Change the decimal to a fraction: 0.123123123….

Question

A. 123/999
B. This decimal is Irrational and cannot be written as a fraction
C. 12 13/10
D. 123/1000
E. 1 23/100

Answer 1

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

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

2. Notice that the decimal repeats every 3 digits. Therefore, multiply both sides by \(1000\) (since the repeating part is 3 digits long):

   \[ 1000x = 123.123123123... \]

3. Now we can set up an equation using these two expressions:

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

4. Subtract the second equation from the first:

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

   This simplifies to:

   \[ 999x = 123 \]

5. Now, solve for \(x\):

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

6. We can simplify \(\frac{123}{999}\) by finding the greatest common divisor (GCD) of 123 and 999. The GCD is 123, so:

   \[ \frac{123 \div 123}{999 \div 123} = \frac{1}{8.12} = \frac{1}{8} \]

7. Thus, \(x = 0.123123123... = \frac{1}{8}\) or it remains as \(x = \frac{123}{999}\) for the exact representation of the repeating decimal.

Since it appears you need to choose from the given options:

The correct answer is:

A. \(\frac{123}{999}\).