How do you show the group of digits that repeat in a repeating decimal?(1 point)
Responses
One way to show the group of digits that repeat in a repeating decimal is by using a bar notation. The repeated group of digits is placed over a bar. For example, if the decimal 0.333... repeats the digit "3," it can be represented as 0.3̅.
How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?(1 point)
Responses
The length of the repeating cycle of a decimal representation of a rational number with a denominator of 7 can be at most 6 digits long. This is because the repeating cycle of the decimal can only include the digits 1 to 6, as dividing any number by 7 can yield a remainder of only those digits.
Convert 0.27¯¯¯¯¯
to a rational number in simplest form.(1 point)
To convert the repeating decimal 0.27¯¯¯¯¯ to a rational number, we can set it up as follows:
Let x = 0.27¯¯¯¯¯
Multiply both sides of the equation by 100 to eliminate the repeating decimal:
100x = 27.¯¯¯¯¯
Next, subtract x from 100x to eliminate the repeating decimal:
100x - x = 27.¯¯¯¯¯ - 0.27¯¯¯¯¯
99x = 27
Now, divide both sides of the equation by 99:
99x/99 = 27/99
Simplifying gives:
x = 3/11
Therefore, the rational number equivalent of the repeating decimal 0.27¯¯¯¯¯ is 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?(1 point)
To convert the repeating decimal 0.264¯¯¯¯¯ to a rational number, you would set x equal to 0.264¯¯¯¯¯ and then multiply both sides of the equation by 1000.
The reason we multiply by 1000 is to eliminate the repeating decimal. Since there are three digits in the repeating part (264), we multiply by 10 raised to the power of the number of repeating digits to shift the decimal point to the right place. Therefore, we multiply by 1000 (10^3).
Hope this helps!
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator? (1 point)
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with a denominator of 999.
The reason for this is because when we have a three-digit repeating cycle, we can express it as a fraction where the numerator represents the repeating digits and the denominator is a string of nines with the same number of digits as the repeating cycle. This is because each digit in the repeating cycle corresponds to a place value in the denominator.
For example, if the repeating cycle is 123, the fraction representation would be 123/999.
To show the group of digits that repeat in a repeating decimal, you can follow these steps:
1. Begin by converting the repeating decimal to a fraction. Let's call the repeating decimal "N". To do this, assign a variable (let's say "x") to N and subtract it from N multiplied by a power of 10. This eliminates the repeating part of the decimal.
Example: Let N = 0.888...
x = N = 0.888...
10x = 8.888...
Subtracting the equation x = N from 10x - x = 8.888... - 0.888... will give you the equation 9x = 8.
2. Solve the equation obtained in the previous step for "x". In this case, divide both sides by 9.
9x = 8
x = 8/9
3. Now you have the fraction equivalent of the repeating decimal. The numerator of the fraction represents the repeating part of the decimal.
In this example, the numerator is 8 and the denominator is 9. Therefore, the group of digits that repeat in the decimal 0.888... is the digit "8".