Change the decimal to a fraction: 0.123123123….(1 point)

Question

Responses

1231000
123 over 1000

This decimal is Irrational and cannot be written as a fraction
This decimal is Irrational and cannot be written as a fraction

123999
123 over 999

121310
12 and 13 tenths

123100

Answer 1

To convert the repeating decimal \(0.123123123...\) into a fraction, we can use the following method:

1. Let \(x = 0.123123123...\).

2. Since the repeating block "123" has 3 digits, multiply \(x\) by \(1000\):

   \[1000x = 123.123123123...\]

3. Now we have two equations:

   • \(x = 0.123123123...\)
   • \(1000x = 123.123123123...\)

4. Subtract the first equation from the second:

   \[1000x - x = 123.123123123... - 0.123123123...\]

   \[999x = 123\]

5. Solve for \(x\):

   \[x = \frac{123}{999}\]

6. We can simplify \(\frac{123}{999}\) by dividing both the numerator and the denominator by 123:

   \[\frac{123 \div 123}{999 \div 123} = \frac{1}{9}\]

So, \(0.123123123... = \frac{1}{9}\).

Thus, among the provided options, the correct representation is:

123999 123 over 999.