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 frame the cabin alone?
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 ???
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.