Question

how do you show the group of dgits that repeat in a repeating decimal?

by putting them in parentheses
by printing them in italics
by putting a bar over them
by underlining them

Answer 1

how many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?

up to six digits long
infinitely long
up to seven digits long
any finite length

Answer 2

convert 0.27 to a rational number in the simplest form.

27/99
3/11
27/100
1/27

Answer 3

The standard way to show the group of digits that repeat in a repeating decimal is by putting a bar over them. For example, in the decimal \( 0.666...\), it is usually represented as \( 0.\overline{6} \).

Answer 4

The repeating cycle of a repeating decimal corresponding to a rational number can be up to one less than the denominator when the denominator is prime. Since 7 is a prime number, the repeating cycle can be up to 6 digits long. Therefore, the correct answer is **up to six digits long**.

Answer 5

To convert the repeating decimal \(0.27\) (which I assume is \(0.\overline{27}\)) into a rational number, you can follow these steps:

1. Let \(x = 0.\overline{27}\).
2. Multiply by 100 to shift the decimal point: \(100x = 27.\overline{27}\).
3. Now, subtract the first equation from the second:
\[
100x - x = 27.\overline{27} - 0.\overline{27}
\]
\[
99x = 27
\]
4. Solve for \(x\):
\[
x = \frac{27}{99}
\]
5. Simplify \(\frac{27}{99}\):
\[
x = \frac{27 \div 9}{99 \div 9} = \frac{3}{11}
\]

So, the simplest form of the rational number for \(0.\overline{27}\) is **\(\frac{3}{11}\)**.