Essential Question: Why is a repeating decimal a rational number?
1) Set the decimal equal to x. In this example, that means that x = 1.833333 repeating.
2) Create two equations using x to represent the decimal so that the repeating part of the decimal lines up in both equations. For this example, my two equations can be 10x = 18.333 and 100x = 183.333. Why can't we use x = 1.8333? (The 8 in the decimal will not line up properly with another 8 in any other equation. Only the 3's can line up with one another.)
3) Subtract the equations from one another. Make sure that like terms are put together. 100x - 10x = 183.333 - 18.333 will equal 90x = 165.
4) Solve the equation by doing the inverse operation to isolate the variable. Here we would divide by 90 on both sides and simplify by dividing by 15 to get 11/6, which is 1 5/6.
1) Set the decimal equal to x. x = 1.267267267
2) Create two equations so the repeating part of the decimal lines up correctly (this helps us when we need to subtract so there is no leftover in the decimal place values - if there were a decimal left there, we would have a weird looking fraction after solving the equation). Here I can use x = 1.267267 and 1000x = 1267.267267 because the part after the decimal is the same for both.
3) Subtract 1000x - x = 1267.267267 - 1.267267 to get 999x = 1266 (Hopefully you realize that the decimal part should always subtract to equal 0, leaving a whole number to work with.)
4) Solve the equation and simplify it from 1266/999 to 422/333, which is 1 89/333.
Using 1.8333 repeating
1) Take the non-repeating part of the decimal and subtract it from the entire decimal. Put the answer as the numerator. 83 - 8 = 75 for the numerator.
2) Count the repeating part of the decimal until it repeats itself again. In this example, we need only 1 place value before it repeats again. This tells you how many 9's to put as your denominator so we should have one 9 in the denominator.
3) Before you are finished, count how many numbers are NOT repeating in the decimal, and add a 0 to the denominator for each one. We have one digit that does not repeat so we have one 0 to add to the 9 in the denominator.
4) All done - your answer will be 75/90.