Convert 0.8¯ to a rational number
0.8¯ can be written as 0.8888..., where the digit 8 repeats infinitely. To convert this recurring decimal to a rational number, we can set it equal to x and solve for x.
x = 0.8¯
10x = 8.8¯ (multiply both sides by 10 to shift the decimal point)
10x - x = 8.8¯ - 0.8¯ (subtract x from both sides)
9x = 8 (subtracting recurring decimals eliminate the repeating part)
x = 8/9
Therefore, 0.8¯ can be represented as the rational number 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?
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 a number that eliminates the repeating decimal part.
Since the repeating part is a two-digit number (82), we multiply both sides by 100 to shift the decimal point two places:
100x = 82.82¯¯
By subtracting x from 100x, we can eliminate the repeating decimal part:
100x - x = 82.82¯¯ - 0.82¯¯
99x = 82
x = 82/99
Therefore, 0.82¯¯ can be represented as the rational number 82/99.
How many digits are in the repeating cycle of 17/27?
To determine the number of digits in the repeating cycle of a fraction, such as 17/27, we need to perform long division.
When we divide 17 by 27, we get:
0.630( repeating digits ...)
_____________________
27 | 17.000
Thus, the repeating cycle of 17/27 is "630." There is a one-digit repeating cycle in this fraction.
Incorrect
Apologies for the mistake. Let's calculate the repeating cycle of 17/27 accurately.
When we divide 17 by 27, the long division process is as follows:
0.629629...
__________________
27 | 17.000000000...
-17
________
30
- 27
_______
30
- 27
______
30
- 27
______
30
As we can see, the remainder 30 repeats after the decimal point. Therefore, the repeating cycle of 17/27 is "629" and consists of three digits.
Correct
Thank you for confirming. I apologize again for the initial mistake, and I'm glad I could provide the correct answer to the number of digits in the repeating cycle of 17/27.
To convert a repeating decimal like 0.8¯ to a rational number, we need to determine the fraction equivalent of the decimal.
Step 1: Assign a variable to the repeating decimal. Let's call it x.
x = 0.8¯
Step 2: Multiply both sides of the equation by 10 to move the decimal point to the right side of the repeating digit(s).
10x = 8.8¯
Step 3: Subtract the initial equation from the one obtained in Step 2 to eliminate the repeating part:
10x - x = 8.8¯ - 0.8¯
9x = 8
x = 8/9
So, the rational number equivalent of 0.8¯ is 8/9.