Assuming the shape of the room is a rectangle :
Area = Width x Length.
Let's call width "w" and length "l" . Now if, the width is 4/5 of the length that means 4/5 times length gives us the width, like this :
\(w=\frac { 4 }{ 5 } \cdot l\)
As given in the question, area of the rectangle is 320 square feet. And as I said before area of a rectangle can be found by multiplying its length and width.
\(w.l=320\)
\(w=\frac { 4 }{ 5 } \cdot l\) , we can plug w's value in the previous equation to find "l".
\(\frac { 4 }{ 5 } \cdot l\cdot l=320\\ \\ \frac { 4 }{ 5 } \cdot { l }^{ 2 }=320\\ \\ { l }^{ 2 }=320\cdot \frac { 5 }{ 4 } \\ \\ { l }^{ 2 }=\frac { 320\cdot 5 }{ 4 } \\ \\ { l }^{ 2 }=\frac { 1600 }{ 4 } \\ \\ { l }^{ 2 }=400\\ \\ \sqrt { { l }^{ 2 } } =\sqrt { 400 } \\ \\ l=20\)
Let's plug "l"s value in the equation.
\(w.l=320\quad \Longrightarrow \quad w\cdot 20=320\quad \Longrightarrow \quad w=\frac { 320 }{ 20 } \quad \Longrightarrow \quad w=16\)
The length is 20 feet and the width is 16 feet.
