Question

Convert 0.27¯¯¯¯¯
to a rational number in simplest form.(1 point)
Responses

127
Start Fraction 1 over 27 end fraction

311
Start Fraction 3 over 11 end fraction

27100
Start Fraction 27 over 100 end fraction

2799

Answer 1

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)
Responses

10
10

999
999

1,000
1,000

100
100

Answer 2

If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator? (1 point)
Responses

99
99

1,000
1,000

999
999

100

Answer 3

Let \( x = 0.27\overline{27} \). This means \( x = 0.272727...\).

To convert this repeating decimal to a fraction, we can set up the equation as follows:

1. Multiply \( x \) by 100 (to shift the decimal point two places to the right):
\[
100x = 27.272727...
\]

2. Now, multiply \( x \) by 1 (keeping the decimal as it is):
\[
x = 0.272727...
\]

3. Subtract the second equation from the first:
\[
100x - x = 27.272727... - 0.272727...
\]
\[
99x = 27
\]

4. Solve for \( x \):
\[
x = \frac{27}{99}
\]

5. Simplify the fraction \( \frac{27}{99} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 9:
\[
x = \frac{27 \div 9}{99 \div 9} = \frac{3}{11}
\]

Thus, \( 0.27\overline{27} \) as a rational number in simplest form is \(\frac{3}{11}\).

So, the correct response is:
\[
\frac{3}{11}
\]

Answer 4

To convert the repeating decimal \( 0.264\overline{264} \) to a rational number, you start by letting \( x = 0.264\overline{264} \).

Since the repeating part ("264") has three digits, you should multiply both sides by \( 1000 \) to shift the decimal point three places to the right:

1. Set \( x = 0.264264264...\).
2. Multiply both sides by \( 1000 \):
\[
1000x = 264.264264264...
\]

Thus, to convert \( 0.264\overline{264} \) to a rational number, you would multiply both sides by **1,000**.

So, the correct response is:
\[
1,000
\]

Answer 5

If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with a denominator of \( 999 \).

For example, if you have a decimal like \( 0.abcabcabc...\) (where "abc" is the three-digit repeating block), when converted to a fraction, the denominator will always be \( 999 \) because \( 999 = 10^n - 1 \) where \( n \) is the number of digits in the repeating block (in this case, \( n = 3 \)).

So, the correct response is:
\[
999
\]