Convert 0.27¯¯¯¯¯

Question

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

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 2

what?

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!