Question

Put the steps in order for changing the repeating decimal, which is rational, to a ratio or fraction. 0.523523... what fraction? (5 points) 999x = 523 Let x = 0.523523... X= 523/999 1000x523.523523... Subtract 1x = 0.523523...

Answer 1

To convert the repeating decimal \(0.523523...\) into a fraction, we can follow these steps in the correct order:

1. **Let \(x = 0.523523...\)**
We start by letting \(x\) represent the repeating decimal.

2. **Multiply by 1000 to shift the decimal point:**
\[ 1000x = 523.523523... \]
This moves the decimal point three places to the right.

3. **Subtract the original \(x\) from the equation:**
\[ 1000x - x = 523.523523... - 0.523523... \]
This gives us:
\[ 999x = 523 \]

4. **Solve for \(x\):**
\[ x = \frac{523}{999} \]

5. **Finalize the fraction:**
Thus, \(0.523523...\) can be written as \(\frac{523}{999}\).

So, in order, the steps are:

1. Let \(x = 0.523523...\)
2. Multiply by 1000: \(1000x = 523.523523...\)
3. Subtract \(x\) from \(1000x\): \(999x = 523\)
4. Solve for \(x\): \(x = \frac{523}{999}\)

So the final answer is \(\frac{523}{999}\).