This was given to me as a practice problem as preparation for the final exam. The only thing is I have no idea how to do the question, as we did not go over generating fractions in class. I googled what a generating fraction was, and I found some information about generating functions, but nothing too helpful.
The hint says to find the sum of the series. But I'm not sure how to find the series in the first place.
Any help would be much appreciated, thank you.
On
The series they are referring to is $0.00062\cdot 0.01^{n} = \frac{62}{100\,000}\cdot 0.01^{n}$, with $n$ going over the natural numbers from $0$ to $\infty$. The summation formula for geometric series says $$ \sum_{n = 0}^\infty\frac{62}{100\,000}\cdot 0.01^n = \frac{62}{100\,000}\sum_{n = 0}^\infty 0.01^n\\ = \frac{62}{100\,000}\cdot\frac{1}{1-0.01} \\ \frac{62}{100\,000}\cdot \frac{100}{99}\\ =\frac{6200}{9\,900\,000} = \frac{62}{99\,000} $$ This represents the number $0.000626262\ldots$, though, which is $0.215 = \frac{215}{1000}$ too small. So, we just add that and we get $$ 0.215626262\ldots = 0.215 + 0.000626262\ldots\\ = \frac{215}{1000} + \frac{62}{99\,000}\\ = \frac{215\cdot 99 + 62}{99\,000} = \frac{21347}{99\,000} $$ One may also use the same approach on $0.2156262\ldots = 0.262626\ldots - 0.047$, or any number of other options.
On
You could write $0.26262626 \dots=0.2156262626\dots +0.047$
And since $\frac 1{99}=0.0101010101\dots$ this gives $0.2156262626\dots = \frac {26}{99}-\frac {47}{1000}$.
If you need to do a series the one here is $\frac 1{100}+\frac 1{(100)^2}+\frac 1{(100)^3}+\dots =\frac 1{99}$
but samjoe's solution is the simplest.
I don't really know what this refers to, but I think its highly probable that it means to find equivalent fraction. We have pretty standard method of dealing with this. Let this number be $x$. Then
$$1000x = 215.6262... \\ 100000x = 21562.6262... \\$$
Subtracting these two, we get:
$$ x = \frac{21347}{99000}$$