If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator?
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with a denominator of 999.
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)
10
999
1,000
100
How do you show the group of digits that repeat in a repeating decimal?
Bot GPT 3.5
To determine the denominator of a rational number, we need to consider the length of the repeating cycle in the decimal representation. If the repeating cycle has three digits, the denominator should be 999.
Here's why:
Let's assume our repeating decimal number is represented as x = 0.abcabcabc..., where a, b, and c are the repeating digits.
To convert this decimal to a rational number, we need to multiply it by a power of 10 to shift the repeating digits to the left of the decimal point.
Let's denote the number without the repeating cycle as y. It would be y = 0.abc.
Now, subtracting y from x, we have:
x - y = 0.abcabcabc... - 0.abc = 0.abc(xyz - 1)
Notice that xyz - 1 is equivalent to 999, since x, y, and z are the repeating digits and their place values are 100, 10, and 1, respectively.
So, we have:
x - y = 0.abc(xyz - 1) = 0.abc * 999
Since x - y is a rational number, it can be represented as p/q, where p and q are integers. Therefore, we can set:
0.abc * 999 = p/q
By multiplying both sides by 1/999, we obtain:
0.abc = p/(999q)
This shows that the repeating decimal 0.abc is a rational number with a denominator of 999.
To find the denominator of a rational number equivalent to a repeating decimal with a repeating cycle of three digits, you need to know the number of digits in the repeating cycle.
In this case, the repeating cycle is made up of three digits.
To convert the repeating decimal to a rational number, you can use the following steps:
Step 1: Let's represent the repeating decimal using a variable. Let's use "x" for this example.
x = 0.abcabcabc...
Step 2: Multiply both sides of the equation by a power of 10 that will move the decimal point to the right of the repeating cycle. Since the repeating cycle has three digits, multiply by 1000 (10 raised to the power of 3).
1000x = abc.abcabcabc...
Step 3: Subtract the original equation (step 1) from the modified equation (step 2) to eliminate the repeating part.
1000x - x = abc.abcabcabc... - 0.abcabcabc...
Simplifying the equation gives:
999x = abc
Dividing both sides of the equation by 999:
x = abc / 999
Step 4: Since the denominator of a rational number is the number that cannot be reduced further, the denominator will be 999.
Therefore, a repeating decimal with a repeating cycle of three digits will convert to a rational number with a denominator of 999.