Question

Convert 0.27¯¯¯¯¯
to a rational number in simplest form.
Responses

27100
27 over 100 - incorrect

311
3 over 11 - not selected, this is the correct answer

127
1 over 27 - no response given

2799

Answer 1

what?

Answer 2

To convert the repeating decimal \( 0.27\overline{27} \) to a fraction, we can use the following method:

Let \( x = 0.27\overline{27} \). This means that \( x = 0.2727272727... \).

1. Multiply \( x \) by 100 to shift the decimal point two places to the right:
\[
100x = 27.27272727...
\]

2. Now, we also know \( x \) but want to isolate the repeating part. Multiply \( x \) by 10000 (to shift four decimal places) as there are two digits in the repeating part:
\[
10000x = 2727.27272727...
\]

3. Now we have two equations:
\[
100x = 27.27272727... \quad (1)
\]
\[
10000x = 2727.27272727... \quad (2)
\]

4. Subtract equation (1) from equation (2):
\[
(10000x - 100x) = (2727.27272727... - 27.27272727...)
\]
\[
9900x = 2700
\]

5. Now solve for \( x \):
\[
x = \frac{2700}{9900}
\]

6. Simplify the fraction:
- Find the GCD of 2700 and 9900. The GCD is 900.
\[
x = \frac{2700 \div 900}{9900 \div 900} = \frac{3}{11}
\]

Thus, the repeating decimal \( 0.27\overline{27} \) can be expressed as the rational number \( \frac{3}{11} \).

The correct response from the options provided is:
\[
\boxed{\frac{3}{11}}
\]

Answer 3

Apologies for the confusion in my previous response. Let's clarify the conversion of the repeating decimal \(0.27\overline{27}\) step by step correctly:

1. **Identify the repeating decimal:**
Let \( x = 0.272727...\)

2. **Multiply to eliminate the repeating part:**
To eliminate the repeating portion, we can multiply by \(100\) because the decimal repeats every two digits:
\[
100x = 27.272727...
\]
This equation shows that \(100x\) is merely the decimal shifted two places to the right.

3. **Subtract the original equation from this new equation:**
We then write another equation for \(x\):
\[
x = 0.272727...
\]
Now we subtract the second equation from the first:
\[
100x - x = 27.272727... - 0.272727...
\]
This simplifies to:
\[
99x = 27
\]

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

5. **Simplify the fraction:**
Both the numerator and denominator can be divided by \(9\):
\[
x = \frac{27 \div 9}{99 \div 9} = \frac{3}{11}
\]

Thus, the repeating decimal \(0.27\overline{27}\) converts to the rational number \(\frac{3}{11}\).

So the correct answer is:
\[
\frac{3}{11} \quad \text{(311)}
\]

Thank you for your patience, and sorry for the oversight earlier!