Solving for Recurring Numbers: Techniques and Applications

[FateMaster]

Hihi... I am stumped at this question... I know there is a techique in doing this question... But i forget oredi... Help please...


Express this recurring number at a fraction of a/b...
Recurring number ---> 0.1454545454545...

Plz explain to me the technique used... No calculators allowed...

 

[Zurtex]

The general trick is to equalise it x multiply by 10^(the number of digits recuring) then take away the orriginal number.

Using your problem as an example.

x = 0.1454545454545...

2 digits are recurring so multiply it be 100

100x = 14.54545454545...

Take away the orriginal number:

100x - x = 14.54545454545... - 0.1454545454545...

Take this a digit at a time:

99x = 14.40000000000...

99x = 14.4

That should be a little easier to solve now

 

[FateMaster]

Stupid me...

Thankx... lol... So its this easy...

Hehe... Thankx for your help mate...

 

[HallsofIvy]

Slightly different way:

Because there is that "1" before the recurring "45", first multiply by 10:

10x= 1.454545...

Now multiply that by 100: 1000 x= 145.454545... and subtract

1000x- 10x= 900x= 144 so x= 144/900

Of course, that gives exactly the same result.

 

[recon]

1000x- 10x= 900x= 144 so x= 144/900

Of course, that gives exactly the same result.

Oh, no. Minor snag here. 1000x - 10x = 990x. So the answer really is 144/990.

You may simplify 144/990, of course.

 

[recon]

I like the fact that there are many different ways for solving problems. Here's another method. Not that it is really different: it just differs slightly from all the others.

x = 0.1454545...
10x = 1.454545...

10x - 1 = 0.454545...

We can work out 0.454545... to be 45/99 (100z - z = 45).

10x - 1 = 45/99
10x = (99 + 45)/99
10x = 144/99
x = 144/990

 

[mathwonk]

also can do it by "geometric series", really the same again, but done once for all:

i.e. .1 + .045454545...

is .1 Plus the geometric series with initial term a= .045 and ratio r= 1/100, so the sum

is a + ar + ar^2 +...= a/(1-r) i.e. .045/(99/100) = (4.5)/99, so the answer is

.1 + this, as before.

I really do not like this answera s the others's answers are more elementary. but at least it shows how to algebraize their methods.