Converting repeating decimals to fractions? Please help.?

Question

This is what my book says:


Set the variable, n. equal to the decimal.
In order to get the same fractional part as n, multiply both sides of the equation by the same power of 10, This operation allows an integer to remain as the difference in Step 3.
Subtract both sides of the equation by the same value. Since Step 1 stated that n is equal to the decimal, subtract n from one side of the equation and subtract the decimal from the other side of the equation.
Solve the equation for the variable, n, using division.
Simplify.

I'm so confused! Can you please explain how to do this in 8th grade language? Maybe an example? Anything helps. Thanks

Answer 1

I must admit that is a very confusing way to explain it.



Let me start of with an example:



Convert 0.367367367... to a fraction.

Note: The 367 is repeating forever.



Let X = 0.367367367...

1000X = 367.367367367...

If you multiply a number by 10 you move the decimal point by 1 place to the right. So if you multiply it by 1000 = 10x10x10 it will have moved 3 places to the right.



Now 1000X - 1X = 367.367367367... - 0.367367367...



=> 999X = 367



Because the numbers after the decimal point are the same.



=> X = 367/999



I hope that you can see a general way of how to do this from the example and if you have more problems just ask.



EDIT: Also there is a case where the repeating part of the decimal doesn't start straight after the 0:



0.14545... , where 45 is repeating



0.14545... = 0.1 + 0.04545...



0.1 = 1/10



Let X = 0.04545...

10X = 0.4545...

1000X = 45.4545...



990X = 45

=> X = 1/22



now 0.14545... = 1/10 + X = 1/10 + 1/22 = 32/220 = 8/55

Answer 2

i'm sorry, but books are never helpful when it comes to this sort of stuff...i don't understand the book's version. i had the same problem last year.



there's a little secret, if your repeating decimal is say 0.4(repeating) your answer is 4 over 9

if your repeating decimal is say 0.06 then your fraction is 6 over 99

if your repeating decimal is 0.4593 then your fraction is 4593 over 9999

so you see, you take the number that is repeating and count how many numbers there are in it. that is how many nines are in your denominator then you put your repeating number as the numerator. (that is the word for the top number on a fraction right?0

Answer 3

Geometric series is the easiest way.



For example, 0.235...(3 and 5 recurring)

=0.2 + 0.035 + 0.00035 +...

Sum to infinity =a/(1-r)

a=0.2 r=1/100

= 7/198

So 0.235 (3 and 5 recurring) = 0.2+(7/198)=233/990



Edit: The method above doesn't work all of the time!