Question

1/(1+1/(1-1/x))=4 ,then find x.

Answer 1

For the short answer:

$x = \frac{3}{7}$

Step-by-step explanation:

To visualise this equation let us first show it as this:

$\frac{1}{1+\frac{1}{1-\frac{1}{x} } } =4$

First, let us simplify the fraction itself (this is the hardest bit so I'll write a full explanation in case of confusion) :

1. First, we should solve the lowermost denominator, $1-\frac{1}{x}$ as it contains the $x$ term.
   If we consider the 1 as $\frac{x}{x}$ this is the same as saying $\frac{x}{x} -\frac{1}{x}$  or  $\frac{x-1}{x}$ .
2. Now we solve $\frac{1}{\frac{x-1}{x} }$  using the general rule of dividing fractions:
   $1\div\frac{x-1}{x}\\= 1\times \frac{x}{x-1} \\=\frac{x}{x-1}$
3. Now, we solve $1+\frac{x}{x-1}$ .
   If we consider 1 as $\frac{x-1}{x-1}$ this results in the answer of $\frac{x-1+x}{x-1}$ or $\frac{2x-1}{x-1}$.
4. We then lastly work out $\frac{1}{\frac{2x-1}{x-1} }$, again using the rule of dividing fraction:
   $1\div\frac{2x-1}{x-1} \\=1\times\frac{x-1}{2x-1} \\=\frac{x-1}{2x-1}$
   $\therefore$ The simplified answer is $\frac{x-1}{2x-1}$.

Now we do simple maths to find out when $\frac{x-1}{2x-1}=4$ :

1. $x-1=4(2x-1)$
2. $x-1=8x-4$
3. $x-1-8x=-4$
4. $x-8x=-4+1$
5. $-7x=-3$ or same as saying $7x=3$
6. $x=\frac{3}{7}$

Hope this helps!!!