Turn 82.693 with the 9 and 3 repeating, into a mixed number.
82.693939393..........=82 229/330
+129839 Turn 82.693 with the 9 and 3 repeating, into a mixed number.
Forget about the 82 for a second
.693 with the 9 and 3 repeating can be turned into a fraction thusly:
Writh the 693 without the decimal......from this, subtract the non-repeating part, 6.....[693 - 6] = 687 ... this will be the numerator of our fraction.........for the denominator......writte the same number of 9's as the number of repeating digits = 99 and append the same munber of 0's as the non-repating part [ i.e, one "0" ]....this gives us 990
So we have :
[693 - 6] / 990 = 687/990 and we can reduce this by dividing the numerator and denominator by 3 → 229/330
And our mixed number is 82 + 229/330........check this for yourself .....https://www.wolframalpha.com/input/?i=82+%2B687%2F990
It would be a lot easier if the questioner learned to use "continued fractions method", which for this particular fraction are: [0; 1, 2, 3, 1, 2, 1, 6] =229/330.
+129839 Guest...I see how your answer of [0; 1, 2, 3, 1, 2, 1, 6] will produce the necessary fraction.......could you explain how this works????......I don't think I have seen this method before.....!!!
CPhill: If I haven't forgotten it!!. You start with the last number 6. You take the reciprocal of 6, which is:0.16666666, then add to it the next to last number, which is 1 and you get 1.1666666. Then multiply this by the last number or 6, which will give you 7. Then take 1.1666666... and take its reciprocal which will give you 0.857142........, then add the 3rd from the last which is a 2 and you get
2.857142......... and then multiply this by the last whole number we got, which was 7 and you should get 20. Then you would take the reciprocal of this last 2.857142 which will give you 0.35, then add the next number from the end which is a 1 and you get 1.35. Then multiply this by the last whole integer we got, which was 20 and you will get 27. and so on until you get 229 as numerator and the last step should give you 330 which is the denominator.
Of course, the opposite of this is to obtain the integers of the continued fraction such as we had here, i.e. 0.69393939393. You would take the reciprocal of this and you would get 1.4410480349... and you subtract the integer part, which is a 1. Then you will take the reciprocal of the remaining fraction or, 0.4410480........and subtract the integer part, which a 2........and so on. That is how W/A gets those numbers, i.e. [0; 1, 2, 3, 1, 2, 1, 6].
P.S. I have programmed this into my computer which does it almost intantaneously. I don't know if you know how to program, which you could do easily. Even if you are familiar with spreadsheets like Excel, you could program it into it and it will automate this whole process very rapidly.
+37084 A similar way:
x = 82.6939393...
10x= 826.939393....
1000x=82693.939393....
1000x 82693.9393....
- 10x 826.9393...
____________________
990x = 81867 Solve for x
x = 81867/990 = 82 687/990 = 82 229/330
The "continued fraction method" is very powerful for all kinds of fractions as well as irrational numbers such as Pi: The continued fraction of Pi is:[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, ...]. From these numbers I can obtain the value of Pi to dozens of decimal digits, when using a computer programmed for this purpose.
The "continued fraction method" is very powerful for all kinds of fractions as well as irrational numbers such as Pi: The continued fraction of Pi is:[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, ...]. From these numbers I can obtain the value of Pi to dozens of decimal digits, when using a computer programmed for this purpose.
