decimal to a fraction: 0.123123123
To convert a repeating decimal to a fraction, we can use the following steps:
Step 1: Let x = the repeating decimal.
Step 2: Multiply x by a power of 10 to shift the repeating part to the left of the decimal point. Let's determine the power of 10 by counting the number of digits after the decimal point until the repeating sequence starts. In this case, the repeating part starts at the 4th decimal place, so we will multiply by 1000 (10^3).
1000x = 123.123123123...
Step 3: Subtract x from 1000x to eliminate the repeating part:
1000x - x = 123.123123123... - 0.123123123...
999x = 123
Step 4: Solve for x by dividing both sides of the equation by 999:
x = 123/999
Step 5: Simplify the fraction. In this case, the numerator (123) and the denominator (999) have no common factors other than 1, so the fraction is already in its simplest form.
Therefore, 0.123123123... is equal to 123/999.