How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?
The repeating cycle of a repeating decimal is at most one less than the denominator of the rational number. Therefore, if the denominator is 7, the repeating cycle can be at most 6 digits long.
Convert 0.27¯¯¯¯¯
to a rational number in simplest form.
Let x = 0.27¯¯¯¯¯
Multiplying both sides of the equation by 100, we get:
100x = 27.¯¯¯¯¯
Subtracting the original equation from the new equation gives:
100x - x = 27.¯¯¯¯¯ - 0.27¯¯¯¯¯
99x = 27
Simplifying, we find:
x = 27/99
Now we can simplify the fraction:
x = 3/11
Therefore, 0.27¯¯¯¯¯ is equivalent to the rational number 3/11.
To convert 0.264¯¯¯¯¯¯¯¯ to a rational number, you would set x equal to 0.264¯¯¯¯¯¯¯¯ and then multiply both sides of the equation by what number?
To convert 0.264¯¯¯¯¯ to a rational number, we would set x equal to 0.264¯¯¯¯¯ and then multiply both sides of the equation by a power of 10.
The number of repeating digits in 0.264¯¯¯¯¯ is 3 (since the repeating cycle is 264).
To eliminate the repeating decimal, we multiply both sides by 1000 (10^3), as it has 3 digits, which is the same as the number of repeating digits.
Thus, we multiply both sides of the equation x = 0.264¯¯¯¯¯ by 1000.
1000x = 264.264¯¯¯¯¯
Now, we subtract the original equation from the new equation:
1000x - x = 264.264¯¯¯¯¯ - 0.264¯¯¯¯¯
Simplifying the equation:
999x = 264
Now we can simplify the fraction:
x = 264/999
Therefore, 0.264¯¯¯¯¯ is equivalent to the rational number 264/999, which can be further simplified to 88/333.