Question

Convert 0.09¯¯¯¯¯%0D%0A0.%0D%0A09%0D%0A¯%0D%0A to a rational number in simplest form.%0D%0A%0D%0AShow your work

Answer 1

no unnecessary words or symbols

Answer 2

in simplist form

Answer 3

dumb it down

Answer 4

less steps

Answer 5

so a 8th grader can understand is

Answer 6

To convert the repeating decimal \(0.09\overline{09}\%\) to a rational number, we'll first convert the percentage to a decimal and then express it as a fraction. Follow these steps:

1. **Convert the percentage to a decimal**:

Since \(0.09\overline{09}\%\) means \(0.09\overline{09}\) percent, we should first convert it to a decimal form. Knowing that 1% is equal to \(0.01\), we convert the percentage by dividing by 100:
\[
0.09\overline{09}\% = 0.09\overline{09} \times 0.01 = 0.0009\overline{09}
\]

2. **Let \(x = 0.0009\overline{09}\)**.

3. **Eliminate the repeating part**:
We recognize the repeating decimal \(0.0009\overline{09}\) has two parts:
- The non-repeating part: \(0.0009\)
- The repeating part: \(0.0000\overline{09}\)

To express \(0.0000\overline{09}\) as a fraction, we can solve it separately.

Let \(y = 0.0000\overline{09}\).
Multiply both sides by \(1000\) (to shift the decimal three places to the right):
\[
1000y = 0.09\overline{09}
\]

Now, multiply both sides by \(100\) (to shift the decimal two places to the right again):
\[
10000y = 0.9\overline{09}
\]

4. **Set up an equation**:
Now, we have two equations:
\[
1000y = 0.09\overline{09}
\]
\[
10000y = 0.9\overline{09}
\]

The repeating part can be expressed as:
\[
0.1y = 0.09\overline{09}
\]
Now, we can subtract the first equation from the second:
\[
10000y - 1000y = (0.9\overline{09} - 0.09\overline{09})
\]
This gives us:
\[
9000y = 0.9
\]

5. **Solving for \(y\)**:
Dividing both sides by \(9000\):
\[
y = \frac{0.9}{9000} = \frac{9}{90000} = \frac{1}{10000}
\]

6. **Combine \(y\) with the non-repeating part**:
Now recall:
\[
0.0009 = \frac{9}{10000}
\]

So, the final fraction is:
\[
x = 0.0009 + 0.0000\overline{09} = \frac{9}{10000} + 0.0000\overline{09}
\]
Since \(0.0000\overline{09}\) we found as \(\frac{1}{10000}\).

7. **Add the fractions**:
Finally, we sum our two fractions:
\[
x = \frac{9}{10000} + \frac{1}{10000} = \frac{10}{10000} = \frac{1}{1000}
\]

Thus, the repeating decimal \(0.09\overline{09}\%\) expressed as a rational number in simplest form is \(\frac{1}{1000}\).

Answer 7

To convert \(0.09\overline{09}\%\) to a rational number:

1. Convert percentage to decimal:
\[
0.09\overline{09}\% = \frac{0.09\overline{09}}{100} = 0.0009\overline{09}
\]

2. Let \(x = 0.0009\overline{09}\).

3. Break it into:
\[
x = 0.0009 + 0.0000\overline{09}
\]
Let \(y = 0.0000\overline{09}\).

4. For \(y\):
\[
100y = 0.009\overline{09}
\]
\[
10000y = 0.9\overline{09}
\]
Subtract:
\[
9900y = 0.9
\]
\[
y = \frac{0.9}{9900} = \frac{9}{99000} = \frac{1}{11000}
\]

5. Add \(0.0009\):
\[
0.0009 = \frac{9}{10000}
\]
Combine:
\[
x = \frac{9}{10000} + \frac{1}{11000}
\]

6. Find common denominator (110000):
\[
= \frac{99}{110000} + \frac{10}{110000} = \frac{109}{110000}
\]

Final answer:
\[
\frac{109}{110000}
\]

Answer 8

Let's simplify \( \frac{109}{110000} \).

1. **Check for GCD (Greatest Common Divisor)**:
- The numerator is \( 109 \).
- Since \( 109 \) is a prime number, its only divisors are \( 1 \) and \( 109 \).
- Now, check if \( 109 \) divides \( 110000 \):
- \( 110000 \div 109 \approx 1010.09 \) (not an integer).

2. **Conclusion**:
- Since \( 109 \) does not divide \( 110000 \) evenly, the fraction \( \frac{109}{110000} \) is already in simplest form.

Final answer:
\[
\frac{109}{110000}
\]

Answer 9

To convert \(0.09\overline{09}\%\) to a simple fraction:

1. **Convert percentage to decimal**:
\[
0.09\overline{09}\% = 0.0009\overline{09}
\]

2. **Let \(x = 0.0009\overline{09}\)**.

3. **Break it into parts**:
- Non-repeating part: \(0.0009\)
- Repeating part: \(0.0000\overline{09}\)

4. **Find the repeating part**:
- Let \(y = 0.0000\overline{09}\).
- Multiply by \(1000\):
\[
1000y = 0.09\overline{09}
\]
- Multiply by \(100\):
\[
10000y = 0.9\overline{09}
\]
- Subtract the two equations:
\[
9000y = 0.9 \quad \Rightarrow \quad y = \frac{0.9}{9000} = \frac{9}{90000} = \frac{1}{10000}
\]

5. **Combine parts**:
- \(0.0009 = \frac{9}{10000}\)
- So:
\[
x = \frac{9}{10000} + \frac{1}{10000} = \frac{10}{10000} = \frac{1}{1000}
\]

Final answer:
\[
\frac{1}{1000}
\]

Answer 10

To convert \(0.09\overline{09}\%\) to a fraction:

1. **Convert to decimal**:
\[
0.09\overline{09}\% = 0.0009\overline{09}
\]

2. **Let \(x = 0.0009\overline{09}\)**.

3. **Express repeating part**:
- \(0.0009 + 0.0000\overline{09}\)
- The repeating part: \(0.0000\overline{09} = \frac{1}{10000}\)

4. **Non-repeating part**:
\[
0.0009 = \frac{9}{10000}
\]

5. **Combine**:
\[
x = \frac{9}{10000} + \frac{1}{10000} = \frac{10}{10000} = \frac{1}{1000}
\]

Final answer:
\[
\frac{1}{1000}
\]

Answer 11

Sure! Here's a simple way to convert \(0.09\overline{09}\%\) into a fraction:

1. **Understand the percentage**: \(0.09\overline{09}\%\) means the same as saying "0.09 repeating percent."

2. **Convert to decimal**:
- First, change it to a decimal:
\[
0.09\overline{09}\% = 0.0009\overline{09}
\]

3. **Break it down**:
- We can think of it as:
\[
0.0009 + \text{the repeating part}
\]

4. **Find the repeating part**:
- The repeating part \(0.0000\overline{09}\) can be calculated to be \(0.0001\).

5. **Add them together**:
- So, you have:
\[
0.0009 + 0.0001 = 0.0010
\]
- Which is the same as:
\[
\frac{1}{1000}
\]

**Final answer**:
\[
\frac{1}{1000}
\]