and then multiply both sides of the equation by what number?(1 point)
To convert 0.82¯¯¯¯¯ to a rational number, you would set x equal to 0.82¯¯¯¯¯ and then multiply both sides of the equation by 100.
So, you would multiply both sides of the equation x = 0.82¯¯¯¯¯ by 100:
100x = 82.82¯¯¯¯¯
This is because when we multiply a decimal by a power of 10, we move the decimal point to the right by the same number of places as the power of 10.
Therefore, to eliminate the repeating part, we multiply by a power of 10 that has a number of 9s equal to the number of repeating digits in the decimal. In this case, since there are two repeating digits (82), we multiply by 100 (which is 10 raised to the power of 2) to move the repeating part to the left of the decimal point.
To determine the number of digits in the repeating cycle of a fraction, like 17/27, we need to look at the decimal representation of the fraction.
To find the decimal representation of a fraction, we divide the numerator (17) by the denominator (27).
Dividing 17 by 27 gives us 0.629629... (the ellipsis (...) indicates that the digits repeat infinitely).
To find the length of the repeating cycle, we can start by looking for any patterns in the digits. We observe that the digits 62 repeat in the decimal representation.
Therefore, the repeating cycle of 17/27 consists of the digits "62", and it repeats indefinitely.
So, there are 2 digits in the repeating cycle of 17/27.