How do you show the group of digits that repeat in a repeating decimal??
To show the group of digits that repeat in a repeating decimal, you can use a bar over the digits that repeat. This bar, known as a vinculum, indicates that the digits are repeating infinitely. For example, if you have the decimal 0.33333..., you can write it as 0.3̅, where the digit 3 repeats infinitely. Similarly, 0.153153153... can be written as 0.15̅3̅.
How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?
The length of the repeating cycle of a repeating decimal depends on the prime factors of the denominator. In this case, since the denominator is 7, which is a prime number, the repeating cycle can be at most 6 digits long.
Let's take an example to illustrate this. Suppose we have the rational number 1/7. When we divide 1 by 7, the decimal representation is 0.142857142857..., and the cycle of repeating digits is 142857. As 7 is a prime number, the repeating cycle cannot be longer than the number of digits in the denominator (7 in this case), which means that the repeating cycle can be at most 6 digits long.
Therefore, for a rational number with a denominator of 7, the repeating cycle can be 1 to 6 digits long.
Convert 0.27 repeating, to a rational number in simplest form
To convert the repeating decimal 0.27̅ (where the digits 27 repeat infinitely) to a rational number in simplest form, we can use algebraic manipulation.
Let's represent the repeating decimal as x:
x = 0.27̅
To eliminate the repeating digits, we can multiply both sides of the equation by 100 (to shift the decimal point two places to the right):
100x = 27.27̅
Now, let's subtract the original equation from the new equation to eliminate the repeating part:
100x - x = 27.27̅ - 0.27̅
99x = 27
To get x by itself, divide both sides of the equation by 99:
x = 27/99
Now, simplify the fraction by finding the greatest common divisor (GCD) of 27 and 99, which is 9:
x = (27/9) / (99/9)
x = 3/11
Therefore, the rational number equivalent to 0.27̅ is 3/11 in simplest form.
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.
Let's take an example to understand this. Suppose we have the repeating decimal 0.abc̅, where the digits abc repeat infinitely (c̅ indicates that c is the repeating digit). To convert this repeating decimal to a rational number, we can write it as:
0.abc̅ = abc/999
The denominator 999 is used because it is a three-digit number with all 9s, and when written as a fraction, it has three 9s in the numerator and three 9s in the denominator.
Therefore, if a repeating decimal has a repeating cycle of three digits, its equivalent rational number will have a denominator of 999.