Many of us do not think twice about repeating decimals. We either put a bar over the repeating portion of round off, but that does not always allow for accurate calculations.
If you were to see .666666…. or .66667, you’d probably go “that’s 2/3”. However that’s from previous knowledge.
What happens if the number is 7.1111111….?
Some might recognize the .111111… as 1/9, and correctly assume this to be 7 + 1/9, or 64/9.
Well that’s fine and dandy, but what if it’s something like 136.78678678678….?
Not so easy now is it? There is cut-and-dry method of figuring it out. First locate the repeating pattern (678).
Then let x be the full number.
x = 136.78678….
If you can somehow remove the repeating portion, it wouldn’t be so hard to figure out the fraction. But how do you take the repeating portion out?
Multiply x by a multiple of 10 that results in a number with the same repeating decimals in the same spot. Since the repeating number is 678 – a 3 digit number – I will conveniently use 1000.
1000x = 136786.78678…
x = 136.78678…
This is a system of equations. Notice how the repeating end match? Now you can subtract the equations, which gives you:
999x = 136650
Solve for x
x = 136650 / 999
There’s your answer!
Or you can simplify it to get 45550/333, which is 136 + 262/333.
See? As long as you manipulate so he repeating part subtracts itself, you’ll get a fraction. I used 1000, but I could have used .001 or 1000,000.