Lee can frame a cabin in 4 days less than Ron. When they work together, they will do the job in 4 days. How long would each of them take to

Lee can frame         $$\frac{1\; cupboard }{x\; days}$$

Ron can frame   $$\frac{1\; cupboard }{x+4\; days}$$

So together they can frame    

  $$\\\frac{1\; cupboard }{x\; days}+\frac{1\; cupboard }{x+4\; days}\\\\ =\frac{(x+4)\; cupboard}{ x*(x+4)days}+\frac{x\; cupboard }{x(x+4)\; days}\\\\ =\frac{(2x+4)\; cupboard}{ x*(x+4)days}\\\\\\$$

Now we know that together they can fram one cupboard in 4 days so

$$\\\frac{(2x+4)\; cupboard}{ x*(x+4)days}\times \frac{4\;days}{1}=1 cupboard \qquad $NOTE: The days cancel out$\\\\\\$$

This gives the equation

$$\\\frac{(2x+4)}{ x*(x+4)}\times \frac{4}{1}=1 \\\\ \frac{4(2x+4)}{ x*(x+4)}=1 \\\\ 4(2x+4)=x*(x+4) \\\\ 8x+16=x^2+4x \\\\ x^2-4x-16=0 \\\\$$

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{16}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{2}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{5}}}}\\ {\mathtt{x}} = {\mathtt{2}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{5}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{2.472\: \!135\: \!954\: \!999\: \!579\: \!4}}\\ {\mathtt{x}} = {\mathtt{6.472\: \!135\: \!954\: \!999\: \!579\: \!4}}\\ \end{array} \right\}$$

The first  answer is invalid

So individually Lee can fram a cupboard in  6.47 days and Ron will take 10.47  days

Exactly the same as CPhill got.     These are really tricky.    

Let x be the number of days that takes Ron to frame the cabin by himself. Then, the number of days it takes Lee to frame the cabin is (x - 4)

Rate *Time  = Work Done

So we have

Ron's Rate * Time  + Lee's Rate * Time  = 1 job done

(1/x)(4) + (1/(x-4))(4) = 1    simplify

4/x + 4/(x-4) = 1                  get a common denominator

[4(x-4) + 4x] / [x(x-4)] = 1    multiply both sides by x(x-4)

4x - 16 + 4x = x(x-4)             simplify

8x - 16 = x^2 - 4x                 rearrange

x^2 - 12x + 16 = 0                using the on-site solver, we have

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{12}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{16}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{6}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{5}}}}\\ {\mathtt{x}} = {\mathtt{2}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{5}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{1.527\: \!864\: \!045\: \!000\: \!420\: \!6}}\\ {\mathtt{x}} = {\mathtt{10.472\: \!135\: \!954\: \!999\: \!579\: \!4}}\\ \end{array} \right\}$$

Reject the first answer

So Ron takes about 10.47 days and Lee takes 4 fewer days = about 6.47 days

The answers seem funky.....anyone else want to take a stab ???